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Population Ecology: Population Dynamics

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Global human population

United Nations projections (2004)

(red, orange, green)

U. S. Census Bureau modern (blue)

& historical (black) estimates

The demographic processes that can change population size:Birth, Immigration, Death, Emigration

B. I. D. E. (numbers of individuals in each category)

Population Dynamics

Nt+1 = Nt + B + I – D – E

For an open population, observed at discrete time steps:

For a closed population, observed through continuous time:

dN

dt = (b-d)N

dN = rN

(b-d) can be considered a proxy for average per capita fitness

dt

Population Dynamics

5 main categories of population growth trajectories:

Exponential growthLogistic growth

Population fluctuationsRegular population cycles

Chaos

Population Dynamics

Cain, Bowman & Hacker (2014), Fig. 11.5

Invariant density-dependent vital rates

Stable equilibrium carrying capacity

Deterministic logistic growth

r

dN

dt = rN

N

K 1 –

Population Dynamics


Deterministic vs. stochastic logistic growth

Invariant density-dependent vital rates

“Fuzzy” density-dependent vital rates

Stable equilibrium carrying capacity

Fluctuating abundance within a range of values for carrying

capacity

r ri

Population Dynamics


dN

dt = rN

N(t-)

K 1 –

Instead of growth tracking current population size (as in logistic), growth

tracks density at units back in time

Time lags can cause delayed density dependence,

which can result in population cyclesIf r is small,

logistic

If r is intermediate,damped oscillations

If r is large,stable limit cycle

Sir Robert May, Baron of Oxford

Population Dynamics

Time lags can cause delayed density dependence, which can result in population cycles or chaos

Photo from http://www.topbritishinnovations.org/PastInnovations/BiologicalChaos.aspx

Population Dynamics

Per capita rate of increase

Pop

ulat

ion

size

(s

cale

d to

max

. si

ze a

tta

inab

le)

Population cycles & chaos

Is the long-term expected per capita growth rate (r) of a population simply an average across years?

At t0, N0=100t1 is a bad year, so N1 = N0 + (rbad* N0) = 50t2 is a good year, so N2 = N1 + (rgood*N1) = 75

Consider this hypothetical example:rgood = 0.5; rbad = -0.5

If the numbers of good & bad years are equal, is the following true?rexpected = [rgood + rbad] / 2

Variation in r and population growth

The expected long-term r is clearly not 0 (the arithmetic mean of rgood & rbad)!

Variation in and population growth

Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258

Nt+1 = Nt

=Nt

Nt+1

1.21

0.87

1.17

1.02

1.13

Arithmetic mean = 1.02

Geometric mean = 1.01

A fluctuating population



Nt+1 = Nt

=Nt

Nt+1

1.02

1.02

1.02

1.02

1.02



A steadily growingpopulation

1.02

1.02

1040

1061

1082

1104

1126

1020

1000

1148



Nt+1 = Nt

=Nt

Nt+1

1.01

1.01

1.01

1.01

1.01



A steadily growingpopulation

1.01

1.01

1020

1030

1040

1051

1061

1010

1000

1072

Which mean (arithmetic or geometric) best

captures the trajectory of the fluctuating population (the example given in the

textbook)?

Deterministic r < 0

Genetic stochasticity & inbreeding

Small populations are especially prone to extinction from both deterministic and stochastic causes

Population Size & Extinction Risk

Demographic stochasticity individual variability around r (e.g., variance at any given time)

Environmental stochasticity temporal fluctuations of r (e.g., change in mean with time)

Catastrophes

Each student is a sexually reproducing, hermaphroditic, out-crossing annual plant. Arrange the plants into small

sub-populations (2-3 plants/pop.).

In the first growing season (generation), each plant mates (if there is at least 1 other individual in the population)

and produces 2 offspring.

Offspring have a 50% chance of surviving to the next season. flip a coin for each offspring; “head” = lives, “tail” = dies.

Note that average r = 0; each parent adds 2 births to the population and on average subtracts 2 deaths [self & 1 offspring – since 50% of offspring live and

50% die] prior to the next generation.

Demographic stochasticity


Environmental stochasticity


How could the previous exercise be modified to illustrate environmental stochasticity?

Natural catastrophes


What are the likely consequences to populations of sizes: 10; 100; 1000; 1,000,000

if 90% of individuals die in a flood?

Density (N)K

Zone of Allee Effects

Birth (b)

Death (d)

Rate

?

?


Allee Effects occur when average per capita fitness declines as a population becomes smaller

Spatially-Structured Populations

Patchy population(High rates of inter-patch dispersal, i.e., patches are well-connected)


Mainland-island model(Unidirectional dispersal from mainland to islands)


Classic Levins-type metapopulation (collection of populations) model(Vacant patches are re-colonized from occupied patches

at low to intermediate rates of dispersal )

Original metapopulation idea from Levins (1969)

occupied

occupied

occupied

occupied

unoccupied

unoccupied

Assumptions of the basic model:

1. Infinite number of identical habitat patches

2. Patches have identical colonization probabilities

(spatial arrangement is irrelevant)

3. Patches have identical local extinction (extirpation)

probabilities

4. A colonized patch reaches K instantaneously (within-patch

population dynamics are ignored)


Classic Levins-type metapopulation (collection of populations) model(Vacant patches are re-colonized from occupied patches

at low to intermediate rates of dispersal )

Original metapopulation idea from Levins (1969)

occupied

occupied

occupied

occupied

unoccupied

unoccupied

dp

dt = cp(1 - p) - ep

c = patch colonization rate

e = patch extinction rate

p = proportion of patches occupied

Key result:metapopulation persistence

requires (e/c)<1

Habitats vary in habitat quality;occupied sink habitats broaden the realized niche

Source-Sink Population Dynamics

Original source-sink idea from Pulliam (1988)

source

source

source

sink

sink

sink

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