Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in...

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

Transcript of Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in...

Population EcologyI.Attributes of PopulationsII.DistributionsIII. Population Growth – change in size through time

A. Calculating Growth Rates1. Discrete Growth

With discrete growth, N(t+1) = N(t)λOr, Nt = Noλt

Population EcologyI.Attributes of PopulationsII.DistributionsIII. Population Growth – change in size through time

A. Calculating Growth Rates2. Exponential Growth – continuous reproduction

With discrete growth:N(t+1) = N(t)λ orNt = 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

Population EcologyI.Attributes of PopulationsII.DistributionsIII. Population Growth – change in size through time

A. Calculating Growth Rates3. Equivalency

• If λ is between zero and 1, the r < 0 and the population will decline.

• 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 EcologyI.Attributes of PopulationsII.DistributionsIII. Population Growth – change in size through time

A. Calculating Growth Rates3. Equivalency

The rate of population growth is measured as:

The derivative of the growth equation: Nt = Noert

dN/dt = rNo

III. Population Growth – change in size through timeA. Calculating Growth RatesB. 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 timeA. Calculating Growth RatesB. The Effects of Age Structure

1. Life Table

III. Population Growth – change in size through timeA. Calculating Growth RatesB. The Effects of Age Structure

1. Life Table

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

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age Structure1. Life Table - dynamic (or “cohort”) – follow a group of individuals through

their life

x nx lx dx qx Lm ex

0 115 1.00 90 0.78 70.0 1.01

1 25 0.22 6 0.24 22.0 1.86

2 19 0.17 7 0.37 15.5 1.29

3 12 0.10 10 0.83 7.0 0.75

4 2 0.02 1 0.50 1.5 1.00

5 1 0.01 1 1.00 0.5 0.50

6 0 0 - - - -

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

Age classes (x): x = 0, x = 1, etc. Initial size of the population: nx, at x = 0.

x nx lx dx qx Lm ex

0 115

1

2

3

4

5

6

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

x nx lx dx qx Lm ex

0 115

1 25

2 19

3 12

4 2

5 1

6 0

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.

x nx lx dx qx Lm ex

0 115 1.00

1 25 0.22

2 19 0.17

3 12 0.10

4 2 0.02

5 1 0.01

6 0 0

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.

x nx lx dx mx Lm ex

0 115 1.00 90 0.78

1 25 0.22 6 0.24

2 19 0.17 7 0.37

3 12 0.10 10 0.83

4 2 0.02 1 0.50

5 1 0.01 1 1.00

6 0 0 - -

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

x nx lx dx qx Lm ex

0 115 1.00 90 0.78 70.0

1 25 0.22 6 0.24 22.0

2 19 0.17 7 0.37 15.5

3 12 0.10 10 0.83 7.0

4 2 0.02 1 0.50 1.5

5 1 0.01 1 1.00 0.5

6 0 0 - - -

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.

x nx lx dx qx Lm ex

0 115 1.00 90 0.78 70.0 1.01

1 25 0.22 6 0.24 22.0 1.86

2 19 0.17 7 0.37 15.5 1.29

3 12 0.10 10 0.83 7.0 0.75

4 2 0.02 1 0.50 1.5 1.00

5 1 0.01 1 1.00 0.5 0.50

6 0 0 - - - -

III. Population Growth – change in size through timeA. Calculating Growth RatesB. The Effects of Age Structure

1. Life Tables2. Age Class Distributions

III. Population Growth – change in size through timeA. Calculating Growth RatesB. The Effects of Age Structure

1. Life Tables2. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth Potential

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

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 20 10 10

2 0 - - -

R = 10 T = 1 r = 2.303

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth PotentialD. Life History Redux

Net Reproductive Rate = Σ(lxbx) = 10Generation Time – T = Σ(xlxbx)/ Σ(lxbx) = 1.0rm (estimated) = ln(Ro)/T = 2.303

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 20 10 10

2 0 - - -

R = 10 T = 1 r = 2.303

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 22 11 11

2 0 - - -

R = 11 T = 1 r = 2.398

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth PotentialD. Life History Redux

- increase fecundity, increase growth rate (obvious)

x lx bx lxbx xlxbx

0 1.0 2 2 0

1 0.5 20 10 10

2 0 - - -

R = 12 T = 0.833 r = 2.983

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 22 11 11

2 0 - - -

R =11T = 1 r = 2.398

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth PotentialD. Life History Redux

- increase fecundity, increase growth rate (obvious) - decrease generation time (reproduce earlier) – increase growth rate

R = 20 T = 1.5 r = 2.00

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 22 11 11

2 0 - - -

R = 11T = 1 r = 2.398

x lx bx lxbx xlxbx

0 1.0 0 0 0

1 0.5 20 10 10

2 0.5 20 10 20

3 0 - - -

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth PotentialD. 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 = 230 T = 2.30 r = 2.36

Original r = 2.303

x lx bx lxbx xlxbx

0 1.0  0 0 0

1 0.9 0 0 0

2 0.8 200 160 320

3 0.7 100 70 210

4 0 - - -

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth PotentialD. 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

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth Potential

D. Life History Redux E. Limits on Growth: Density Dependence

Robert Malthus1766-1834

Premise: - as population density increases, resources become limiting

and cause an increase in mortality rate, a decrease in birth rate, or both...

DENSITY

RATE

BIRTH

DEATH

r > 0

r < 0

Premise: - as population density increases, resources become limiting

and cause an increase in mortality rate, a decrease in birth rate, or both...

DENSITY

RATE

BIRTH

DEATH

r > 0

r < 0

Premise: - as population density increases, resources become limiting

and cause an increase in mortality rate, a decrease in birth rate, or both...

DENSITY

RATE

BIRTH

DEATH

r > 0 r < 0

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....

DENSITY

RATE

BIRTH

DEATH

r = 0

K

1. Premise 2. Result3. The Logistic Growth Equation:

Exponential:

dN/dt = rN

N

t

1. Premise 2. Result3. The Logistic Growth Equation:

Exponential: Logistic:

dN/dt = rN dN/dt = rN [(K-N)/K]

N

t

N

t

K

N

t

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).

1. Premise 2. Result3. The Logistic Growth Equation (Pearl-Verhulst Equation, 1844-45):

Logistic:

dN/dt = rN [(K-N)/K]

N

t

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).

1. Premise 2. Result3. The Logistic Growth Equation:

Logistic:

dN/dt = rN [(K-N)/K]

N

t

K

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

1. Premise 2. Result3. The Logistic Growth Equation:

Logistic:

dN/dt = rN [(K-N)/K]

N

t

K

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

m

1. Premise 2. Result3. The Logistic Growth Equation:

Logistic:

dN/dt = rN [(K-N)/K] [(N-m)/K]

III. Population Growth – change in size through timeA. Calculating Growth Rates

B. The Effects of Age StructureC. Growth Potential

D. Life History Redux E. Limits on Growth: Density Dependence F. Temporal Dynamics

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

N

t

K

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

Nt

F. Temporal Dynamics - DISCRETE GROWTH

N

t

K

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

Nt

F. Temporal Dynamics - DISCRETE GROWTH

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

N

t

K

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

Nt

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

F. Temporal Dynamics - DISCRETE GROWTH

N

t

K

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

Nt

N

t

K

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

Nt

N

t

K

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.

Nt

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

F. Temporal Dynamics - DISCRETE GROWTH

F. Temporal Dynamics - DISCRETE GROWTH

N

t

K

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

Nt

Breed here, and may be out of balance with resources

F. Temporal Dynamics - CONTINUOUS GROWTH

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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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 timeA. Calculating Growth Rates

B. The Effects of Age StructureC. 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.

If e > c, pop goes extinct.

p = proportion of suitable habitats occupiede = 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

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?

Extinction probability is strongly influenced by population size.

Degree of patch isolation affects colonization probability

Closer patches mean higher c

Low e/c

High e/c

??

??

Why is this important?