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Population Ecology Attributes of Populations Distributions

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Population Ecology Attributes of Populations Distributions III. Population Growth – change in size through time Calculating Growth Rates 1. Discrete Growth. With discrete growth, N (t+1) = N (t) λ Or, N t = N o λ t. Population Ecology Attributes of Populations Distributions

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Population Ecology

• Attributes of Populations
• Distributions
• III. Population Growth – change in size through time
• Calculating Growth Rates
• 1. Discrete Growth

With discrete growth, N(t+1) = N(t)λ

Or, Nt = Noλt

Population Ecology

• Attributes of Populations
• Distributions
• III. Population Growth – change in size through time
• Calculating Growth Rates
• 2. Exponential Growth – continuous reproduction

With discrete growth:

N(t+1) = N(t)λ or

Nt = Noλt

Continuous growth:

Nt = Noert

Where r = intrinsic rate of growth

(per capita and instantaneous)

and e = base of natural logs (2.72)

So, λ = er

• If λ = 1, then r = 0 and the population size will not change.
• If λ >1, then r > 0 and the population will increase.
• Population Ecology
• Attributes of Populations
• Distributions
• III. Population Growth – change in size through time
• Calculating Growth Rates
• 3. Equivalency

The rate of population growth is measured as:

The derivative of the growth equation: Nt = Noert

dN/dt = rNo

• Population Ecology
• Attributes of Populations
• Distributions
• III. Population Growth – change in size through time
• Calculating Growth Rates
• 3. Equivalency

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

- static: look at one point in time and survival for one time period

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

Why λ ? (discrete breeding season and discrete time intervals)

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

- dynamic (or “cohort”) – follow a group of individuals through their life

Song Sparrows Mandarte Isl., B.C. (1988)

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Number reaching each birthday are subsequent values of nx

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Survivorship (lx): proportion of population surviving to age x.

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Survivorship (lx): proportion of population surviving to age x.

Mortality: dx = # dying during interval x to x+1.

Mortality rate: mx = proportion of individuals age x that die during interval x to x+1.

Survivorship Curves:

Describe age-specific probabilities of survival, as a consequence of age-specific mortality risks.

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Survivorship (lx): proportion of population surviving to age x.

Number alive DURING age class x: Lm = (nx + (nx+1))/2

Age classes (x): x = 0, x = 1, etc.

Initial size of the population: nx, at x = 0.

Survivorship (lx): proportion of population surviving to age x.

Number alive DURING age class x: Lm = (nx + (nx+1))/2

Expected lifespan at age x = ex

- T = Sum of Lm\'s for age classes = , > than age (for 3, T = 9)

- ex = T/nx (number of individuals in the age class) ( = 9/12 = 0.75)

- ex = the number of additional age classes an individual can expect to live.

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Tables

2. Age Class Distributions

III. Population Growth – change in size through time

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Tables

2. Age Class Distributions

When these rates equilibrate, all age classes are growing at the same single rate – the intrinsic rate of increase of the population (rm)

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential

Net Reproductive Rate = Σ(lxbx) = 2.1

Number of daughters/female during lifetime.

If it is >1 (“replacement”), then the population has the potential to increase multiplicatively (exponentially).

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential

Generation Time – T = Σ(xlxbx)/ Σ(lxbx) = 1.95

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential

Intrinsic rate of increase:

rm (estimated) = ln(Ro)/T = 0.38

Pop growth dependent on reproductive rate (Ro) and age of reproduction (T).

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential

Intrinsic rate of increase:

rm (estimated) = ln(Ro)/T = 0.38

Pop growth dependent on:

reproductive rate (Ro) and age of reproduction (T).

Doubling time = t2 = ln(2)/r = 0.69/0.38 = 1.82 yrs

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential

Northern Elephant Seals: <100 in 1900  150,000 in 2000. r = 0.073, λ = 1.067

Malthus: All populations have the capacity to expand exponentially

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux

R = 10

T = 1

r = 2.303

Net Reproductive Rate = Σ(lxbx) = 10

Generation Time – T = Σ(xlxbx)/ Σ(lxbx) = 1.0

rm (estimated) = ln(Ro)/T = 2.303

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• - increase fecundity, increase growth rate (obvious)

R = 10

T = 1

r = 2.303

R = 11

T = 1

r = 2.398

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• - increase fecundity, increase growth rate (obvious)
• - decrease generation time (reproduce earlier) – increase growth rate

R =11

T = 1

r = 2.398

R = 12

T = 0.833

r = 2.983

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• - increase fecundity, increase growth rate (obvious)
• - decrease generation time (reproduce earlier) – increase growth rate
• - increasing survivorship – DECREASE GROWTH RATE (lengthen T)

R = 11

T = 1

r = 2.398

R = 20

T = 1.5

r = 2.00

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• - increase fecundity, increase growth rate (obvious)
• - decrease generation time (reproduce earlier) – increase growth rate
• - increasing survivorship – DECREASE GROWTH RATE (lengthen T)
• - survivorship adaptive IF:
• - necessary to reproduce at all
• - by storing E, reproduce disproportionately in the future

R = 230

T = 2.30

r = 2.36

Original r = 2.303

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• E. Limits on Growth: Density Dependence

Robert Malthus

1766-1834

Premise:

- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...

r > 0

DEATH

BIRTH

RATE

r < 0

DENSITY

Premise:

- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...

r > 0

RATE

r < 0

DEATH

BIRTH

DENSITY

Premise:

- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...

r > 0

r < 0

DEATH

RATE

BIRTH

DENSITY

As density increases, successful reproduction declines

And juvenile suvivorship declines (mortality increases)

Lots of little plants begin to grow and compete. This kills off most of the plants, and only a few large plants survive.

Premise:

Result:

There is a density at which r = 0 and DN/dt = 0.

THIS IS AN EQUILIBRIUM....

r = 0

DEATH

RATE

BIRTH

DENSITY

K

Premise

• 2. Result
• 3. The Logistic Growth Equation:
• Exponential:
• dN/dt = rN

N

t

Premise

• 2. Result
• 3. The Logistic Growth Equation:
• Exponential: Logistic:
• dN/dt = rNdN/dt = rN [(K-N)/K]

K

N

N

t

t

Premise

• 2. Result
• 3. The Logistic Growth Equation (Pearl-Verhulst Equation, 1844-45):
• Logistic:
• dN/dt = rN [(K-N)/K]

When a population is very small (N~0), the logistic term ((K-N)/K) approaches K/K (=1) and growth rate approaches the exponential maximum (dN/dt = rN).

K

N

t

Premise

• 2. Result
• 3. The Logistic Growth Equation:
• Logistic:
• dN/dt = rN [(K-N)/K]

As N approaches K, K-N approaches 0; so that the term ((K-N)/K) approaches 0 and dN/dt approaches 0 (no growth).

K

N

t

Premise

• 2. Result
• 3. The Logistic Growth Equation:
• Logistic:
• dN/dt = rN [(K-N)/K]

Should N increase beyond K, K-N becomes negative, as does dN/dt (the population will decline in size).

K

N

t

Premise

• 2. Result
• 3. The Logistic Growth Equation:
• Logistic:
• dN/dt = rN [(K-N)/K] [(N-m)/K]

Minimum viable population size add (N-m)/N

K

N

m

t

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• E. Limits on Growth: Density Dependence
• F. Temporal Dynamics

F. Temporal Dynamics

- DISCRETE GROWTH

ROUGHLY, growth per generation is: log(Nt+1) - log(Nt) = R[log(K) - log(Nt)]

SO:

when R < 1, the population will grow only a fraction of the difference between K and Nt. Asymptotic approach to K.

K

N

Nt

t

F. Temporal Dynamics

- DISCRETE GROWTH

ROUGHLY, growth per generation is: log(Nt+1) - log(Nt) = R[log(K) - log(Nt)]

SO:

If R = 1, then the population reaches K exactly.

K

N

Nt

t

F. Temporal Dynamics

- DISCRETE GROWTH

ROUGHLY, growth per generation is: log(Nt+1) - log(Nt) = R[log(K) - log(Nt)]

SO:

If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation.

K

N

Nt

t

SO:

If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation.

K

N

Nt

t

SO:

If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation.

K

N

Nt

t

F. Temporal Dynamics

- DISCRETE GROWTH

ROUGHLY, growth per generation is: log(Nt+1) - log(Nt) = R[log(K) - log(Nt)]

SO:

when 2 < R < 2.5, oscillations increase each time interval; divergent oscillation initially.

But at low N, the linearity breaks down and this equation is not descriptive. End up with a stable limit cycle.Over 2.5? Chaotic.

K

N

Nt

t

F. Temporal Dynamics

- DISCRETE GROWTH

F. Temporal Dynamics

- CONTINUOUS GROWTH

Breed here, and may be out of balance with resources

Lags occur because of developmental time between acquisition of resources FOR breeding and the event of breeding itself.

K

N

Nt

t

F. Temporal Dynamics

- CONTINUOUS GROWTH

Population cycles are a function of R and the lag time (t).

Either high R or long lags increase the amplitude

<0.37 < 1.6 > 1.6

F. Temporal Dynamics

- CONTINUOUS GROWTH

Growth curves are not an intrinsic characteristic of a species – they can change with environmental conditions. Daphnia

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• E. Limits on Growth: Density Dependence
• F. Temporal Dynamics
• G. Spatial Dynamics and Metapopulations

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• E. Limits on Growth: Density Dependence
• F. Temporal Dynamics
• G. Spatial Dynamics and Metapopulations

Dealing with small populations with increase chance of stochastic extinction (e).

The survival of the metapopulation is dependent on a rate of migration that can ‘rescue’ extinct at a compensatory rate.

III. Population Growth – change in size through time

• Calculating Growth Rates
• B. The Effects of Age Structure
• C. Growth Potential
• D. Life History Redux
• E. Limits on Growth: Density Dependence
• F. Temporal Dynamics
• G. Spatial Dynamics and Metapopulations

p = proportion of suitable habitats occupied

e = extinction rate

ep = prop of habitats being vacated

Rate of colonization depends on fraction of patches that are empty (1-p), and number of occupied patches dispersing colonists.

c = migration rate, and so rate of colonization = cp(1-p)

Change in patch occupancy:

dp/dt = cp(1-p) - ep

Peq = 1 – e/c

Dealing with small populations with increase chance of stochastic extinction (e).

The survival of the metapopulation is dependent on a rate of migration that can ‘rescue’ extinct at a compensatory rate.

If e > c, pop goes extinct.

G. Spatial Dynamics and Metapopulations

What influences e and c?

Extinction probability is strongly influenced by population size.

G. Spatial Dynamics and Metapopulations

What influences e and c?

Degree of patch isolation affects colonization probability

Extinction probability is strongly influenced by population size.

Low e/c

??

??

High e/c

Closer patches mean higher c