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Title Pathogen transmission models in clonal plant population : Analysis on the effects of superinfection and seedreproduction

Author(s) 酒井, 佑槙

Citation 北海道大学. 博士(環境科学) 甲第12487号

Issue Date 2016-12-26

DOI 10.14943/doctoral.k12487

Doc URL http://hdl.handle.net/2115/64730

Type theses (doctoral)

File Information Yuma_Sakai.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Pathogen transmission models

in clonal plant population

– Analysis on the effects of superinfection

and seed reproduction –　

Yuma Sakai

Graduate School of Environmental Science

Hokkaido University

2016

CONTENTS

Summary 4

Chapter 1 Introduction 10

1.1 Clonal plant & pathogen . . . . . . . . . . . . . . . . . . . 10

1.2 Mathematical models of pathogen propagation . . . . . . . 13

1.3 Approximation methods . . . . . . . . . . . . . . . . . . . 16

1.4 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 2 Superinfection model 24

2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 1-strain model . . . . . . . . . . . . . . . . . . . . . 32

2.2.2 Multiple-strain models . . . . . . . . . . . . . . . . 40

2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 3 Seed propagatin model 62

3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Single population . . . . . . . . . . . . . . . . . . . 69

3.2.2 Optimal proportion of vegetative propagation (Mixed

population) . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Chapter 4 Conclusion 89

Acknowledgements 97

Referances 98

Appendix 109

A. Simplification of the master equation . . . . . . . . . . . . 109

B. Analysis in superinfection model . . . . . . . . . . . . . . . 112

B.1 Mean-field approximation . . . . . . . . . . . . . . 112

B.2 Pair approximation . . . . . . . . . . . . . . . . . . 114

C. Analysis in seed propagation model . . . . . . . . . . . . . 122

C.1 Mena-field approximaation . . . . . . . . . . . . . . 122

C.2 Extinction phase in pair approximation . . . . . . . 123

Summary

Many clonal plants have two breeding systems, vegetative and seed

propagation. In seed propagation, plants reproduce genetically differ-

ent offspring that have a high mortality rate because their long-distance

dispersal and lack of a physical connection does not allow them to be

supported by their parents. In vegetative propagation, plants reproduce

genetically identical offspring that have lower mortality rates because re-

sources are supplied to the offspring from other individuals through inter-

connected ramets. However, vegetative propagation assists the pathogen’

s spread because the vascular system in the ramets acts as a transmission

pathway to other ramets. Thus, the disease becomes epidemic in a colony

of vegetative propagules. Increasing seed propagation is an effective de-

fensive behavior against the spread of pathogens because the plants will

reproduce in areas distant from the infection site.

Pathogens take several actions to increase their fitness within plant

populations. A diversity of infections, a superinfection, influences the

ability of pathogens to spread. A superinfection involves different pathogens

infecting (secondary infections) already infected individuals, either se-

4

quentially or concurrentl, and it leads to an increase in pathogen fitness

levels relative to a single infection. In this thesis, the focus is on two

characteristic phenomena, superinfection of pathogens, which increase

the fitness of the pathogen, and seed propagation in plants, which is an

effective defensive behavior against pathogen spread. Thus, two mod-

els, superinfection and seed propagation, were constructed applying the

two-stage contact process on lattice space to explore the effects of spa-

tial structures and each phenomenon on plant and pathogen populations.

The two-stage contact process expresses the dynamics of the contact in-

fection process simply on a graph and is applied to express the plant

reproduction process. However the analysis of the model is too complex,

thus we adopted two approximation methods, mean-field approximation

and pair approximation, to analyze the model analytically. Additionally,

we examined the effect of spatial structure through the comparison of the

result among the approximation methods and Monte Carlo simulation.

In the superinfection model, the effects of superinfection events on

the genetic diversity of pathogens using several models, including the

1-strain and multiple-strain models, were examined. In the analysis of

5

the 1-strain model, an equilibrium value was derived using the mean-

field approximation and pair approximation, and its local stability using

the Routh–Hurwitz stability criterion. In the multiple-strain models, the

dynamics using numerical simulations and Monte Carlo simulation were

analyzed. Through these analyses, the effects of parameter values, such

as the density of individuals, transition of a dominant pathogenic strain,

and competition between plants and pathogens, on the dynamics of the

models were shown. As a result, the superinfection event is one of the

important factors to maintenance of genetic diversity of pathogens. (i)

The strain with an intermediate cost became dominant, when both the

superinfection and growth rates were low; (ii) The competition among

strains occurred in the coexistence of various strains phase; (iii) Too high

a growth rate led to occupation by the strain with the lowest cost. Thus,

competition between the strain and the hosts occurred, and, therefore,

the host population decreased in all of the models; (iv) Pathogens easily

maintained their genetic diversity when there was a low superinfection

rate. However, if they did not superinfect, such maintenance became dif-

ficult; and (v) When the growth rate of a plant was low, an individual at

6

a local site was strongly interconnected by distant individuals.

In the seed propagation model, the dynamics of plant reproduction and

pathogen propagation, and the effects of seed propagation on the defense

responses to pathogen spread in single and mixed (coexistence of sev-

eral plant types) plant populations were examined. Thus, the change of

relative merit in the breeding system caused by the invasion of a plant

population by systemic pathogens was expressed. In the analysis, the

equilibrium and its local stability were derived using pair approximation

in the case of single populations. Additionally, using the Monte Carlo

simulation, the effects of spatial structure through a comparison with the

results of the pair approximation was examined. In mixed populations,

two situations were assumed, infected and uninfected populations, and

they were analyzed using only the Monte Carlo simulation because other

analyses of the model are too complex to obtain analytical results, having

too many variables,. The efficacy of seed propagation on the suppression

of epidemic infections was examined by comparing the results in the two

situations. As a result, seed propagation is an effective defensive behavior

against systemic pathogens. Generally, when the pathogen infectivity is

7

low, relative to plant fecundity, plants can escape from infected individ-

uals through the vegetative propagation, and the feature is expressed in

the presented model. However, the effect of pathogen abilities (infectiv-

ity and virulence) on the optimal balance of breeding systems becomes

decrease with the increase in the fecundity of pathogen in analysis of the

model. Thus, the adjustment of the breeding systems has an important

role to block of pathogen transmission when the plants have low fecundity.

In Chapter 4, the results of Chapter 2 and Chapter 3 were collected

to discuss the effect of the focal phenomena and the spatial structure on

plant reproduction and pathogen spread within clonal plant population.

As a result, the intrinsic fecundity of plant makes a major impact on the

evolution of pathogen within plant population and on the optimal balance

of breeding systems in clonal plant population, and the both the adjust-

ment of the balance of breeding systems and the selection of virulence

level through superinfection have influence on each other. Additionally,

the spatial structure impacts the dynamics of a plant population infected

by a systemic pathogen. In particular, when the plant ’s growth rate

is slower than the pathogen ’s infection rate, then the influence of the

8

spatial structures increases as indicated by the increasing quantitative

discrepancy between the approximation method and Monte Carlo sim-

ulation. However, the pair approximation can analyze the qualitative

characteristics of dynamics well.

9

Chapter 1 Introduction

1.1 Clonal plant & pathogen

Approximately 70% of terrestrial plants [1] and most aquatic plants,

such as sea grass [2], are clonal. Many clonal plants have two breeding

systems, vegetative and seed propagation [Fig. 1]. In seed propagation,

plants reproduce genetically different offspring that have a high mortality

rate because their long-distance dispersal and lack of a physical connec-

tion do not allow them to be supported by their parents. Thus, plants

maintain their genetic diversity and increase their habitat range, even

though the seedlings have a greater mortality rate. By contrast, in veg-

etative propagation, plants reproduce genetically identical offspring that

have lower mortality rates because resources are supplied to the offspring

from other individuals through interconnected ramets [3, 4]. However, if

a systemic pathogen invades the population, then it spreads rapidly and

the plants suffer serious damage because the vegetative propagules are

growing so close together [5].

According to Stuefer et al. (2004), systemic pathogens have diverse

10

Fig 1. Breeding systems of clonal plants. Plants can widely

disperse their offspring through seed propagation. Through vegetative

propagation, plants reproduce physically interconnected offspring. The

vegetatively propagated offspring can share resources through

interconnected ramets; however, pathogens can also spread through the

vascular system of the ramets.

11

negative effects on plants, which result in severe damage or death. For

example, they can lead to leaf deformations [6], growth rate reductions [7,

8, 9], growth-form changes [10, 11] and reduced reproduction [12, 13, 14].

Pathogens take several actions to increase their fitness within plant pop-

ulations. A diversity of infections, a superinfection, influences the ability

of pathogens to spread [15, 16, 17]. A superinfection involves differ-

ent pathogens infecting (secondary infections) already infected individ-

uals, either sequentially or concurrentl [16], and it leads to an increase

in pathogen fitness levels relative to a single infection [18]. Additionally,

the selection of pathogen’s virulence level also increases their fitness, de-

pending on the plant’s life cycle. For instance, if the host plant has a

long lifespan, then a low level of virulence is beneficial to the pathogen

because the host survives for a long period. However, if the plant repro-

duces vegetatively, then many susceptible individuals are produced. In

this situation, a high virulence level is beneficial because there are plenty

of other plants to infect. Thus, the methods of plant reproduction and

pathogen propagation influence each other.

Plants have diverse defense responses to systemic pathogens [19], such

12

as (i) deliberately detaching the infected ramets or tissues [20], (ii) in-

creasing their clonal growth rate [1, 10, 11, 21, 22], and (iii) limiting the

infection risk and pathogen spread by severing the physical connections

of ramets [9, 23] or by long-distance dispersal through seed propagation.

The detaching action blocks the spread of the pathogen in a popula-

tion, although the benefits of vegetative propagules decrease due to the

reduction in the genet size. The increase in the growth rate of vege-

tative propagation is an effective escape behavior from the pathogen’s

spread. However, vegetative propagation assists the pathogen’s spread

because the vascular system in the ramets acts as a transmission path-

way to other ramets. Thus, the disease becomes epidemic in a colony of

vegetative propagules [5]. Increasing seed propagation is an effective de-

fensive behavior against the spread of pathogens because the plants will

reproduce in areas distant from the infection site.

1.2 Mathematical models of pathogen propagation

Spatial structures play important roles in the evolution of both plants

and pathogens. The interactions between a plant and a pathogen de-

13

pend on the spatiotemporal dynamics, such as pathogen dispersal and

the spatial positioning of ramets [24, 19]. According to Koubek and Her-

ben (2008), features of the host assist local pathogen transmission and

the evolution of the pathogen towards lower virulence levels [25] because

clonal growth increases the probability of finding susceptible hosts in the

vicinity of the initially infected host. To explain these dynamics, mod-

els were constructed based on the contact process (CP) [26], especially

the two-stage contact process (TCP) [27, 28], described in mathemat-

ics. These are simple models that graphically express the dynamics of

the contact infection process. Thus, the models represent the spatiotem-

poral dynamics of pathogen propagation depending on the state of the

connected vertices.

In the CP, a vertex of the graph can represent either state, healthy or

infected, and the state of the vertex transitions to the other state proba-

bilistically with time. A vertex representing the healthy state transitions

to the infected state at a rate of nIβ, which indicates infection rate (β)

proportional to the number of connected vertices of the infected state (nI)

(infection), and a vertex representing the infected state transitions to the

14

healthy state at a rate of 1 (recovery) at the next time t + ∆t [Fig. 2].

The transition process is represented by a master equation as follows:

PSS =2 (PSI − nIβIPISS) ,

PSI =nIβIPISS + PII − nIβIPISI − PSI,

PII =2(nIβIPISI − PII).

(1)

Here, let Pσiσj(t) be the probability that two randomly chosen connected

sites are of state σi and state σj at time t (Pσiσj= Pσjσi

). Pσiσjσk(t) is the

probability that a randomly chosen site is of state σj and two randomly

chosen connected sites are of state σi and σk at time t (Pσiσjσk= Pσkσjσi

).

The positive and negative terms indicate transitions from one state and

to any other state. For instance, PSS transitions to PSI because of infection

and transitions from PSI because of recovery. The model is applied to

express the plant reproduction process, specifically, the names of the two

states are changed from“ healthy”and ”infected”to ”empty”and

”occupied”, respectively.

The TCP assumes three states, empty, healthy and infected. The

empty state transitions to the healthy state at a rate proportional to

15

the number of vertices having the healthy state (reproduction) and from

the healthy state or the infected state at a rate of 1 (death) [Fig. 3]. The

healthy state transitions to the infected state at a rate proportional to

the number of vertices having the healthy state (infection), and the in-

fected state does not transition to the healthy state (no recovery), which

is different from the CP. Thus, healthy individuals reproduce new off-

spring into neighboring open areas, and infected individuals increase only

by the infection’s transmission into neighboring healthy individuals. In-

fected individuals do not recover to the healthy state. The plant offspring

remain close to the parents, and the pathogens transmit to close individ-

uals. Therefore, this model is suited to describe the features of both the

vegetative and pathogen propagation processes.

1.3 Approximation methods

Eq. (1) is not closed because the probability relevant to a triple vertex is

necessary to express the dynamics of the probability relevant to a pair of

vertices. Additionally, to express the dynamics of the probability relevant

to a triple vertex, the probability relevant to a quad vertex is necessary.

16

t t+ t rate

S I nIβ

I S 1

S I S

S S S

1

tt ∆+

t I S I

I I I

β2

S S I

S I I

β

Fig 2. Dynamics of the contact process. (a) transition rule, (b and

c) reproduction process, (d) death process. The symbols 0 and S

indicate empty and occupied by an individual, respectively. Parameter

β indicates fecundity and nS indicates the number of individuals in the

nearest-neighbor sites of the empty area.

Thus, to analyze the Eq. (1) exactly, the probability relevant to infinite

connected vertices is necessary. Therefore, neither an explicit solution for

the equilibrium nor a threshold for the phase transition were obtained in

the CP and TCP models.

Several approximation methods have been applied to analytically dis-

cuss the behavior of the system, such as the mean-field approximation

(MA) and the pair approximation (PA). The MA is the simplest approx-

imation method, and it does not take into account the effects of other

sites. Thus, the dynamics of each vertex are independent in this method.

17

t t+ t rate

0 S nS βS

S I nI βI

S,I 0 1

S 0 I

0 0 0

1

tt ∆+

t S S I

S I I

Iβ

S 0 I

S S I

Sβ

Fig 3. Dynamics of the two-stage contact process. (a) transition

rule, (b) reproduction process, (c) infected process, and (d)death

process. The symbols 0, S and I indicate empty, healthy and infected,

respectively. The parameters are: βS, fecundity; βS, infectivity; nS, the

number of individuals in the nearest-neighbor sites of the empty area;

and nS, the number of infected individuals in the nearest-neighbor sites

of healthy individuals.

18

Thus, Pσiσjσkis approximated to ρσi

ρσjρσk

. Here, ρσi(t) is the probabil-

ity that a randomly chosen site is of the state σi at time t) [Fig. 4 (a)].

The PA assumes that the effects of distant sites will be less important

than those of the nearest neighbor sites. Thus, Pσiσjσkis approximated

to PσiσjPσjσk

/ρσj[Fig. 4 (b)]. Specifically, the probability relevant to the

connected triple vertex is expressed by the multiplication of the probabil-

ity relevant to the two pairs of vertices. Here, in the multiplication, the

chosen probability of the center vertex (in the triplet vertex) is included

in both probabilities relevant to the pairs. Thus, the probability relevant

to the triplet vertex is approximated by dividing the multiplication of

the probability relevant to the pairs of vertices by the chosen probability

of the center vertex. Therefore, this is a valid approximation method

for analyzing the effects of local connections. Additionally, the analyzing

the model on lattice space is suited to express plant reproduction pro-

cesses. The mathematical model on lattice space has been analyzed in

mathematics, physics and ecology (including the Ising model, percolation

model and contact process). The model create discrete spaces, and the

framework (the configurations of sites and distances between each site)

19

of the lattice does not change. Plants are distributed discretely in space

and plants cannot move from the established place during their lifetime.

Thus, the lattice model is better suited to express plant dispersal.

0 S S

S0 S S0 S

S0

S S

Fig 4. Definitions of the approximation methods. (a) mean-field

approximation, (b) pair approximation. These methods do not take into

account the effects of distant sites.

1.4 Purpose

In this thesis, the focus is on two characteristic phenomena, superin-

fection of pathogens, which increase the fitness of the pathogen, and seed

propagation in plants, which is an effective defensive behavior against

pathogen spread. There are several theoretical studies pertinent to the

evolution of pathogen virulence through superinfection [29, 30, 31, 32,

25, 33, 34], and of the optimal balance between seed and vegetative prop-

20

agation [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Previous superinfection

models did not consider spatial structure or the dynamics of host repro-

duction, and the previous seed propagation models did not examine the

effects of pathogen spread on the optimal propagation balance. Thus, in

this thesis, two models, superinfection and seed propagation, were con-

structed applying the TCP on lattice space to explore the effects of spatial

structures and each phenomenon on plant and pathogen populations.

In the superinfection model, the pathogen propagation process related

to superinfections in vegetatively propagated populations (seed propa-

gation was not considered) and the effects of superinfection events on

the genetic diversity of pathogens using several models, including the

1-strain and multiple-strain models, were examined. In the analysis of

the 1-strain model, an equilibrium value was derived using the MA and

PA, and its local stability using the Routh–Hurwitz stability criterion.

In the multiple-strain models, the dynamics using numerical simulations,

including the MA, PA and the Monte Carlo simulation (MCS), were an-

alyzed. Through these analyses, the effects of parameter values, such as

the density of individuals, transition of a dominant pathogenic strain,

21

and competition between plants and pathogens, on the dynamics of the

models were shown.

In the seed propagation model, the dynamics of plant reproduction and

pathogen propagation, and the effects of seed propagation on the defense

responses to pathogen spread in single and mixed (coexistence of sev-

eral plant types) plant populations were examined. Thus, the change of

relative merit in the breeding system caused by the invasion of a plant

population by systemic pathogens was expressed. In this model, the

superinfection process was not included because the focus was on de-

termining the optimal balance of the breeding system against pathogen

propagation. Additionally, the genetic diversity of pathogens has a less di-

rect impact than the reproductive distance on the strategy was assumed.

In the analysis, the equilibrium and its local stability were derived using

pair approximations in the case of single populations. Additionally, using

the MCS, the effects of spatial structure through a comparison with the

results of the PA was examined, and the case of a mixed population was

analyzed. In mixed populations, two situations were assumed, infected

and uninfected populations, and they were analyzed using only the MCS

22

because other analyses of the model are too complex to obtain analytical

results, having too many variables,. The efficacy of seed propagation on

the suppression of epidemic infections was examined by comparing the

results in the two situations.

23

Chapter 2 Superinfection model

There are several theoretical studies pertinent to the evolution of pathogen

virulence during superinfections that use an ordinary differential equa-

tion [29, 30, 31, 32, 25, 33, 34]. These studies defined the already infected

individual as superinfected and as being taken over by a second pathogen

strain of higher virulence. Thus, the strains do not permanently share

the host. Additionally, most of the studies assumed a trade-off between

the infection rate and the virulence of the pathogen. They explored the

evolution of virulence within a host population using the host–parasite

model, and they analyzed the model in the cases where a host is either

infected by only one strain of a pathogen (single infection) or by sev-

eral strains (superinfection) of a pathogen. According to these studies,

the virulence of the superinfection model evolves towards a higher value

compared with that of a single infection. However, they did not consider

the effects of spatial structures.

In contrast, there are several studies using the lattice model [45, 46, 47].

Sato et al. (1994) analyzed the TCP in detail using the PA; however,

they could not analytically determine the stability of the epidemic equi-

24

librium (coexistence of healthy and infected individuals). Satulovsky and

Tome (1994) studied a predator–prey system using an improved TCP

and obtained analytical results with respect to the coexistence equilib-

rium, although they assumed only one predatory species. Haraguchi and

Sasaki (2000) considered mutants of pathogens, which have a different

mortality rate, based on the TCP and analyzed the evolutionarily stable

strategy (ESS) of the mortality and transmission rates using computer

simulations. However, they did not include the superinfection events in

the propagation processes of the pathogens. In summary, the models in

previous studies are insufficient to examine the effects of superinfection

events on pathogen spread within clonal plant populations, which are

affected by spatial structures. Thus, further modifications of the mod-

els are necessary to express a pathogen spread process that incorporates

superinfection events.

2.1 Model

A model of plant growth and pathogen propagation processes, includ-

ing superinfection, was constructed. In the model, a single plant species

25

and multiple pathogen strains were assumed. It was also assumed that a

healthy individual plant is infected by a pathogen strain and an already

infected plant is superinfected by other pathogen strains. The model’s

dynamics is a continuous Markov process on a lattice space. The state of

each site, and the transition and the mortality rates of each state are rep-

resented by a vector Ω = (σ0, σ1, σ2, · · · , σn), B = (βσ0 , βσ1 , βσ2 , · · · , βσn)

and D = (dσ0 , dσ1 , dσ2 , · · · , dσn), respectively, where the total number of

states is n+1. ρσi(t) is the probability that a randomly chosen site is of

the state σi at time t. Thus, ρσi(t) indicates the global density of the site

at the state σi. σ0 = ”0”, σ1 = ”S”, and σi+1 = ”Ii”(i = 1, 2, · · · , n− 1)

represent empty, susceptible (healthy) individuals and individuals in-

fected with i pathogen strains, respectively. In addition, it was assumed

that the already infected individuals having i-strains (”Ii”) are superin-

fected (and taken over) by the more virulent j-strain, because the strain

with higher virulence often wins within-host competitions [25, 48]. Based

on the definitions of d0 and β0, d0 = 0 and β0 =1n

∑ni=1 dσi

because the

empty site does not die, and a transition to ”0” indicates the deaths of

healthy and infected individuals.

26

Four demographic processes were configured: (i) plant growth, (ii) first

infection, (iii) superinfection and (iv) death. The growth process is repre-

sented by a transition from state ”0” to ”S”, which indicates that plants

grow their ramets into an open area. Thus, empty sites are occupied by

healthy individuals. The first infection process is represented by a transi-

tion from state ”S” to ”Ii”, which indicates that healthy individuals have

been infected by pathogens of the i-strain. The superinfection process is

represented by a transition from state ”Ii” to ”Ij” (i > j). The death

process is represented by a transition from ”S” or ”Ii” to ”0”, which rep-

resents the deaths of healthy and infected individuals from natural causes

and the pathogen, respectively. In addition, infected individuals are not

able to recover to a healthy one. Thus, these processes are described

using the following notation, which is often used to explain the TCP:

(i) 0 → S at rateβSn (S)

z

(ii) S → Ii at rateβIi

n (Ii)

z

(iii) Ii → Ij at rate sβIj

n (Ij)

z

(iv) S, Ii → 0 at rate dS, dIi

(TP)

27

Parameter n(σi) is the number of σi-sites among the nearest neighbors

of the focal sites, z is the number of nearest-neighbor sites (e.g. z = 4

for a von Neumann neighborhood on the two-dimensional square lattice),

and s is the superinfection rate. Thus, sβIjdescribes the ratio of the

superinfection to the first infection [30]. Growth and infection events

occur at a rate proportional to the number of the healthy and infected

states, respectively, among the nearest-neighbor sites.

Here, let qσj/σi(t) be the conditional probability that a randomly chosen

nearest neighbor of a σi-site is a σj-site. In particular, qσi/σirepresents the

local density of σi-sites. Pσiσj(t) is the probability that a randomly chosen

site is of the state σi and a randomly chosen nearest-neighbor site is of the

state σj at time t. These variables have the following relationship [49, 45]:

Pσiσj= ρσi

qσj/σi. (2)

Thus, using the above dynamics, the following describes a set of master

equations, which is referred to as the general model (GM):

28

P00 =2

(n−1∑i=1

dIiPIk0 + dSPS0 −

βS (z − 1) qS/00z

P00

),

PS0 =βS (z − 1) qS/00

zP00 + dSPSS −

βS

zPS0 −

βS (z − 1) qS/0Sz

PS0

− dSPS0 +n−1∑i=1

[dIiPSIi

−βIi

(z − 1) qIi/S0z

PS0

],

PSS =2

(βS

zPS0 +

βS (z − 1) qS/0Sz

PS0 −n−1∑i=1

βIi(z − 1) qIi/SS

zPSS − dSPSS

),

˙PIi0 =βIi

(z − 1) qIi/S0z

PS0 + dSPSIi−

βS (z − 1) qS/0Iiz

PIi0 +n−1∑j=1

dIjPIiIj

− dIiPIi0

+ s

(n−1∑

j=i+1

βIi(z − 1) qIi/Ij0

zPIj0 −

i−1∑j=1

βIj(z − 1) qIj/Ii0

zPIi0

),

˙PSIi=βS (z − 1) qS/0Ii

zP0Ii +

βIi(z − 1) qIi/SS

zPSS −

βIi

zPSIi

−n−1∑j=1

βIj(z − 1) qIj/SIi

zPSIi

− (dS + dIi)PSIi

+ s

(n−1∑

j=i+1

βIi(z − 1) qIi/IjS

zPSIj

−i−1∑j=1

βIj(z − 1) qIj/IiS

zPSIi

),

˙PIiIj=βIi

(z − 1) qIi/SIjz

PSIj+

βIj(z − 1) qIj/SIi

zPIiS

−(dIi

+ dIj

)PIiIj

+ s

(n−1∑

k=i+1

βIi(z − 1) qIi/IkIj

zPIkIj

+n∑

k=j+1

βIj(z − 1) qIj/IkIi

zPIiIk

−i−1∑k=1

βIk(z − 1) qIk/IiIj

zPIiIj

−j−1∑k=1

βIk(z − 1) qIk/Ij Ii

zPIiIj

−βIj

zPIiIj

)(i > j) ,

˙PIiIi=2

[βIi

zPSIi

+βIi

(z − 1) qIi/SIiz

PSIi− dIi

PIiIi

+s

(n−1∑

j=i+1

(βIi

zPIiIj

+βIi

(z − 1) qIi/Ij Iiz

PIj Ii

)−

i−1∑j=1

βIj(z − 1) qIj/IiIi

zPIiIi

)].

(3)

29

Here, for example, in the right side of the fifth equation in the set (the

differential equation of PSIi), the first term describes the transition from

P0Ii to PSIi, which indicates that healthy individuals’ offspring take up

empty sites (state 0 → S). In this term, the transition rate is determined

by TP(i), and n(S) is (z − 1) qS/0Ii . The transition begins from P0Ii ; there-

fore, one of the nearest-neighbor sites of the 0-state site is of the state

Ii, and at least one of the other nearest-neighbor sites of the 0-state site

(in (z − 1) sites) should be S for a transition from 0 to S. Thus, the

probability is qS/0Ii , and the expectation of n (S) is equal to (z − 1) qS/0Ii .

In subsequent terms, the n (σ) (σ ∈ Ω) is obtained in a similar process,

except for the third term.

The second and third terms indicate that healthy individuals are in-

fected by i-strains of pathogens (state S → Ii). These terms describe

the transition from PSS to PSIiand from PSIi

to PIiIi, respectively, and the

transition rate of these terms is determined by the TP(ii). In particular,

the value of n(Ii) in the third term is equal to 1. The transition begins

from PSIi; therefore, there is already a Ii-state site among the nearest-

neighbor sites of the S-state site. Here, the state of the other sites among

30

the nearest-neighbor sites is also Ii, which is included in the fourth term.

The fourth term indicates that healthy individuals are infected by j-

strains of pathogens (state S → Ij). This term describes the transition

from PSIito PIj Ii

(i ∈ j), and sums the transition rates for all strains (from

I1 to In−1 ).

The fifth term indicates the death of healthy or infected individuals

(state S, Ii → 0). The term describes the transition from PSIito P0Ii or

PS0, and the transition rate of each process is determined by the TP(iv).

The last term indicates that the already infected individuals are super-

infected by other strains of the pathogen (e.g. state I2 → I1). The first

and second terms in parentheses describe the transition from PSIjto PSIi

and from PSIito PSIj

(j 6= i), respectively. The transition rates of these

terms are determined by the TP(iii). The first term indicates that the

infecting pathogens superinfect already infected individuals with another

strain, and the second term indicates that an already infected individual

is superinfected by another strain of the pathogen. Thus, the range of

the summation in the first term is from i+ 1 to n− 1, and in the second

term it is from 1 to i− 1, based on our assumptions.

31

2.2 Results

In this part, a new parameter mi (mortality cost) was introduced and

defined as βIi/dIi

for n− 1 strains(mi := βIi/dIi

, m1 < m2 < · · · < mn−1),

and set dIi= 1 (∀Ii ∈ Ω) was used to standardize the parameter, for

ease of analysis. The mortality cost represents the expectation of the

number of newly infected individuals produced during the lifetime of an

infected individual. Thus, a more highly virulent strain has a lower mor-

tality cost. Consequently, already infected individuals are superinfected

by strains with lower mortality costs. In addition, dS ≈ 0, because the

plant mortality is generally less than the plant growth rate in long-lived

clonal plants.

2.2.1 1-strain model

Initially, the simplest case (n = 2) for the GM was analyzed using

the MA and PA. The state of each site was denoted by σi ∈ S ≡

0, S, I (I := I1) from the assumption of only one pathogen strain. There-

fore, the following set of master equations to rewrite Eq. (3) was obtained:

32

P00 =2PI0 − 2βS (z − 1) qS/00

zP00,

PS0 =PSI +βS (z − 1) qS/00

zP00

−[βS

z+

βS (z − 1) qS/0Sz

+mI (z − 1) qI/S0

z

]PS0,

PI0 =PII +mI (z − 1) qI/S0

zPS0 −

[βS (z − 1) qS/0I

z+ 1

]PI0,

PSS =2

[βS

z+

βS (z − 1) qS/0Sz

]PS0 − 2

[mI (z − 1) qI/SS

z

]PSS,

PSI =βS (z − 1) qS/0I

zP0I +

mI (z − 1) qI/SSz

PSS

−[mI

z+

mI (z − 1) qI/SIz

+ 1

]PSI,

PII =2

[mI

z+

mI (z − 1) qI/SIz

]PSI − 2PII.

(4)

In this model, superinfection does not occur because there is only one

pathogen strain.

2.2.1.1 Mean-field Approximation

To close the set of Eqs. (4), several variables, qσi/σj≈ ρσi

(e.g. PS0 =

ρSq0/S ≈ ρSρ0) and qσi/σjσk≈ ρσi

, were approximated using the MA. In

addition, the equations were simplified, using the definition of variables

from Eq. (13) (Appendix. A.).

33

ρ0 = 1− ρ0 − ρS (1 + βSρ0) ,

ρS = ρS (βSρ0 −mI (1− ρ0 − ρS)) .

(5)

The system has three equilibrium states.

EM ≡ (ρ∗0 , ρ∗S , ρ

∗I ) = (1, 0, 0) ,

EM ≡ (ρ∗0 , ρ∗S , ρ

∗I ) = (0, 1, 0) ,

EM ≡ (ρ∗0 , ρ∗S , ρ

∗I )

=

(mI − 1

βS +mI

,1

mI

,βS (mI − 1)

mI (βS +mI)

).

EM, EM and EM indicate the states of extinction, disease-free and epi-

demic, respectively. From a local stability analysis of the each equilibrium

(see Appendix.B.1), EM is always unstable, which means that plants do

not become extinct at the positive parameter range in the system. When

mI < 1, EM is stable, then the pathogen is not able to survive if it has a

low mortality cost. By contrast, when mI exceeds 1, EM becomes unsta-

34

ble and EM is always stable, the epidemic occurs because the pathogen

spreads within the plant population. Thus, mI = 1 is the threshold value

for stability shifting. In conclusion, the stability of the equilibrium states

and the equilibrium density of healthy individuals (ρ∗S ) in the epidemic

state depended only on mI, regardless of βS in the MA.

2.2.1.2 Pair Approximation

To close the set of Eqs. (4) and consider the effect of local connec-

tions on the dynamics, several variables were approximated using the PA,

qS/0σ ≈ qS/0 and qI/Sσ ≈ qI/S. In addition, the equations were simplified by

the definitions of the variables (see Appendix.A.), and the following three

equilibrium states were obtained:

35

EP ≡(ρ∗0 , ρ

∗S , ρ

∗I , q

∗0/0, q

∗S/0, q

∗I/0

)=(1, 0, 0, 1, 0, 0) ,

EP ≡(ρ∗0 , ρ

∗S , ρ

∗I , q

∗0/S, q

∗S/S, q

∗I/S

)=(0, 1, 0, 0, 1, 0) ,

EP ≡(ρ∗0 , ρ

∗S , ρ

∗I , q

∗0/0, q

∗S/0, q

∗I/0, q

∗0/S, q

∗S/S, q

∗I/S, q

∗0/I, q

∗S/I, q

∗I/I

)=(see B.2) .

EP indicates that the plants become extinct. Therefore, qσ/S and qσ/I are

non-existent because ρS and ρI are equal to 0. EP represents the disease-

free state. Thus, qσ/0 and qσ/I are non-existent because ρ0 and ρI are equal

to 0. EP represents the epidemic state, at which all 12 variables exist and

have positive values. The local stability of each equilibrium state was ex-

amined using the Routh–Hurwitz stability criterion (see Appendix. B.2).

From the stability analysis, the three stable-equilibrium phases, disease-

free, epidemic and periodic oscillation [Fig. 5], were obtained. In par-

ticular, EP is always unstable (plants do not become extinct), and two

36

thresholds, epidemic and bifurcation, were derived. The stability of EP

and EP depends on whether the parameter values exceed each thresh-

old, especially the epidemic condition that depends only on the mortality

cost, irrespective of the growth rate because the epidemic threshold is

(mI)c = z/ (z − 1), which is similar to that of the MA. Thus, if mI is low,

then pathogens become extinct [panels (a) and (b) in Fig. 6], because the

low mI means a high virulence or low infection rate. Therefore, pathogens

die within the infected hosts before infecting other hosts. In addition, a

large mI leads to a decrease in both healthy and infected individuals. βS

affects the equilibrium value in the epidemic phase. For example, a large

βS leads to an increase in the equilibrium density of infected individuals

[panel (c) in Fig. 6], because pathogens can spread within the hosts sup-

plied by the fast growth rate. In the epidemic phase, when βS is large,

the equilibrium density of healthy individuals (ρ∗S ) decreases. Thus, a

slow growth rate had an advantage over a high growth rate for plants in

this phase. In addition, a Hopf bifurcation occurs when the parameter

values exceeded the bifurcation threshold [panel (b) in Fig. 6]. Thus, the

stability of EP shifted from stable to unstable, which was different from

37

under the MA.

0 20 40 60 80 100

0

20

40

60

80

100

Epidemic

Oscillation

Disease-free

Mortality cost (mI)

Growth

rate

(βS)

Bifurcation threshold

Epidemic

threshold

Fig 5. Phase diagram of the pair approximation. This figure

shows the three phases of the equilibrium state. In the epidemic phase,

plants and pathogens coexist and the equilibrium is stable. In the

oscillation phase, plants and pathogens coexist but the equilibrium is

unstable, and oscillation is observed. Therefore, Hopf bifurcation

occurs. The solid and dashed lines indicates the bifurcation and the

epidemic thresholds, respectively. In the disease-free phase, pathogens

become extinct through a too low mortality cost.

To check the validity of each approximation method, the equilibrium

values of the MA, PA and MCS were compared. The MCS was conducted

100 times for each given parameter set: (a) βS = 10 and mI = 0 ∼ 30, (b)

βS = 30 and mI = 0 ∼ 30, and (c) mI = 15 and βS = 0 ∼ 25 in Fig. 6. A

two-dimensional square lattice torus was used, and the average values of

38

(a). βS = 10 (b). βS = 30 (c). mI = 15

Den

sity

of

healthyindividual(ρ∗ S)

Den

sity

of

infected

individual(ρ∗ I)

Variance

ofρ∗ I

Mortality cost (mI) Grwoth rate (βS)

(i)

(ii)

(iii)

Fig 6. Comparisons among the numerical simulation of the

mean-field approximation (MA) and the pair approximation

(PA) and Monte Carlo simulation (MCS), which include the

mortality cost and growth rate, βS(= 10; 30; 50). (i) equilibrium

densities of healthy individuals (ρ∗S)(ii) equilibrium densities of infected

individuals (ρ∗I ). The dotted, solid and dashed lines represent the results

of the MA, PA and MCS, respectively. These lines show that when mI

is low, both approximation methods express similar trends to that of

the MCS, although the equilibrium value is overestimated. (iii) the

variance among 100 trials using the MCS. The high variance indicates

that the oscillatory solution is observed.

39

100 trials at each parameter value were calculated [Fig. 6]. There were

several discrepancies with respect to the equilibrium and threshold val-

ues. These included that the MA and PA overestimated the equilibrium

values, and that the periodic solution, were observed in the PA under

higher parameter values compared with those of the MCS. In Fig. 6, the

discrepancies between the threshold values determined by the MCS and

MA/PA were great when the mortality cost was large ((a) , (b)) or the

growth rate was low ((c)). The use of these approximation methods,

which neglected the effects from remote sites, may be the cause of the

discrepancies. However, a periodic solution was observed using the re-

sults of the PA and MCS [47], which were different from that of the MA.

Thus, the PA could explain the basic behavior of the system better than

the MA.

2.2.2 Multiple-strain models

Four sub-models (n = 3, 4, 11 and 26) of the GM (Eq. (3)) were exam-

ined; however, the analyses were too complex, having too many variables

to obtain an analytical result (the MA and PA require at least n and

40

Σni=1i−1 variables, respectively). Thus, the equilibrium value was derived

using the MA in the 2- and 3-strain models, and all of the models were

analyzed using computer simulations (the MCS and numerical simula-

tions of the MA and PA). In the simulations, mi = mi−1+∆mi,i−1 (i = 2,

3, . . ., n − 1), s = 0, 0.5, 1.0 and 1.5 and the value of βS varied. In this

paper, it was assumed that ∆m := ∆mi,i−1 was constant to maintain the

simplicity of the model. As a result of the numerical simulations of the

MA and PA, healthy individuals did not become extinct in all of the mod-

els. However, the healthy and infected individuals did become extinct in

the MCS, especially when n was large and βS was small. A comparison

among the MA, PA and MCS, indicated that the discrepancies increased

as the number of strains increased.

2.2.2.1 The 2-strain model (n = 3)

Five equilibrium states were obtained using the MA (Table 1 in detail):

E1: extinction (ρ∗0 = 1), E2: disease-free (ρ∗I1 = 0, ρ∗I2 = 0), E3: occu-

pation of a strain with high cost (ρ∗I1 = 0, ρ∗I2 > 0), E4: occupation of

a strain with low cost (ρ∗I1 > 0, ρ∗I2 = 0), and E5: coexistence (ρ∗I1 > 0,

ρ∗I2 > 0). In particular, in the (equilibrium) phase of coexistence (E5 is

41

stable), the dominant strain changed depending on the plant growth rate.

Table 1. Equilibria of the 2-strain model in MA.

E1 (1, 0, 0, 0)

E2 (0, 1, 0, 0)

E3

(m2−1βS+m2

, 1m2

, 0, βS(m2−1)m2(βS+m2)

)E4

(m1−1βS+m1

, 1m1

, βS(m1−1)m1(βS+m1)

, 0)

E5

(m2−m1

sβSm1, m1(sβS+1)−m2

βS(m2+m1(s−1)), sβSm1(m2−1)−(m2−m1)(βS+m2)

sβSm1(m2+m1(s−1)), m1(m2−m1)+βS(m2−m1−sm1(m1−1))

sβSm1(sm1+m2−m1)

)E ≡

(ρ∗0, ρ

∗S, ρ

∗I1, ρ∗I2

).

When s > 0, Figs. 7 and 8 show that the equilibrium density of each

state depended on the growth rate (βS) of a given superinfection rate (s)

and that the difference in mortality cost (∆m) was simulated by the MA

[panels (a)], PA [panels (b)] and MCS [panels (c)], as well as the tran-

sition of the equilibrium phase [panels (d)]. As a result, when βS was

small, the strain with a high cost was dominant. Then, the dominant

strain shifted to the strain with a low cost as βS increased, and the strain

occupied the pathogen population at a large βS. The increase in s and

the decrease in ∆m led to lower threshold values of the phase transitions

[panels (d) in Figs. 7 and 8]. Thus, when s was small or ∆m was large,

the parameter range of the phase of coexistence increased. Notably, when

∆m was large (moderate competition), multiple strains were more likely

42

to coexist , as in the ’limiting similarity of niche’ proposed by the com-

petition theory [50, 51]. For healthy individuals, their densities increased

at the coexistence phase and decreased at other phases in the PA. Thus,

if the growth rate was out of the range value of the coexistence phase,

the healthy population did not increase.

When s = 0 (no superinfection), the shift of the equilibrium phase

along the gradient of βS was different from the case of s > 0 in the MCS.

For example, when βS was large, the strain with the highest mortality cost

occupied and coexisted with healthy plants (strains with lower costs be-

come extinct). In addition, the range of the coexistence phase decreased

compared with when s > 0 [Fig. 9 (i)]. In the MA and PA, the strain

with the higher cost always occupied the host, regardless of the values of

the other parameters [panels (a) and (b) in Fig. 9 (i)]. Thus, when s = 0

and βS was large, the pathogen population was occupied by the strain

with highest cost, contrary to when s > 0.

In a comparison of the numerical simulations (the MA and PA) with

the MCS, when βS was small, the result was at extreme variance with the

MCS. Thus, in the parameter range, the approximation method could not

43

0 20 40 60 80 100

0.5

1

0 20 40 60 80 100

0.5

1

0 20 40 60 80 100

0.5

1

0 20 40 60 80 100

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

s = 0.5(a)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(d)

MA

E5

E4

E3

PA E4 E5 E3

MCS E4 E5

s = 1.0(a)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0ρ∗S

ρ∗I1ρ∗I2ρ∗I1

(d)

MA E4 E5 E3

PA E4 E5 E3

MCS

E2

E3 E5 E4

E5

E3

s = 1.5(a)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

(d)MA E4

E5

E3

PA E4 E3

MCS E3 E5 E3

Growth rate βS

Equilibrium

Density

ofeach

strain

Fig 7. The equilibrium value of each state and the transition

from the equilibrium phase depends on the superinfection rate

in the 2-strain model. m1 = 5 and ∆m = 5. The I, II and III differ

in their values of s (=0.5, 1.0 and 1.5, respectively). (a-c) the variation

of the equilibrium density of each state (”0”, ”S”, ”I1” and ”I2”.

Σσρσ = 1) with the growth rate in each simulation: (a) MA, (b) PA and

(c) MCS. (d) the transition from the equilibrium phase with βS in the

MA, PA and MCS.44

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10 12 14

0.5

1

0 2 4 6 8 10 12 14

∆m = 5(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0

ρ∗Sρ∗I1ρ∗I2

(d)

MA E4

E5

E3

PA E4 E3

MCS E3 E5 E3

∆m = 10(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0 ρ∗Sρ∗I1 ρ∗I1ρ∗I2

(d)

MA E4E5 E3

PA E4 E5 E3

MCS E3 E5 E4

E5

E3

∆m = 15(a)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(b)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

(c)

ρ∗0

ρ∗S

ρ∗S

ρ∗I1ρ∗I1 PP ρ∗I2

ρ∗I2

(d)

MA E4 E5 E3

PA E4 E5 E3

MCS E3 E5

E3

E2 E4 E5 E3

Growth rate βS

Equilibrium

Density

ofeach

strain

Fig 8. The equilibrium value of each strain and transition from

the equilibrium phase depends on the mortality cost in the

2-strain model. mI1 = 5 and ∆m = 5 in the figures. The I, II and III

differ in their values of s (=0.5, 1.0 and 1.5, respectively). The (a) MA,

(b) PA, (c) MCS and (d) the transition from equilibrium phase.

45

0 5 10 15 20 25

0.5

1

0 5 10 15 20 25

0.5

1

0 5 10 15 20 25

0.5

1

0 2 4 6 8 10

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

(i). 2-strain (n = 3)(a)

ρ∗0

ρ∗S ρ∗I2

(b)

ρ∗0

ρ∗S

ρ∗I2

(c)

ρ∗0ρ∗S

ρ∗I2

(d)

MA E4

PA E5

MCS

E2

E3

E2

E4

E5

E4

(ii). 3-strain(n = 4)(a)

ρ∗0

ρ∗S ρ∗I2

(b)

ρ∗0

ρ∗S

ρ∗I2

(c)

ρ∗0

ρ∗S

ρ∗I2

ρ∗I3

(d)

MA E5

PA E5

MCS

E2

E3

E6 E4 E8 E5

Growth rate βS

Equilibrium

Density

ofeach

strain

Fig 9. No superinfection (s = 0) in the 2-strain and 3-strain

models using the MA, PA and MCS. The global density of each

strain of the pathogen is plotted at the equilibrium state depending on

βS. The values of the parameters are: (a) n = 3, mI1 = 5, ∆m = 10, and

(b) n = 4, mI1 = 5, ∆m = 5.

46

be applied. When βS was large enough, the discrepancy between them

increased as s [Fig. 7] and ∆m decreasd [Fig. 8]. However, the MA and

PA could explain the transition process of the equilibrium phase [panels

(d) in Figs. 7 and 8]. In addition, the oscillatory solution, in which the

solution oscillates for a long time, although unproven, was not observed

in the coexistence phase [Fig. 10]. However, in the occupation phase of a

strain, the oscillatory solution was observed because the behavior of the

model then followed that of the 1-strain model.

5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

20 40 60 80 100

0.2

0.4

0.6

0.8

1.0

20 40 60 80 100

0.2

0.4

0.6

0.8

20 40 60 80 100

0.2

0.4

0.6

0.8

Time (t)

Global

density

ofeach

state

I.(a)

II.(a)

(b)

(b)

(c)

(c)

Fig 10. Time series in the 2-strain model. The equilibrium value

of the global density of each state is plotted. In the figures, s = 1.0 and

∆m = 15. I. mI1 = 5 and II. mI1 = 30 with (a) βS = 10, (b) βS = 15 and

(c) βS = 25.

2.2.2.2 The 3-strain model (n = 4 )

Nine equilibrium states were obtained using the MA (Table 2 in detail):

47

E1: extinction, E2: disease-free, E3−−5: occupation of a strain (ρ∗Ii > 0,

ρ∗Ij = 0, ρ∗Ik = 0), E6−−8: coexistence of two strains (ρ∗Ii > 0, ρ∗Ij > 0,

ρ∗Ik = 0), and E9: coexistence of all strains (ρ∗Ii > 0, ρ∗Ij > 0, ρ∗Ik > 0).

Table 2. Equilibria of the 3-strain model in MA.

E1 (1, 0, 0, 0, 0)

E2 (0, 1, 0, 0, 0)

E3

(m1−1βS+m1

, 1m1

, βS(m1−1)m1(βS+m1)

, 0, 0)

E4

(m2−1βS+m2

, 1m2

, 0, βS(m2−1)m2(βS+m2)

, 0)

E5

(m3−1βS+m3

, 1m3

, 0, 0, βS(m3−1)m3(βS+m3)

)E6

(m2−m1

sβSm1, m1(sβS+1)−m2

βS(m2+m1(s−1)), sβSm1(m2−1)−(m2−m1)(βS+m2)

sβSm1(m2+m1(s−1)), m1(m2−m1)+βS(m2−m1−sm1(m1−1))

sβSm1(sm1+m2−m1), 0)

E7

(m3−m1

sβSm1, m1(sβS+1)−m3

βS(m3+m1(s−1)), sβSm1(m3−1)−(m3−m1)(βS+m3)

sβSm1(m3+m1(s−1)), 0, m1(m3−m1)+βS(m3−m1−sm1(m1−1))

sβSm1(sm1+m3−m1)

)E8

(m3−m2

sβSm2, m2(sβS+1)−m3

βS(m3+m2(s−1)), 0, sβSm2(m3−1)−(m3−m2)(βS+m3)

sβSm2(m3+m2(s−1)), m2(m3−m2)+βS(m3−m2−sm2(m2−1))

sβSm2(sm2+m3−m2)

)E9

(m1m3−m2

βSm2+m1m3, m2

m1m3, βSm2(s(m1m3−m2)−m3+m2)−m1m3(m3−m2)

sm1m3(βSm2+m1m3),

m2(m3−m1)−sm1(m2+m1m3)sm1m3(βSm2+m1m3)

, m1m3(sβSm1−m2+m1)−βSm2(sm1+m2−m1)sm1m3(βSm2+m1m3)

)E ≡

(ρ∗0, ρ

∗S, ρ

∗I1, ρ∗I2 , ρ

∗I3

).

When s > 0, Figs. 11 and 12 show the equilibrium densities of each state

based on βS being in a given s and the difference in the mortality costs

(∆m), respectively, using the MA [panel (a)], PA [panel (b)] and MCS

[panel (c)]. The occupation phase of a single strain and the coexistence

phase of two strains showed generally similar responses to the parameter

values in the single-strain and 2-strain models. The effects of s and∆m on

48

equilibrium values, threshold values and discrepancies among simulation

results were similar to those of the 2-strain model. Thus, the increase or

decrease in the density of healthy individuals depended on βS in the PA,

which was different from in the MA, in which the growth rate negatively

affected the density of the healthy individuals. However, when s and

∆m were both large, the transition of the equilibrium phase in the MA

(PA) was different from in the MCS, because the strain with the highest

cost (I3) became extinct in the MCS. In addition, the oscillatory solution

was observed in a parameter range [Fig. 13], and the strain with the

middle cost (I2) was dominant, which was similar to the results of previous

studies [29, 30, 31, 32, 25][Figs. 11 and 12]. In addition, when s = 0

[Fig. 9(ii)], the response to βS was the same as in the 2-strain model (the

range of the coexistence phase decreased and the pathogen population

was occupied by the strain with the highest cost at a large βS).

2.2.2.3 Multiple-strain models

In the multiple-strain models, there are many equilibrium states: ex-

tinction, disease-free, occupation of a strain, coexistence of various strains,

and coexistence of all strains.

49

0 10 20 30 40 50 60 70

0.5

1

0 10 20 30 40 50 60 70

0.5

1

0 10 20 30 40 50 60 70

0.5

1

0 5 10 15 20 25

0.5

1

0 5 10 15 20 25

0.5

1

0 5 10 15 20 25

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

s = 0.5(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

ρ∗I3

(b)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

ρ∗I3

(c)

ρ∗0

ρ∗S ρ∗I1ρ∗I2

s = 1.0(a)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2 ρ∗I3ρ∗I3

(b)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

ρ∗I3

(c)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

s = 1.5(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

(b)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2 ρ∗I3ρ∗I3

(c)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

ρ∗I3P

Growth rate βS

Equilibrium

Density

ofeach

strain

Fig 11. The equilibrium value of each strain depends on the

superinfection rate in the 3-strain model. mI1 = 5 and ∆m = 5 in

the figures. The I, II and III differ in their values of s (=0.5, 1.0 and

1.5, respectively). (a) MA, (b) PA and (c) MCS

50

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 2 4 6 8 10

0.5

1

0 5 10 15 20 25 30

0.5

1

0 5 10 15 20 25 30

0.5

1

0 5 10 15 20 25 30

0.5

1

0 5 10 15 20 25 30

0.5

1

0 5 10 15 20 25 30

0.5

1

0 5 10 15 20 25 30

0.5

1

∆m = 5(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

(b)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2 ρ∗I3ρ∗I3

(c)

ρ∗0

ρ∗S

ρ∗I1ρ∗I2

ρ∗I3PP

∆m = 10(a)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2ρ∗I3 ρ∗I3

(b)

ρ∗0ρ∗S

ρ∗I1ρ∗I2 ρ∗I3

ρ∗I3

(c)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2 ρ∗I3

∆m = 15(a)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3

ρ∗I3

(b)

ρ∗0ρ∗S

ρ∗I1ρ∗I2

ρ∗I3ρ∗I3

(c)

ρ∗0 ρ∗S

ρ∗I1ρ∗I2 ρ∗I3

Growth rate βS

Equilibrium

Density

ofeach

strain

Fig 12. The equilibrium value of each strain depends on the

mortality cost in the 3-strain model. mI1 = 5 and s = 1.5 in the

figures. The I, II and III differ in their values of ∆m (=5, 10, 15). (a)

MA, (b) PA and (c) MCS.

51

200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000

0.2

0.4

0.6

0.8

200 400 600 800 1000

0.2

0.4

0.6

0.8

200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000

0.2

0.4

0.6

0.8

200 400 600 800 1000

0.2

0.4

0.6

0.8

Time (t)

Global

density

ofeach

state

I.(a)

II.(a)

(b)

(b)

(c)

(c)

Fig 13. Time series for the global density of each state in the

3-strain model. The equilibrium value of the global density of each

state is plotted. The parameter values are set as: I. s = 1.0, mI1 = 10,

∆m = 15 and II. s = 1.5, mI1 = 10, ∆m = 20 with (a) βS = 3, (b)

βS = 5 and (c) βS = 7. In the 3-strain model, the oscillatory solution is

observed in a particular parameter range during the coexistence phase

[panel (b)].

The results of the 10-strain [Fig. 14] and 25-strain models [Fig. 15]

were plotted at ∆m = 5 with varying βS and s values. As a result, a

smaller value of βS or s led to the dominance of the strain with the higher

cost [Figs. 14 and 15], and the cost of the dominant strain shifted to a

lower value as these parameter values increased, which was similar to the

results of the 2-strain and 3-strain models. In addition, when the βS was

small, the oscillatory solution was observed [Fig. 16], and when the s was

also small, there was a possibility of extinction in the MCS when the n

was too large [Fig. 15].

52

Number of strain

Globaldensity

I. s = 0.5, βS = 25

II.s = 1.0, βS = 25

III.s = 1.5, βS = 25

IV.s = 1.5, βS = 5

MA PA MCS

Fig 14. The equilibrium density distribution of strains in the

10-strain model. The left, center and right panels show the results of

simulations using the MA, PA and MCS, respectively. mI1 = 5 and

∆m = 5. The other parameter values are: I. s = 0.5, βS = 25, II.

s = 1.0, βS = 25, III. s = 1.5, βS = 25 and IV. s = 1.5, βS = 5.

53

Number of strain

Globaldensity

I. s = 0.5, βS = 25

II.s = 1.0, βS = 25

III.s = 1.5, βS = 25

IV.s = 1.5, βS = 5

MA PA MCS

Fig 15. The equilibrium density distribution of strains in the

25-strain model. The left, center and right panels show the results of

simulations using the MA, PA and MCS, respectively. mI1 = 5 and

∆m = 5. The other parameter values are: I. s = 0.5, βS = 5, II. s = 0.5,

βS = 25, III. s = 1.0, βS = 25 and IV. s = 1.5, βS = 25.

200 400 600 800 1000

0.2

0.4

0.6

0.8

20 40 60 80 100

0.2

0.4

0.6

0.8

time t

Globalden

sity

Fig 16. Time series in the 25-strain model. The equilibrium value

of the global density of each state is plotted. In the figures, mI1 = 5,

∆m = 5 and βS = 5. Additionally, (a) s = 1.5 and (b) s = 0.5.

54

2.3 Discussion

Superinfection events increase the fitness of pathogens by widely spread-

ing them, through the additional associated transmission pathways, within

a plant population. Several studies have analyzed the process of pathogen

spread by approximating the lattice space [45, 47, 46] and the superinfec-

tion process using mathematical models without spatial structures [30].

Sato et al. (1994) studied the case of dS = 1 in our single-strain model.

Their model had three equilibrium states, (disease-free, endemic and epi-

demic), and they derived equilibrium values at two states (disease-free

and endemic) along with their explicit local stability conditions. How-

ever, they did not obtain the epidemic equilibrium value and its local

stability. Haraguchi and Sasaki (2000) considered that there is no trade-

off between the infection rate and the virulence of a pathogen, and they

assumed that multiple pathogens have different virulence levels. They

examined the ESS of the infection rate using a numerical simulation and

discussed the evolution of the pathogen’s infection rate. Their simulation

suggested that pathogens evolve to an intermediate infection rate. Sat-

ulovsky and Tome (1994) considered a transition rule similar to that in

55

our model and assumed a correlation between the transition rates of each

state (βS +mI + dI = 1). According to their PA and MCS results, there

are four equilibrium phases (three stationary and one oscillation phase),

and if the transition rate to state 0 (dI) is small, then the Hopf bifurca-

tion occurs. Nowak and May (1994) examined the superinfection events

using an ordinary differential equation. As a result, superinfection leads

to the maintenance of the pathogen strain’s polymorphism, and the os-

cillatory solution (competition among plants and pathogens) is observed

when there is more than one strain. However, they did not consider spa-

tial structures or host reproductive dynamics. They assumed that the

host constantly increases.

To consider the effects of spatial structures, plant reproduction and

pathogen propagation dynamics during superinfection in a clonal plant

population on a lattice space were analyzed. Five models (the single-

strain model and four multiple-strain models) that included interactions

among the host plant and several strains of pathogen, were analyzed, and

the MA and PA were adopted to analyze the dynamics of the models. In

addition, the validity of the approximation methods in comparison with

56

the MCS was checked.

In the single-strain model (n = 2), the value of the epidemic thresh-

old depends only on the mortality cost, which means that establishing a

pathogen within a plant population depends only on the pathogen’s abil-

ity, regardless of the plant. In addition, the density of healthy individuals

decreases with increasing growth rates and mortality costs [panel (i) in

Fig. 6]. By contrast, the density of infected individuals increases with the

growth rate and decreases with an increase in the mortality cost [panel

(ii) in Fig. 6]. Therefore, plants should not increase their growth rate to

maintain a large population size when infected by a systemic pathogen,

and the pathogen should evolve a low mortality cost that is higher than

the epidemic threshold to maintain their population and that of their

hosts.

Additionally, if the parameter values exceed the bifurcation thresh-

old, then the Hopf bifurcation occurs and the periodic solution, in which

plants and pathogens continue to compete forever, is observed using the

PA and MCS, but not the MA. Thus, the effects of local interactions are

important when expressing the dynamics of the pathogen propagation

57

process. However, the MA (which neglects the local interactions) is use-

ful for analyzing the dynamics within a measured situation, such as the

number of equilibrium states in the model.

In the multiple-strain models (n > 2), it was assumed that the multi-

ple pathogen strains had different mortality costs, and that the already

infected individuals were superinfected by strains with lower mortality

costs. The analytical results of the MA showed that there are a lot of

equilibrium states: extinction, disease-free, occupation of a strain, co-

existence of various strains, and coexistence of all strains (Table 1 and

2). Based on the MA, PA and MCS results, the equilibrium phase and

the dominance of a strain in the coexistence phase depend on parameter

values, and the oscillatory solution is observed in the coexistence phases,

except for in the 2-strain model using the PA and MCS [Figs. 10 and 13].

In addition, the genetic diversity of a pathogen is maintained by a

decrease in superinfection events. In fact, the parameter range of the

coexistence phase increases with a decrease in superinfection rates, even

when the difference in the mortality cost is small. This is because a su-

perinfection event is conducive to a strong competition among strains,

58

which is caused by additional transmission routes that decrease the dif-

ferences in infection rates among the strains. However, if superinfection

does not occur (s = 0), then the range of the coexistence phase decreases

[Fig. 9]. Thus, superinfection is important to maintain genetic diversity.

Additionally, when the plant growth rate increases, the pathogen popu-

lation is eventually occupied by a strain, regardless of the superinfection

rate [Figs. 7, 9 and 11]. Thus, the increase in the plant growth rate

causes a decrease in the genetic diversity of the pathogen. For healthy

individuals, too high of a growth rate provides them no benefit. Healthy

individuals can increase their abundance through the growth rate in the

coexistence phase of several strains. However, if the growth rate is too

high, an equilibrium phase shifts to the phase of occupation by one strain,

and the density of healthy individuals then decreases as the growth rate

increases. The dynamics follow those of the single-strain model in this

phase.

The results are summarized as follows: (i) The strain with an inter-

mediate cost became dominant, similar to the results of previous stud-

ies [29, 30, 31], when both the superinfection and growth rates were low.

59

However, a high superinfection or growth rate led to the dominance of

the strain with the lowest cost in our model. Additionally, the pathogen

received more benefit due to a low mortality cost when the hosts grow

rapidly [5]; (ii) The competition among strains occurred in the coexis-

tence of various strains phase when using the PA and MCS in the n > 3

models; (iii) Too high a growth rate led to occupation by the strain with

the lowest cost. Thus, competition between the strain and the hosts oc-

curred, and, therefore, the host population decreased in all of the models;

(iv) Pathogens easily maintained their genetic diversity when there was a

low superinfection rate. However, if they did not superinfect, such main-

tenance became difficult; and (v) When the growth rate of a plant was

low, an individual at a local site was strongly interconnected by distant

individuals because the MA and PA did not apply in this case.

In conclusion, pathogens maintain their genetic diversity through su-

perinfection events and a moderate mortality cost relative to growth rate.

Thus, their mortality costs and superinfection rates evolve based on the

host plant’s ability to maintain their populations. By contrast, the num-

ber of healthy individuals (plants) increases in (all and several strains)

60

coexistence phases with the growth rate. Thus, when systemic pathogens

invade the plant population, the plant’s growth rate evolves to be slightly

lower than the threshold value at which the equilibrium phase shifts to

the phase of occupation by a strain to increase their population.

61

Chapter 3 Seed propagatin model

Many clonal plants have two breeding systems, vegetative and seed

propagation [Fig. 1]. The seed-propagated offspring (seedlings) have a

higher mortality rate because of their long-distance dispersal, which does

not allow them to be supported by their parents (no physical connec-

tion). The vegetatively propagated offspring (vegetative propagules) have

a lower mortality rate because they supply resources among themselves

through interconnected ramets [3, 4]. However, if systemic pathogens

invade the population, the interconnected ramets become pathways of

pathogen spread. Thus, the balance between the two breeding systems

has an important role in the defensive behavior against systemic pathogens.

The balance between the breeding systems, vegetative and seed prop-

agation, has been studied experimentally [52, 53, 54, 55]. According

to these studies, the balance is determined by several functions, such as

resource allocation, competitive ability and colonization capacity. For

instance, if the resource is distributed heterogeneously in space, then

vegetative propagation has an advantage over seed propagation because

the vegetative propagules can be supplied resources from local colonies

62

of clones [56]. On the contrary, if the resource is distributed homoge-

neously, then seed propagation has an advantage over vegetative propa-

gation because seed propagation can spread the offspring long distances

and distribute them over an entire habitat [57].

Several mathematical models, such as transition matrix model [35],

reaction-diffusion equation model [36], lattice model [37, 38, 39] and

individual-based model [40, 41, 42, 43, 44], are used to express the plant

reproductive process. They analyze the optimal balance of the breed-

ing systems depending on several functions, such as resource distribu-

tion [41, 56], distance from the parents [39] and the density of individu-

als [35, 44]. Harada et al. (1996) analyzed the plant reproduction process

with seed propagation using the TCP on lattice space. They assumed that

the optimal balance of the breeding systems depends on the distance from

the parents. Ikegami et al. (2012) examined the effects of plant density

and mortality on the adoption of a breeding system using a computer

simulation on lattice space. However, these studies did not consider the

effects of pathogen propagation on the optimal breeding system balance.

63

3.1 Model

A model that included the plant reproductive process (both vegetative

and seed propagation) and the pathogen transmission process was con-

structed. The dynamics of the model is a continuous Markov process on

a lattice space. The states of each site are presented as empty (”0”),

susceptible (healthy) individual (”S”), infected individual (”I”), the in-

trinsic reproduction rate of the plant by mS, the intrinsic transmission

rate of the pathogen by mI, the proportion of vegetative propagation by

α, and the mortality rate of individuals of each state by 1. Additionally,

ρσ (t) (σ ∈ 0, S, I) is the probability that a randomly chosen site has

state σ at time t. Thus, ρσ (t) indicates the global density of the site with

state σ.

Four demographic processes were configured: (i) vegetative propaga-

tion (VP); (ii) seed propagation (SP); (iii) infection (IP); and (iv) death

(DP). The plant reproductive processes (VP and SP) are represented by

transitions from state ”0” to ”S”, which indicates that plants reproduce

offspring by either breeding system into an open area (an empty site is

then occupied by a healthy individual). IP is represented by the transi-

64

tion from state ”S” to ”I”, which indicates that healthy individuals are

infected by pathogens. DP is represented by the transition from ”S” or

”I” to ”0”, which represents the death of healthy or infected individuals,

respectively, from natural causes and the virulence of the pathogen, re-

spectively. In addition, infected individuals can not return to health and

reproduce their offspring.

Plant offspring inhabit distant and close open areas through seed and

vegetative propagation, respectively. Thus, an empty site becomes oc-

cupied by a healthy individual through reproduction from a randomly

chosen healthy site by seed propagation or from nearest-neighbor healthy

sites by vegetative propagation. Additionally, the pathogens transmit

from infected individual to surrounding healthy individual. Thus, in the

vegetative propagation and infection processes, the transition rate de-

pends on the states of nearest-neighbor sites. For instance, a healthy

individual is likely to become infected if the individual is surrounded by

infected individuals. These processes were described using the following

notation that is often used to explain the CP:

65

(i) 0 → S at rate (1− α)mSρS

(ii) 0 → S at rateαmSn (S)

z

(iii) S → I at ratemIn (I)

z

(iv) S, I → 0 at rate 1

(TP)

Parameter n(σ) is the number of σ-sites in the nearest neighbors of the

focal sites, z (= 2) is the number of nearest-neighbor sites (e.g. z = 4 for

a von Neumann neighborhood on a two-dimensional square lattice). The

vegetative propagation and infection events occur at rates proportional

to the number of the healthy and infected states in the nearest-neighbor

sites, respectively.

The above dynamics were described using a master equation that incor-

porates the additional variables Pσiσjand Pσiσjσk

(σi, σj, σk ∈ Σ), which

are referred to as pair density and triplet density, respectively. The vari-

ables represent the probability that the randomly chosen two or three

neighboring sites have the state σi, σj and σk at time t. Thus, Pσiσj=

Pσjσiand Pσiσjσk

= Pσkσjσi(master equation) Here, the positive and neg-

ative terms indicate transition probabilities from any state and to any

66

other state, respectively.

P00 =2

− (1− α)mSρSP00︸ ︷︷ ︸(i)SP

−αmS (z − 1)

zPS00︸ ︷︷ ︸

(ii)VP

+PS0 + PI0︸ ︷︷ ︸(iv)DP

P0S = (1− α)mSρS (P00 − P0S)︸ ︷︷ ︸

(i)

+αmS

[(z − 1)

z(PS00 − PS0S)−

1

zP0S

]︸ ︷︷ ︸

(ii)

−(z − 1)mI

zPIS0︸ ︷︷ ︸

(iii)IP

+(PSS − P0S) + PSI︸ ︷︷ ︸(iv)

PI0 =− (1− α)mSρSPI0︸ ︷︷ ︸(i)

−αmS (z − 1)

zPS0I︸ ︷︷ ︸

(ii)

+mI (z − 1)

zPIS0︸ ︷︷ ︸

(iii)

+PIS + (PII − PI0)︸ ︷︷ ︸(iv)

PSS =2

(1− α)mSρSPS0︸ ︷︷ ︸(i)

+αmS

(1

zPS0 +

(z − 1)

zPS0S

)︸ ︷︷ ︸

(ii)

+mI (z − 1)

zPISS︸ ︷︷ ︸

(iii)

−PSS︸︷︷︸(iv)

PIS = (1− α)mSρSPI0︸ ︷︷ ︸

(i)

+αmS (z − 1)

zPS0I︸ ︷︷ ︸

(ii)

+mI

[(z − 1)

z(PISS − PISI)−

1

zPIS

]︸ ︷︷ ︸

(iii)

−PIS − PIS︸ ︷︷ ︸(iv)

PII =2

mI

1

zPIS +

(z − 1)

zPISI

︸ ︷︷ ︸

(iii)

−PII︸︷︷︸(iv)

(6)

Here, the positive and negative terms indicate transition probabilities

67

from any state and to any other state, respectively.

3.2 Result

The two models, single population and mixed population models, were

analyzed in homogeneous environment to ease the analysis although the

merit of vegetative propagation through resource sharing disappears (i.e.

we can examine the effect of long distance dispersal through seed propaga-

tion on pathogen spread directly). In the single population model, there

are one type of plant and pathogen, and the plant adjusted the breed-

ing system’s balance to block the spread of the pathogen. In contrast,

in the mixed population, there are several types of plants and one type

of pathogen, and it was expected that the optimal balance of the breed-

ing systems is different from that of the single population model because

of competition among the plant types and the block to the pathogen’s

spread. The system was analyzed using the MA, PA (in single population

model) and MCS (in both models).

The MCS was conducted 100 times at each parameter set in a two-

dimensional square lattice torus (whose size is 100×100), and the average

68

value of the 100 trials was calculated.

As a result, the master equation using the MA does not depend on

the parameter α (Appendix. C.1). Thus, the PA and MCS were used

in subsequent analyses. As a result of the PA, the system has three

equilibrium states (extinction, disease-free and epidemic), similarly to a

previous study [45] (corresponding to the case of α = 1 in our model),

and two thresholds [Fig. 17(a)], which are referred to as the extinction

and epidemic threshold, were derived from the local stability analysis

(Appendix. C.1). Additionally, the results of the PA and MCS were com-

pared to examine the effects of the distant sites (i.e. spatial structure).

Here, the new parameter µ = mI/mS was introduced, and it represented

the relative scale to plant fecundity of pathogen infectivity.

3.2.1 Single population

3.2.1.1 Extinction phase

In this phase, both the healthy and infected plants went to extinction.

Thus, all of the sites converted to the empty state; ρ∗0 = 1, ρ∗S = 0, and

ρ∗I = 0. The extinction equilibrium became stable when the values of the

69

parameters were below the extinction threshold [Fig. 17(a)]. If the mS

was large enough, then the extinction equilibrium was always unstable,

regardless of α. Thus, plants with low fecundity could prevent extinction

by increasing the proportion of seed propagation in their reproductive

strategy. Using the PA, the following extinction threshold was analyti-

cally derived (Appendix C.2):

(mS)c =α− z +

√(z + α)2 − 4zα2

2α (1− α)

The threshold did not depend on mI; therefore, the infectivity of the

pathogen was irrelevant to the extinction of the host plants.

In a comparison of the MCS with the PA [Fig. 18], the discrepancy in

the equilibrium value increased with an increase in α and a decrease inmS.

Thus, the importance of the spatial structure (the effects of distant sites)

increased with an increase in the proportion of vegetative propagation

and a decrease in fecundity.

3.2.1.2 Disease-free phase

In this phase, the pathogens disappeared with the infected individuals

70

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.1

1.2

1.3

1.4(a)

Extin tion

Disease-free

Extin tion threshold

Proportion of vegetative propagation ()

F

e

u

n

d

i

t

y

(

m

S

)

0.0 0.2 0.4 0.6 0.8 1.0

0.35

0.36

0.37

0.38

0.39

0.40(b)

Disease-free

Epidemi

Epidemi threshold

Proportion of vegetative propagation()

P

r

o

p

o

r

t

i

o

n

o

f

i

n

f

e

t

i

v

i

t

y

t

o

f

e

u

n

d

i

t

y

(

)

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.3

0.4

0.5

0.6 (c)mS = 4

mS = 5

mS = 6

Proportion of vegetative propagation(α)

Proportion

ofinfectivityto

fecundity(µ)

Fig 17. The extinction and epidemic thresholds in the pair

approximation. (a) The solid line indicates the extinction threshold.

When the values of the parameters are below the threshold, the

extinction equilibrium is stable, and when the value exceeds that of the

threshold, it becomes unstable, and the disease-free equilibrium is

stable. Thus, if the fecundity of the plant is low, then the plant becomes

extinct. Additionally, when the value of the fecundity is close to that of

the threshold, the decrease in the proportion of vegetative propagation

is effective in protecting the plant from extinction. (b) The solid line

indicates the epidemic threshold at mS = 5. When the value of

parameters is below the threshold, the disease-free equilibrium is stable,

and when the value exceeds the threshold, the disease-free equilibrium

becomes unstable and epidemic equilibrium is stable. Thus, if invading

pathogens have low infectivity levels, the pathogens do not spread

within the plant population. Additionally, when the values of the

parameters are close to that of the threshold, the increase in the

proportion of vegetative propagation is effective in protecting the plant

from the epidemic. (c) The epidemic threshold shifts downward with an

increase in mS. Thus, a plant population with a high fecundity is likely

to lead to an epidemic, even when pathogens having a low infectivity

level invade the population. 71

0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

Proportion of vegetative propagation (α)

Global

density

ofhealthyindividual(ρ

∗ S) (a) (b)

(c) (d)

Fig 18. The comparison of the equilibrium values of ρSbetween the pair approximation and Monte Carlo simulation.

The solid and dotted lines indicate the the pair approximation and

Monte Carlo simulation, respectively. The figure shows the ρ∗S in the

extinction and disease-free phases. (a-d) differ in the value of mS (=1.2,

1.3, 1.4 and 1.5, respectively). From the comparison between the pair

approximation and Monte Carlo simulation, the discrepancy increases

with an increase in α and a decrease in mS.

72

(ρ∗I = 0). The disease-free equilibrium became stable when the values of

the parameters were between the values of the extinction and epidemic

thresholds [Fig. 17 (a )and (b)]. The ρ∗S increased with a decrease in α

[Figs. 18 and 19] or an increase in µ [Fig. 20]. However, an increase in

the mS led to a lower value for the epidemic threshold [Fig. 17 (c)]. Thus,

the increase of mI led to a transition to the epidemic phase, even when

mS or α was low.

3.2.1.3 Epidemic phase

In this phase, pathogens could invade and spread within a plant pop-

ulation. The value of ρ∗S reached its maximum at α = 1 or 0, which

depended on µ. In particular, the increase in µ led to a shift in the value

of α from 1 to 0 [Fig. 21 (I.)]. However, the value of ρ∗I increased with

a decrease in α, regardless of µ (the ρ∗I reached its maximum value at

α = 0) [Fig. 21 (II.)]. In addition, the proportion of healthy individuals,

out of all of the individuals (ρ∗S/ (ρ∗S + ρ∗I )), also increased with a decrease

in α, regardless of µ [Fig. 21 (III.)]. The increase of µ led to a shift to the

disease-free phase and then a shift to the extinction phase [Fig. 17(a)].

In the model of a previous study [45], in which seed propagation was

73

0.2 0.4 0.6 0.8 1.0

0.788

0.790

0.792

0.794

0.796

0.798

0.2 0.4 0.6 0.8 1.0

0.780

0.785

0.790

0.795

0.2 0.4 0.6 0.8 1.0

0.76

0.77

0.78

0.79

0.2 0.4 0.6 0.8 1.0

0.824

0.826

0.828

0.830

0.832

0.834

0.2 0.4 0.6 0.8 1.0

0.80

0.81

0.82

0.83

0.2 0.4 0.6 0.8 1.0

0.76

0.78

0.80

0.82

0.2 0.4 0.6 0.8 1.0

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.2 0.4 0.6 0.8 1.0

0.005

0.010

0.015

0.020

0.2 0.4 0.6 0.8 1.0

0.01

0.02

0.03

0.04

0.2 0.4 0.6 0.8 1.0

0.0005

0.0010

0.0015

0.0020

0.0025

0.2 0.4 0.6 0.8 1.0

0.01

0.02

0.03

0.04

0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

Proportion of vegetative propagation(α)

Global

density

ofhealthyindividual(ρ

∗ S)

Global

density

ofinfected

individual(ρ

∗ I)

I.

II.

(a) (b) (c)

Epidemic Disease-free

Fig 19. The equilibrium values of ρS and ρI depend on α.

Groups I and II are plots of the values of ρ∗S and ρ∗I , respectively, in

disease-free and epidemic phases. The upper panels in each group show

the pair approximation, and the lower panels show the Monte Carlo

simulation. (a-c) differ in the value of µ (=3.6, 3.7 and 3.8, respectively)

and the appropriate value of fecundity (mS) was selected to show the

disease-free and epidemic phases (mS = 5 in the pair approximation and

mS = 6 in the Monte Carlo simulation). The ρ∗S increases with a

decrease in α during the disease-free phase and with an increase in α

during the epidemic phase.

74

0.2 0.4 0.6 0.8 1.0

0.45

0.50

0.55

0.60

0.65

0.70

0.2 0.4 0.6 0.8 1.0

0.4

0.5

0.6

0.7

0.8

0.2 0.4 0.6 0.8 1.0

0.5

0.6

0.7

0.8

0.2 0.4 0.6 0.8 1.0

0.4

0.5

0.6

0.7

0.8

0.9

Proportion of infectivity to fecundity(µ)

Global

density

ofhealthyindividual(ρ

∗ S) (a) (b)

Fig 20. The equilibrium values of ρS depend on µ in the

disease-free and epidemic phases. The upper panels show the pair

approximations, and the lower panels show the Monte Carlo

simulations. α = 0.8, and (a) mS = 4 and (b) mS = 5. The increase in

the mS leads to a lower epidemic threshold value.

75

0.0 0.2 0.4 0.6 0.8 1.0

0.197

0.198

0.199

0.200

0.201

0.2 0.4 0.6 0.8 1.0

0.1835

0.1840

0.1845

0.1850

0.2 0.4 0.6 0.8 1.0

0.172

0.173

0.174

0.175

0.0 0.2 0.4 0.6 0.8 1.0

0.16

0.17

0.18

0.19

0.20

0.2 0.4 0.6 0.8 1.0

0.25

0.26

0.27

0.28

0.2 0.4 0.6 0.8 1.0

0.130

0.135

0.140

0.145

0.150

0.2 0.4 0.6 0.8 1.0

0.20

0.21

0.22

0.23

0.24

0.2 0.4 0.6 0.8 1.0

0.19

0.20

0.21

0.22

0.23

0.24

0.2 0.4 0.6 0.8 1.0

0.19

0.20

0.21

0.22

0.23

0.2 0.4 0.6 0.8 1.0

0.26

0.28

0.30

0.32

0.34

0.36

0.2 0.4 0.6 0.8 1.0

0.20

0.25

0.30

0.2 0.4 0.6 0.8 1.0

0.25

0.30

0.2 0.4 0.6 0.8 1.0

0.46

0.48

0.50

0.52

0.2 0.4 0.6 0.8 1.0

0.46

0.48

0.50

0.0 0.2 0.4 0.6 0.8 1.0

0.44

0.46

0.48

0.50

0.2 0.4 0.6 0.8 1.0

0.45

0.50

0.55

0.2 0.4 0.6 0.8 1.0

0.34

0.36

0.38

0.40

0.42

0.44

0.2 0.4 0.6 0.8 1.0

0.32

0.34

0.36

0.38

0.40

0.42

Proportion of vegetative propagation(α)

Global

density

ofhealthyindividual(ρ

∗ S)

Global

density

ofinfected

individual(ρ

∗ I)

Proportion

ofhealthy

individual

inallindividual

I.

II.

III.

(a) (b) (c)

Fig 21. The values of ρ∗S (I.), ρ∗I (II.) and ρ∗S/ρ∗S + ρ∗I (III.)

depend on α in the epidemic phase (mS = 4). The upper panels

show the pair approximation, and the lower panels show the Monte

Carlo simulation. The panels (a-c) differ in the value of µ (=2.1, 2.3 and

2.5, respectively). From the panels in I, the value of α, which maximizes

the value of ρ∗S, depends on the value of µ. When µ is low, the ρ∗Sreaches its maximum at α = 0. However, when µ is high, the value of α

that maximizes the ρ∗S is equal to 1. Panels II and III show that the

value of ρ∗I and the proportion of healthy individuals out of all of the

individuals, respectively, increase with a decrease in α, regardless of µ.

76

not assumed, the epidemic equilibrium shifted to unstable and the oscilla-

tory solution (limit cycle) was observed at a high mS and mI , indicating

that the Hopf bifurcation occurred. The oscillatory solution indicated

that competition between plants and pathogens occurred, and the com-

petition cost was the maintenance of the plant population. However, in

the present model, the epidemic equilibrium was always stable if the val-

ues of the parameters exceed the epidemic threshold, and the bifurcation

was not observed. Thus, the plants could evade competition through seed

propagation.

3.2.2 Optimal proportion of vegetative propagation (Mixed

population)

To explore the defensive behavior through the breeding systems against

pathogen spread by applying the above model (single population), it was

assumed that the mixed plant population consisted of 11 types of plants

having different proportions of vegetative propagation (α = 0, 0.1, 0.2, · · · , 1).

The system was analyzed in two cases, uninfected and infected popula-

tions, using only the MCS.

77

3.2.2.1 Uninfected population

In this case, pathogens did not invade the plant population. Thus, the

initial density of the infected individuals was equal to 0 (ρI (0) = 0). The

plant type with the lower α became dominant when the mS was low, and

the α of the dominant type increased with mS [Fig. 22]. Additionally,

when the mS was low enough, plants became extinct. Therefore, when

plants had a high fecundity, vegetative propagation was advantageous

over seed propagation.

3.2.2.2 Infected population

In this case, plants were infected by pathogens (pathogens invaded the

plant population). The system had three equilibrium phases, extinction,

disease-free and epidemic, and the stable phase switched depending on

the values of the parameters, similar to in a single population. Here,

the pathogen type with the lower α became dominant when mS was low,

regardless of µ, and when mS was high, the α of the dominant type shifted

from an intermediate to lower value with an increase in µ (mI) [Fig. 23].

Thus, seed propagation was advantageous over vegetative propagation

even when the plants had a high fecundity, which was different from in

78

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1.0

Fecundity of plant (mS)

Proportion

ofvegetative

propagationof

dom

inan

ttype(α

)

Fig 22. The transition of the dominant type of plant depends

on mS in mixed population. This shows that pathogens did not

invade the plant population. The initial density of the infected

individual is set equal to 0, and the value of α for the dominant type is

plotted. The type with the lower α becomes dominant when the mS is

low, and the α changes to a higher value with an increase of mS.

Additionally, when the mS is low enough, plants become extinct.

79

an uninfected population.

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

Proportion of infectivity to fecundity(µ)

Proportion

ofvegetative

propagationofdominanttype(α

)

(a) (b)

Fig 23. The transition of the dominant type of plant depends

on µ in a mixed population. This shows the case when plants are

infected by pathogens. (a) mS = 30 and (b) mS = 15, and the value of α

of the dominant type is plotted. From panel (a), the α of the dominant

type shifts from an intermediate to lower value with an increase in µ.

From panel (b), the individual with lower α becomes dominant when

mS is low, regardless of µ.

3.3 Discussion

Adjusting the breeding system’s balance is effective in defending against

the spread of systemic pathogens, which transmit to close individuals,

within a plant population. According to field studies, the vegetative

propagules have lower mortality rates than seedlings because of resource

sharing through interconnected ramets. However, the ramets also trans-

port viruses along with the resources through their vascular systems, and

80

provide space for the hyphal growth of fungal pathogens in their vascu-

lar vessels. Additionally, the pathogens also spread through air and a

dense population of plants assists the pathogen’s spread. Thus, vegeta-

tive propagation has a disadvantage over seed propagation when systemic

pathogens invade a plant population because the breeding system repro-

duces offspring close to the parents.

There are many approaches that use the lattice model to analyze the

breeding dynamics of clonal plants [37, 38, 39, 44], as well as pathogen

transmission dynamics [45, 46]. Among them, studies of breeding dy-

namics examined the effects of spatial structures on the reproductive

strategy, represented in plants by the competition between vegetative

and seed propagation. Harada and Iwasa (1994), Harada et al. (1997)

and Harada (1999) considered two types of plants. One type reproduces

through both seed and vegetative propagation (mixed strategy), and the

other one reproduces through only vegetative propagation (pure strat-

egy). They analyzed the competition dynamics between the two types

by adopting the PA. Additionally, they assumed that the proportion of

both breeding propagation systems in the mixed strategy depends on the

81

distance from the parents, and they examined the ESS of the balance

between vegetative and seed propagation using computer simulations.

Ikegami et al. (2012) considered the effects of plant density and mor-

tality on the adoption of breeding systems. They assumed that each

individual switches between seed and vegetative propagation depending

on the local density of the individuals and that the switching threshold

of the reproductive pattern is affected by mortality. They analyzed, by

computer simulation, the optimal switching strategy based on local den-

sity and mortality. These studies did not consider the effects of pathogens

on the reproductive strategy.

However, studies of pathogen transmission dynamics examined the

transition threshold of the equilibrium phase (mainly extinction, disease-

free and epidemic phases). Sato et al. (1994) analyzed in particular the

phase transition in the TCP using the PA. Haraguchi and Sasaki (2000)

assumed that multiple pathogens have different virulence levels. They ex-

amined the ESS of the infection rate using a numerical simulation. Their

simulation suggested that pathogens evolve to an intermediate infection

rate. However, their models could not express the seed propagation pro-

82

cess because the models were constructed based on the basic TCP. Thus,

it is necessary to modify their models to describe both the plant reproduc-

tion process, including seed propagation, and the pathogen propagation

process.

The new model, as presented here, has three (positive) equilibrium

phases, extinction, disease-free and epidemic, using the PA and the spa-

tial structure affects the system when the fecundity is low, similar to

supreinfectin model (in Section.Chapter 2). The stability condition of

the extinction phase requires that the values of the parameters are be-

low the extinction threshold, which does not depend on the virulence of

the pathogens, as determined by an analysis using the PA. Thus, the

plants become extinct due to low fecundity, and with the death of the

host, the pathogens also become extinct. A stable equilibrium shifts from

extinction equilibrium to disease-free equilibrium when the values of the

parameters exceed the extinction threshold. In the disease-free phase,

the increase in fecundity leads to a large plant population size. Addi-

tionally, the population size increases with the proportion of vegetative

propagation. Thus, seed propagation has an advantage over vegetative

83

propagation in a homogeneous environment. However, high fecundity or

a low proportion of vegetative propagation leads to the downward shift

of the epidemic threshold [Figs. 17(c)]. Therefore, when plant fecundity

or the proportion of vegetative propagation is high, pathogens spread

within the plant population, even if the infectivity of the pathogen is

weak. Thus, the decrease in the production of offspring or aggressive

vegetative propagation (even when plants produce a number of offspring)

makes it difficult to prevent an epidemic because plants assist the spread

of the pathogen by reproducing susceptible individuals.

A stable equilibrium shifts from disease-free to epidemic equilibrium

when the values of the parameters exceed the epidemic threshold. In this

phase, the relative merit of both breeding propagation systems changes

depending on the relative scale to plant fecundity of pathogen infectivity.

Vegetative propagation has an advantage when the relative scale is low,

and seed propagation has an advantage when the relative scale is high. If

the infectivity of the pathogen is weak, then the breeding destinations of

new vegetative propagules become farther and farther away from infected

individuals with time because of a faster reproduction rate relative to the

84

pathogen transmission rate. The breeding destination of seedlings is se-

lected randomly in the entire area, and the seedlings may be reproduced

close to the infected individual (assisting the spread of the pathogen),

even when the habitat of the parents is far from the infected individ-

ual. Thus, plants inhibit pathogen spread by increasing the proportion

of vegetative propagation if the pathogen is weakly infective. If the infec-

tivity of the invading pathogen is strong (pathogen transmission rate is

faster than the plant reproduction rate), then the production of offspring

close to the parents in vegetative propagation can assist the pathogen’s

spread, and the probability of an epidemic within the plant population is

high. Plants can reproduce their offspring far from infected individuals

by seed propagation, even though the pathogen is widely spread. Thus,

plants block the spread of pathogen by increasing the proportion of seed

propagation when the pathogen is strongly infective. The increase in the

reproductive proportion of seed propagation leads to an increase in the

probability of having an infected individual within a plant population,

and this increases the possibility of an epidemic. Seed propagation is not

effective in inhibiting an epidemic in a single population, and plants can

85

avoid competition with pathogens through seed propagation, as seen in

previous studies [45, 47]. It follows that plants should adjust the balance

of the breeding propagation systems when the disease becomes epidemic

within a plant population.

In the mixed population, multiple plant types have different propor-

tions of vegetative propagation (α), and two cases, uninfected and in-

fected populations, were analyzed. In the uninfected population, plants

are not infected by pathogens, and competition among the different types

of plants (competition for breeding destinations) occurs. Consequently,

the plant type with lowest proportion of vegetative propagation becomes

dominant when the fecundity is low, and the proportion of vegetative

propagation in the dominant type changes to a higher value with an in-

crease in fecundity, even though seed propagation has an advantage over

vegetative propagation in homogeneous environments [Fig. 22]. Thus,

when plants are highly fecund, vegetative propagation becomes effective

with increasing spatial competition among different plant types (because

of a decrease in the open area). In the infected population, if plants have

high fecundity, then the optimal proportion of vegetative propagation

86

decreases slightly relative to the uninfected population, and it does not

depend on pathogen infectivity. Thus, in a plant population with high fe-

cundity, the pathogen infection has less of an effect on the optimal balance

than in a plant population with low. By contrast, when the fecundity is

high enough, the dominant type’s proportion of vegetative propagation is

lower than in the uninfected population. Thus, plants should increase the

proportion of seed propagation to escape from pathogens when systemic

pathogens invade a population.

In conclusion, seed propagation is an effective defensive behavior against

systemic pathogens in: (i) single populations: The plants increase their

population by increasing the proportion of seed propagation when the

epidemic pathogen is highly infective. However, the plants cannot reduce

the epidemic using this strategy; and in (ii) mixed populations: When

plants are not infected by the pathogen, and the fecundity is high, plants

increase their reproduction through vegetative propagation. When the

plants are infected by the pathogen, the high fecundity leads to a de-

crease in the effects of the pathogen infection on the optimal balance of

the breeding systems. However, if the fecundity is low, then increasing

87

the proportion of seed propagation is the optimal breeding strategy to

defend against the spread of a systemic pathogen.

88

Chapter 4 Conclusion

In this thesis, simple models, which considered spatial structures, were

constructed to express the relationship between plant reproduction and

pathogen propagation. Additionally, the effects of two characteristic

phenomena, superinfection by pathogens and seed propagation in clonal

plants, were studied. A simple case was analyzed to examine the basic re-

lationships between plants and pathogens, especially the plant fecundity

and pathogen infectivity. Then, the evolution of plants and pathogens

caused by the competition among the multiple types of individuals (of

pathogens in Chapter 2 and plants in Chapter 3) using computer simu-

lations were discussed.

In the superinfection model (in Chapter 2), the superinfection event is

an important factor in the evolution of virulence through the maintenance

of genetic diversity. A lower superinfection rate (but non-zero) leads to

an easing of the coexistence of multiple strains of pathogens. Addition-

ally, the plant reproductive dynamics is also an important factor in the

selection of virulence level. According to the previous studies (assuming

constant supply of the host), the strain of pathogen with intermediate

89

virulence becomes dominant (i.e. the virulence will evolve toward an

intermediate value). However, a high fecundity of the plant led to occu-

pation by the strain with the higher virulence level (or lower infectivity)

in the superinfection model. Thus, considering the both superinfection

and plant reproductive dynamics are necessary to examine the evolution

of pathogens more thoroughly.

In the seed propagation model (in Chapter 3), seed propagation is an

effective defensive behavior against systemic pathogens. Generally, when

the pathogen infectivity is low relative to plant fecundity, plants can es-

cape from infected individuals through the vegetative propagation, and

the feature is expressed in the seed propagation model. However, in analy-

sis of the model, the effect of pathogen abilities (infectivity and virulence)

on the optimal balance of breeding systems decreases with the increase in

the fecundity of pathogen. Thus, the adjustment of the breeding systems

has an important role to block the pathogen transmission when the plants

have low fecundity.

In summation, the intrinsic fecundity of the plant as well as superin-

fectin makes a major impact on the evolution of pathogen within plant

90

population and on the optimal balance of breeding systems in the clonal

plant. If the fecundity of plant is high enough, then the superinfection

event affects the size of plant population rather than the evolution of

the balance of breeding systems in infected clonal plants. The plants

can increase the size of their population when the parameter range of

plant fecundity remains in a place of coexistence of multiple strains of

pathogen, which increase due to lower superinfectin rate. On the con-

trary, the decrease in the size of plants population through the death

from disease is minimized with the increase in the fecundity, because the

pathogen population is occupied by strain with lowest infectivity through

the superinfection.

If the pathogens are not capable of superinfection, then the defensive

behavior through seed propagation is effective. When the pathogens with

high infectivity invade the plant population, plants increase the propor-

tion of seed propagation to escape from the pathogen infection, and then

it is expected that the strain with lower fecundity becomes dominant.

Because, the increase in seed propagation leads to the decrease in the

healthy individual from the neighborhood of the infected individual, and

91

it is expected that the increase in seed propagation yields similar result

to the decrease in the fecundity in the superinfection model. Thus, the

invading pathogen can not spread within the plant population (the fitness

of the pathogen decreases). Contrastingly, if the pathogens are capable

of superinfection, then the pathogens can spread widely within the plant

population in spite that plants escape by seed propagation. Because,

when the plant has low fecundity, the strain which has high infectivity

becomes dominant due to superinfection event. Thus, the superinfection

is an important ability for pathogens to increase in their fitness and plants

can not block the pathogen spread even though plants adjust the balance

of breeding systems due to the superinfection event.

Additionally, both the balance adjustment of breeding systems and

the selection of virulence level through superinfection have influence on

each other. Specifically, it is expected that the increase in proportion

of seed propagation leads to the ease of coexistence of multiple strain

of pathogen. Thus, the seed propagation can assist the maintenance

of genetic diversity of pathogen although it is effective defense behavior

against pathogen spread. Then, the optimal balance of breeding systems

92

might depend on the virulence of dominant strain or virulence and density

of coexisting strains. Therefore, the model including both superinfection

event and seed propagation leads to different result with respect to the

optimal balance of breeding system. Furthermore, the analysis of the

model is necessary to examine the effect of seed propagation in defensive

behavior against systemic pathogen minutely.

A comparison of the three methods, MA, PA and MCS, indicated that

the spatial structure impacts the dynamics of a plant population infected

by a systemic pathogen. In the analysis using the MA (neglecting the

spatial structure), the oscillatory solution (Chapter 2) and the effects of

the balance of the breeding systems on the plant population dynamics

(especially, equilibrium and threshold) (Chapter 3) were not observed,

unlike in the analyses using the PA or MCS. In the superinfection model,

the oscillatory solution indicates that the competition among plants and

pathogen strains occurs and that plants and pathogens do not main-

tain stable populations. Therefore, the competition leads to the further

evolution of the plants and pathogens to stabilize the population. Addi-

tionally, the spatial structure has an important role in the maintenance

93

of the pathogen’s genetic diversity, as indicated by the PA, resulting in a

wider pathogen range in the coexistence phase than the MA. Thus, the

pathogens’ dynamics within a plant population is greatly affected by the

plant’s spatial factors, such as the configuration of the ramets and the size

of the genets. In the seed propagation model, using the MA, there was no

dependence on the proportion of vegetative propagation because the dif-

ference between vegetative and seed propagation was the distance of the

breeding destination in the present model. Thus, the difference between

the breeding systems is nothing. In the escape strategy, the optimal bal-

ance of the breeding systems is profoundly affected by spatial structures

when the systemic pathogen invades the plant population. In particu-

lar, when the plant’s growth rate is slower than the pathogen’s infection

rate, then the influence of the spatial structures increases as indicated by

the increasing quantitative discrepancy between the PA and MCS. How-

ever, the PA can effectively analyze the qualitative characteristics of the

dynamics.

As is usual with CP analyses, the system examined here is very com-

plicated. The PA is very useful to approximately close and solve the

94

system. Fortunately, several analytical solutions for the equilibrium and

phase transition thresholds were obtained. However, the equilibrium and

thresholds were not analytically derived and approximations do not al-

ways work well, compared with numerical calculations, leading to dis-

crepancies. In the future, more details will have to be analyzed, such

as the bifurcation condition, the values of the equilibrium, and the sta-

bility of the equilibrium state. More complicated models will have to

be constructed and analyzed to express the dynamics of real plants and

pathogens. For instance, a model that integrates the superinfection and

seed propagation models, and realistic assumptions, such as those for the

mortality cost and the proportion of vegetative propagation, should be

configured. The mortality cost has a more complex relationship with the

infection rate and virulence level, and the vegetative proportion would be

adjusted depending on several factors (such as the distance from an in-

fected individual). Additionally, the parameter values, notably differences

in morality costs among strains (∆m), should be estimated by compar-

isons among the available quantitative data. However, a more complex

model increases the difficulty of analysis and the discrepancies from nu-

95

merical calculations. Developing the new and convenient approximation

method, such as a higher-order analysis, will be indispensable in the fu-

ture to examine complicated biological systems.

96

Acknowledgements

The author is deeply grateful to Takenori Takada for his practicable

comments and advice in the accomplishment of this study. The au-

thor gratly appreciates Takashi Kohyama, Toshihiko Hara, Masaharu Na-

gayama, Tomonori Sato, and Kazunori Sato for their variable suggestions.

The author thanks Ryo oizumi, Akiko Satake, Motohide Seki and Yuya

Tachiki for checking this study and making it better. The author thanks

our laboratory and Sato’s group members for support and encourage.

97

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108

Appendix

A. Simplification of the master equation

The set of Eq. (4) was simplified by the following process, and the

variables have the following properties from their definitions:

Pσiσj= Pσjσi

, (7)

∑σj∈S

Pσiσj= ρσi

(for any σi) , (8)

∑σi∈S

ρσi= 1, (9)

∑σj∈S

qσj/σi= 1 (for anyσi) . (10)

A differential equation for each variable was obtained from Eqs. (3) and

(8),

109

ρσi=∑σj∈S

Pσiσj, (11)

Pσiσj= ρσi

qσj/σi+ ρσi

qσj/σi,

qσj/σi=

Pσiσj− ρσi

qσj/σi

ρσi

. (12)

In addition, the following variables were replaced using the remaining

variables:(ρS, ρI, q0/S, q0/I, qS/I

)from Eqs. (3), (7), (9) and (10).

ρ0 = 1− ρS − ρI, q0/0 = 1−ρSq0/S

1− ρS − ρI

− qI/0,

qS/0 =ρSq0/S

1− ρS − ρI

, qS/S = 1− q0/S −ρIqS/IρS

,

qI/S =ρIqS/IρS

, qI/0 =ρIq0/I

(1− ρS − ρI),

qI/I = 1−(1− ρS − ρI) qI/0

ρI

− qS/I.

(13)

Thus, a set of equations with five variables was obtained from Eqs. (4),

(8) and (12), using the PA,

110

ρS = P0S + PSS + PIS

= ρS

(βSq0/S −mIqI/S

), (14)

ρI = P0I + PSI + PII

= ρI

(mIqS/I − 1

), (15)

˙qI/0 =PI0 − ρ0qI/0

ρ0

=ρI

ρ0

(qI/I − qI/0

)+

βSqI/0 + (z − 1)mIqI/Sz

qS/0 − qI/0, (16)

˙q0/S =P0S − ρSq0/S

ρS

= qI/S

(mIq0/Sz

+ 1)− βSq0/S

(q0/S +

1− (z − 1)(q0/0 − qS/0

)z

), (17)

˙qS/I =PSI − ρIqS/I

ρI

= qI/S

[(z − 1)mI

(qS/S − qI/S

)− 1

z−mIqS/I

]+

(z − 1) βSqS/0z

. (18)

Thus, the set of simplified equations was obtained by substituting (13)

in Eq. (14)-(18).

111

B. Analysis in superinfection model

B.1 Mean-field approximation

The Jacobian at Eqs. (5) is:

J =

−1− βSρS −βSρ0 − 1

(βS +mI) ρS βSρ0 +mIρS −mI (1− ρ0 − ρS)

.

B.1.1 Extinction region

In the case of the extinction region, the Jacobian is:

JM ≡ J(EM) =

−1 −βS − 1

0 βS

,

thus,

Tr(JM) = βS − 1, Det(JM) = −βS

The stability condition (Tr(JsiM) < 0 andDet(JM) > 0) is 0 > βS. There-

fore, this equilibrium is always unstable under the given assumptions..

B.1.2 Disease-free region

112

In the case of the disease-free region, the Jacobian is:

JM ≡ J(EM) =

−1− βS −1

(βS +mI) mI

,

thus,

Tr(JM) = mI − βS − 1, Det(JM) = βS (1−mI)

The stability condition is:

0 < βS, mI < 1.

Therefore, the stability condition at the equilibrium is mI < 1.

B.1.3 Epidemic region

In the case of the epidemic region, the Jacobian is:

JM ≡ J(EM) =

−βS+mI

βS−mI(1+βS)

βS+mI

βS+mI

βS1

,

thus,

113

Tr(JM) = − βS

mI

, Det(JM) =βS (mI − 1)

mI

The stability condition is:

0 < βS, 1 < mI

Therefore, the stability condition at the equilibrium is 0 < βS and mI > 1.

B.2 Pair approximation

To analyze the local stability at each equilibrium state, we used the

Routh–Hurwitz stability criterion. Let the characteristic polynomial of

the Jacobian of n degrees at equilibrium state be

a0λn + a1λ

n−1 + a2λn−2 + · · ·+ an−1λ+ an,

and the Hurwitz determinant be ∆n.

B.2.1 Extinction region

In the stability analysis of the extinction equilibrium, the same five

114

variables were selected as in Appendix.A. as were the simplified Eqs. (4).

ρS =ρS

(mIqI/S − βSq0/S

),

ρI =mIρSqI/S − ρI,

qI/0 =1

(1− ρS − ρI) z

(ρSq0/S

[βSqI0 + (z − 1)mIqI/S

]+[ρI

(1 + qI/0

)− ρSqI/S − 2 + (d− 2) ρS qI/0

]z),

q0/S =(1 +

mIq0/Sz

)qI/S

− βSq0/S

[q0/S +

1

z− z − 1

z

((1− ρS − ρI)

(1− qI/0

)− 2ρSq0/S

1− ρS − ρI

)],

qI/S =1

z

[βSq0/S

((z − 1) qI/0 − qI/Sz

)+mIqI/S

((z − 2)

(1− qI/S

)− (z − 1) q0/S

)]− qI/S.

(19)

115

The Jacobian of Eqs. (19) at EP (ρ∗0 = 0, ρ∗S = 0, q∗I/0 = 0) is:

JP ≡ J(EP),∣∣∣λI − JP

∣∣∣ = (λ+ 2) (λ+ 1)(λ+mIq

∗I/S − βSq

∗0/S

)∣∣∣∣∣∣∣∣∣λ+

2βS−mIq∗I/S

z+−βS(1− 2q∗0/S) −1−

mIq∗0/S

z

((βS+mI)z−mI)q∗I/S

zλ+ 1 + βS −

m[(z−2)(1−2q∗

I/S)−(z−1)q∗0/S

]z

∣∣∣∣∣∣∣∣∣ .(20)

Two remaining variables have the following equilibrium values:

q∗0/S =mI (z − 2)− z

mI (z − 1), q∗I/S =

βS (mI (z − 2)− z2) (mI (z − 2)− z)

m2I (z − 1) (z − 2) (mI + z)

.

Here,

mIq∗I/S − βSq

∗0/S

=− 2βSz (mI (z − 2)− z)

mIz (z − 2) (mI + z). (21)

From q∗0/S > 0, the Eq. (21) is negative, thus this equilibrium is always

unstable.

116

B.2.2 Disease-free region

In the stability analysis of the disease-free equilibrium, five other vari-

ables(ρS, ρI, q0/0, q0/S, andqI/S

)were selected, as were the simplified Eqs. (4).

ρS =βSρSq0/S −mIρIqS/I,

ρI =ρI

(mIqS/I − 1

),

q0/0 =− 1

z (1− ρS − ρI)

[(z − 2) βSρSq0/0q0/S

+z(2ρS

(1 + q0/Sq0/0

)− ρIq0/0 − 2

(1− ρI − q0/0

))],

q0/S =(1 +

mIq0/Sz

)qI/S − βSq0/S

[q0/S +

1

z− (z − 1)

z

(q0/0 −

ρSq0/S1− ρS − ρI

)],

qI/S =βSq0/Sz

[(z − 1)

(1− q0/0

)− zqI/S −

(z − 1) ρSq0/S1− ρS − ρI

]+

mIqI/Sz

[(z − 2)

(1− qI/S

)− (z − 1) q0/S

]− qI/S.

(22)

117

The Jacobian of Eqs. (22) at EP is:

JP ≡J(EP),∣∣∣λI − JP

∣∣∣ =(λ+ 2 +2 (z − 1) βS

z

)∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λ+ 1 0 0 −mI

0 λ −βS −1

(z−1)βS

z(z−1)βS

zλ+mI

(2− 1

z

)−1

(z−1)βS

z(z−1)βS

z(z−1)βS

zλ+ 1− (z−2)mI

z

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣. (23)

Thus, the coefficients of the characteristic polynomials and the Hurwitz

118

determinants are

a4 =2β2

S (z − 1) (z − (z − 1)mI)

z2,

a3 =βS

z2[z (3z − 2 + 3βS (z − 1))−mI

(z2 (a+ 3)− 3z (a+ 2) + 2 (a+ 1)

)],

a2 =z2 (1 + βS (βS + 5− 2mI)−mI)− z (βS (βS + 3− 5mI)− 2mI)− 2βSmI

z2,

a1 =2 (βS + 1)− βS + (z − 2)mI

z,

a0 =1,

∆2 =1

z3[z (z − 1) (2z − 1) β3

S + β2S

z (3 + z (9z − 10))− (z − 1) (2z − 1)2mI

+ βS

z2 (9z − 5) + (z − 2)2 m2

I − 2zmI (5 + z (4z − 11))

+zmI ((z − 2)mI − 2z) ((z − 2)mI − z)] ,

∆3 =βS

z5[2mI (βS + 1) + z (2 + βS (βS + 3)−mI (5βS + 6))− z2 (2βS + 3) (1 + βS −mI)

][β2

S (z − 1)2 ((z − 2)mI − 3z)− βS (z − 1)((z − 2)2 m2

I − 2zmI (2z − 5) + 5z2)

−z ((z − 2)mI − 2z) ((z − 2)mI − z)] .

Here, from a4 > 0,

mI <z

z − 1, (24)

In addition, we confirmed that all of the coefficients and determinants

119

are positive under these conditions. Thus, the stability condition of the

disease-free equilibrium is Eq. (24).

B.2.3 Epidemic region

In the stability analysis of the epidemic equilibrium, five other variables

(ρ0, ρS, q0/S, q0/I, qS/I

)were selected, as were the simplified Eqs. (4).

ρ0 =1− ρ0 − ρS

(1 + βSq0/S

), (25)

ρS =βSρSq0/S −mI (1− ρS − ρI) qS/I, (26)

q0/S =1

ρ0ρS

[mIρ0 (1− ρ0 − ρS) q0/SqS/I + z (1− ρ0 − ρS) ρ0qS/I

+βSρSq0/Sρ0

(z(1− q0/S

)+ (z − 1) q0/I − 2

)− (z − 1)

(2ρSq0/S + (1− ρS) q0/I

)],

(27)

q0/I =1 +

((z − 1)mIq0/S

ρS

− 1

)qS/I −

q0/I(zρ0

(1 +mIqS/I

)+ (z − 1) βSρSq0/S

)zρ0

,

(28)

qS/I =1

ρ0ρS

[(z − 1) βSρ

2Sq0/Sq0/I

+ρ0qS/ImI

(ρS

[(z − 2)

(1 + qS/I

)− (z − 1) q0/S

]− 2 (z − 1) (1− ρ0) qS/I

)].

(29)

From the above equations (Eqs. (25)-(29)), the following equilibrium

120

value was derived, and all of the other variables were derived by Eq. (13).

ρ∗S =z (mI − 1) + (z − 1)mI − 2zmIq

∗0/I[

z (mI − 1) + (z − 1)mIq∗0/S

](1 + βS)−

[(z − 1) βSq∗0/S + 2z

]mIq∗0/I

,

ρ∗I =βSρ∗S q

∗0/S,

q∗0/S =zmI

(1− q∗0/I

)− (z +mI)

(z − 1) βS

,

q∗S/I =1

mI

,

q∗0/I =1

2zβSmI (βS +mI)

(β2

S (mI − z) + βSm2I (z − 1)− zmI (3βS +mI)

+√

4zβS (βS +mI) [(z − 1)m3I + zmI (βSmI − βS −mI)]

+ (zmI (3βS +mI)− (z − 1) βSm2I − (mI − z) β2

S )2).

Here, the Jacobian at equilibrium was abbreviated as EP, because the

coefficients of the characteristics polynomials and the Hurwitz determi-

nants are too long to write in this paper. In addition, it is too difficult

to derive the stability condition analytically. Thus, the values of the co-

efficients and determinants were confirmed numerically. As a result, all

of the coefficients of ∆2 and ∆3 are always positive, and the sign of ∆4

varies depending on the parameter values. Thus, the Hopf bifurcation

121

occurs to exceed the parameter values at the threshold.

C. Analysis in seed propagation model

C.1 Mena-field approximaation

Using the MA, the set of Eq. (6) was rewritten as follows:

ρ0 = ρS (1−mSρ0) + ρI, (30)

ρS = ρS (mSρ0 −mIρI − 1) , (31)

ρI = ρI (mIρS − 1) . (32)

Thus, the following three equilibria were obtained:

E ≡ (ρ∗0 , ρ∗S , ρ

∗I ) ,

E1 = (1, 0, 0) : extinction,

E2 =(

1mS

, 1− 1mS

, 0)

: disease-free,

E3 =(

mI

mS+mI, 1mI, mSmI−mS−mI

mI(mS+mI)

): epidemic.

These equilibria do not depend on the proportion of vegetative propa-

gation, α. Thus, the effects of seed propagation on the system were not

122

examined using the MA.

C.2 Extinction phase in pair approximation

In this section, The set of Eqs. (6) were simplified and rewritten using

qσ/σ′ and their definitions (Appendix. A.), as follows:

ρS =ρS

(αmSq0/S +mS (1− α) (1− ρS − ρI)−mIqI/S − 1

), (33)

ρI =mIρSqI/S − ρI, (34)

˙q0/0 =αmSρSq0/0 + z(q0/0 (mSρS − 1) + 2

)−

(z − 2)αmSρSq0/0q0/S1− ρS − ρI

, (35)

˙q0/S =1 +mIq0/SqI/S

z− q0/S −mS (1− α)

[(1− ρI) q0/S − (1− ρS − ρI) q0/0

]− αmSq0/siS

[1

z

1− (z − 1)

(q0/0 −

ρSq0/S1− ρS − ρI

)], (36)

˙qI/S =mS (1− α)[(1− ρS − ρI)

(1− q0/0 − qI/S

)− ρSq0/S

]+

1

z

[αmSq0/S

(z − 1)

(1− q0/0 −

ρSq0/S1− ρS − ρI

)− zqI/S

]+

1

z

[mIqI/S

(z − 2)

(1− qI/S

)− (z − 1) q0/S

]. (37)

From Eq.(33), the following two equilibria of ρS were obtained:

ρ∗S = 0,mS

[(1− α) (1− ρ∗I ) + αq∗0/S

]−mIq

∗I/S − 1

(1− α)mS

.

123

The value ρ∗S = 0 was chosen to analyze the extinction equilibrium. Then,

the positive equilibria of the other variables from Eqs.(34)-(37) were de-

rived.

E ≡(ρ∗S , ρ

∗I , q

∗0/0, q

∗0/S, q

∗I/S

)=

0, 0, 0,2αmS (z − 1)− z (mS + 1) +

√(2αmS + z (mS + 1))2 − 8zα2m2

S

2zαmS

, 0

(38)

To analyze the local stability, the eigenvalues of the Jacobian were

124

obtained at the equilibrium. The Jacobian J is:

J =

mS

(1− α

(1− q∗0/S

))− 1 0 0

0 −1 0

−mS

((z−2)αq∗0/S

z+ (1− α)

)− 1 −1 −2

−mS

((z−1)α

(q∗0/S

)2

z+ (1− α)

)−mS (1− α)

(1− q∗0/S

)−mS

((z−1)αq∗0/S

z− (1− α)

)−

mSq∗0/S

(z(1−α)+(z−1)αq∗0/S

)z

0 −mS

((z−1)αq∗0/S

z+ (1− α)

)0 0

0 0

0 0

−mS

((z−1)αq∗0/S

z− (1− α)

)mIq

∗0/S

z

0 −1−mS

(1− α

(1− q∗0/S

))+

mI

((z−2)−(z−1)q∗0/S

)z

.

125

The eigenvalues of the Jacobian are:

λ1 = −1,

λ2 = −2,

λ3 = −mS

(1− α

(1− q∗0/S

))− 1, (39)

λ4 = −mS

((z − 1)αq∗0/S

z− (1− α)

), (40)

λ5 = −1−mS

(1− α

(1− q∗0/S

))+

mI

((z − 2)− (z − 1) q∗0/S

)z

. (41)

Substituting q∗0/S in Eq. (38) into Eqs. (39) and (40),

λ3 =z (mS − 3)− 2αmS +

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

z,

λ4 = −

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

z.

The condition of λ3 < 0 is:

mS <− (z − α) +

√(z + α)2 − 4zα2

2α (1− α):= f(α). (42)

The right-hand side of the inequality was set to f(α). The λ4 is negative

because q0/S is a real number.

126

Then, it was shown that the Eq. (41) is negative. The second term on

the right-hand side of Eq. (41) is negative because mS > 0 and α, q∗0/S ∈

[0, 1]. Thus, if the third term is negative, then Eq. (41) is negative. When

the equilibrium q∗0/S is greater than (z − 2)/(z − 1), Eq. (41) is negative.

The value of the equilibrium decreases with an increase in mS and α from

following analysis:

∂q∗0/S∂mS

= −z(mS + 1) + 2αmS −

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

2αm2S

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

< 0,

(43)

∂q∗0/S∂α

= −(mS + 1)

(2αmS + z (mS + 1)−

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

)2α2mS

√(z (mS + 1) + 2αmS)

2 − 8zα2m2S

< 0.

(44)

Additionally, the f(α) reaches its maximum at α = 1 because the f(α)

is a monotonically increasing function of α. Therefore, the equilibrium is

at its minimum value at α = 1, and the maximum value of mS is within

the parameter range of Eq. (42).

127

Let f(α) = g (α) /h (α),

limα→1

g (α) = limα→1

− (z − α) +

√(z + α)2 − 4zα2 = 0,

limα→1

h (α) = limα→1

2α (1− α) = 0.

Using l’Hopital’s rule,

limα→1

g′(α)

h′(α)= lim

α→1

1 + 2α+2z−8αz

2√

(z+α)2−4α2z

2(1− α)− 2α=

z

z − 1.

Thus,the Eq. (42) at α = 1 is

mS < limα→1

f(α) =z

z − 1,

Therefore, the q∗0/S under the parameter range is

q∗0/S >z − 1

z

(>

z − 2

z − 1

)

Therefore, Eq. (41) is always negative.

In conclusion, the extinction equilibrium became stable under the con-

ditions of Eq. (42).

128