1 / 59

# Population Ecology Attributes of Populations Distributions - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Population Ecology Attributes of Populations Distributions' - wright

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

• 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 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 Ecology

• Attributes of Populations

• Distributions

• III. Population Growth – change in size through time

• Calculating Growth Rates

• 3. Equivalency

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

• 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

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Table

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

• 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)

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

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

Number reaching each birthday are subsequent values of nx

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

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

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.

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

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

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.

• Calculating Growth Rates

B. The Effects of Age Structure

1. Life Tables

2. Age Class Distributions

• 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)

Net Reproductive Rate = Σ(lxbx) = 2.1

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

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

Intrinsic rate of increase:

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

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

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

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

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

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)

• - 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

Robert Malthus

1766-1834

- 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

- 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

- 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

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: off most of the plants, and only a few large plants survive.

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 off most of the plants, and only a few large plants survive.

• 2. Result

• 3. The Logistic Growth Equation:

• Exponential:

• dN/dt = rN

N

t

• Premise off most of the plants, and only a few large plants survive.

• 2. Result

• 3. The Logistic Growth Equation:

• Exponential: Logistic:

• dN/dt = rNdN/dt = rN [(K-N)/K]

K

N

N

t

t

• Premise off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

- 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 off most of the plants, and only a few large plants survive. . 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 off most of the plants, and only a few large plants survive. . 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: off most of the plants, and only a few large plants survive.

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

K

N

Nt

t

SO: off most of the plants, and only a few large plants survive.

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

K

N

Nt

t

F off most of the plants, and only a few large plants survive. . 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 off most of the plants, and only a few large plants survive. . Temporal Dynamics

- DISCRETE GROWTH

F off most of the plants, and only a few large plants survive. . 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. off most of the plants, and only a few large plants survive. 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. off most of the plants, and only a few large plants survive. 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

• 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 off most of the plants, and only a few large plants survive.

What influences e and c?

Extinction probability is strongly influenced by population size.

G. Spatial Dynamics and Metapopulations off most of the plants, and only a few large plants survive.

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