overview of population growth: discretecontinuous density independent density dependent geometric...

Overview ofpopulation growth:

discrete continuous

densityindependent

densitydependent

Geometric Exponential

DiscreteLogistic

LogisticNew Concepts:

- Stability- DI (non-regulating)

vs. DD (regulating) growth

- equilibrium

Variability in growth

(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability

How do populations grow – a derivation of geometric growth

N1 = N0 + rN0

Growth rate (r) = birth rate – death rate

N0 = initial population density (time = 0)

N1 = population density 1 year later (time =1)

(express as per individual)

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

Growth rate (r) = birth rate – death rate

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Growth rate (r) = birth rate – death rate

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Can we rewrite N2 in terms of N0 ???

Growth rate (r) = birth rate – death rate

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Growth rate (r) = birth rate – death rate

substitute

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Growth rate (r) = birth rate – death rate

N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2

substitute

rewrite:

How do populations grow?

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Growth rate (r) = birth rate – death rate

N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2

substitute

or

Nt = N0 (1 + r)t}

= , finite rate of increase

Discrete (geometric) growth

N

time

Nt = N0t

12

3

4

5

= finite rate of increase

Continuous (exponential) growth

N

time

12

3

4

5

Nt = N0ert

r = intrinsic growth rate

Continuous (exponential) growth

N

time

12

3

4

5

1 dNN dt

= rdN dt

= rN;

populationgrowth rate

per capitagrowth rate

N Per capita growth is constant and independent of N

dN dt

Read as change in N (density) over change in time.

1 dNN dt

1 dNN dt

= r

Y = b + mX

Discrete Continuous

Nt = N0t Nt = N0ert

> 1 r > 0 < 1 r < 0

Increasing:Decreasing:

Where: = er r = ln

Every time-step (e.g., generation)Time lag:

None Compounded instantaneously

Applications: Populations w/ discrete breeding season

No breeding season - at any time there are individuals in all stages

of reproduction

Examples: Most temperate vertebrates and plants Humans, bacteria, protozoa

Mathematics: Often intractable;simulations Mathematically convenient

Comparison

Geometric (or close to it)growth in wildebeest populationof the Serengeti following Rinderpest inoculation

Exponential growth in the total human population

Simplest expression of population growth: 1 parameter, e.g., r = intrinsic growth rate

Population grows geometrically/exponentially, but the Per capita growth rate is constant

First Law of Ecology: All populations possessthe capacity to grow exponentially

The Take Home Message:

Exponential/geometric growth is a model to which we build on

Overview ofpopulation growth:

discrete continuous

densityindependent

densitydependent

Geometric Exponential

DiscreteLogistic

LogisticNew Concepts:

- Stability- DI (non-regulating)

vs. DD (regulating) growth

- equilibrium

Variability in growth

(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability

XX

Variability in space In time

No

mig

rati

onm

igra

tion

Variability in space In time

No

mig

rati

onm

igra

tion

Source-sink structure

Variability in space In time

No

mig

rati

onm

igra

tion (arithmetic)

Source-sink structurewith the rescue effect

Source-sink structure

Variability in space In time

No

mig

rati

onm

igra

tion

Source-sink structure

(geometric)

G < A G declines with increasing variance

(arithmetic)

Source-sink structurewith the rescue effect

Variability in space In time

No

mig

rati

onm

igra

tion (arithmetic)

Source-sink structurewith the rescue effect

(geometric)

G < A G declines with increasing variance

Temporal variability reduces population growth rates

Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals

Source-sink structure

(arith & geom)Increase the number of subpopulations increases the growth rate (to a point),and slows the time to extinction