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Lecture 9 Population Growth Models

- I. Exponential Population Growth
- A. What is exponential growth? (FIG. 1)
- B. Does exponential growth ever occur? (FIG. 2)
- C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- A. What is exponential growth? (FIG. 1)
- The number of individuals increases at an

increasing rate because new individuals

contribute to growth of the population. Its

like compound interest. - B. Does exponential growth ever occur? (FIG. 2)
- C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- A. What is exponential growth? (FIG. 1)
- The number of individuals increases at an

increasing rate because new individuals

contribute to growth of the population. Its

like compound interest. - B. Does exponential growth ever occur? (FIG. 2)
- C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- A. What is exponential growth? (FIG. 1)
- The number of individuals increases at an

increasing rate because new individuals

contribute to growth of the population. Its

like compound interest. - B. Does exponential growth ever occur? (FIG. 2)
- Yes, but usually only for a short time in

natural populations. Exponential growth is most

common in introduced populations. - C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- B. Does exponential growth ever occur? (FIG. 2)
- Yes, but usually only for a short time in

natural populations. Exponential growth is most

common in introduced populations. European

rabbits were introduced into Australia by rancher

Thomas Austin in 1859. - C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- B. Does exponential growth ever occur? (FIG. 2)
- Yes, but usually only for a short time in

natural populations. Exponential growth is most

common in introduced populations. European

rabbits were introduced into Australia by rancher

Thomas Austin in 1859. By 1865 he had killed

20,000 rabbits on his ranch and by 1920 they had

spread across the continent. - C. Mathematical models of exponential growth in

closed populations - D. Projecting future population size with the

models - E. Things we can do with population growth

models

(No Transcript)

European rabbit (Oryctolagus cuniculus)

Map of rabbit-proof Fences in Australia

Archive.amol.org.au

library.thinkquest.org

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- B. Does exponential growth ever occur? (FIG. 2)
- Yes, but usually only for a short time in

natural populations. Exponential growth is most

common in introduced populations. European

rabbits were introduced into Australia by rancher

Thomas Austin in 1859. By 1865 he had killed

20,000 rabbits on his ranch and by 1920 they had

spread across the continent. - C. Mathematical models of exponential growth in

closed populations - 1. Continuous population growth model (FIG. 3)
- 2. Discrete population growth model (FIG. 4)
- D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). 2. Discrete population growth model (FIG.

4) - D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). For continuously breeding species

like humans, domesticated animals, and some

wild plants and animals. - 2. Discrete population growth model (FIG. 4)
- D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). For continuously breeding species

like humans, domesticated animals, and some

wild plants and animals. Remember our

population size model? - 2. Discrete population growth model (FIG. 4)
- D. Projecting future population size with the

models - E. Things we can do with population growth

models

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG. 3).

For continuously breeding species like

humans, domesticated animals, and some wild

plants and animals. Remember our population

size model? For a closed population (no I or E)

the model would be Nt1 Nt B - D.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG. 3).

For continuously breeding species like

humans, domesticated animals, and some wild

plants and animals. Remember our population

size model? For a closed population (no I or E)

the model would be Nt1 Nt B - D. - We can express this model as ?N/?t B - D,

which means the change in a closed population

over time t is due to births and deaths in

the population.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG. 3).

For continuously breeding species like

humans, domesticated animals, and some wild

plants and animals. Remember our population

size model? For a closed population (no I or E)

the model would be Nt1 Nt B - D. - We can express this model as ?N/?t B - D,

which means the change in a closed population

over time t is due to births and deaths in

the population. If we reduce the time step, ?t,

to a very small time interval, dt, then B and

D are the number of births and deaths in the

short time interval.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG. 3).

For continuously breeding species like

humans, domesticated animals, and some wild

plants and animals. Remember our population

size model? For a closed population (no I or E)

the model would be Nt1 Nt B - D. - We can express this model as ?N/?t B - D,

which means the change in a closed population

over time t is due to births and deaths in

the population. If we reduce the time step, ?t,

to a very small time interval, dt, then B and

D are the number of births and deaths in the

short time interval. Let b

probability of an individual giving birth and d

probability of an individual dying in this

instant of time.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG. 3).

- We can express this model as ?N/?t B -

D, which means the change in a closed

population over time t is due to births and

deaths in the population. If we reduce the time

step, ?t, to a very small time interval, dt,

then B and D are the number of births and

deaths in the short time interval. Let b

probability of an individual giving birth and

d probability of an individual dying in

this instant of time. - Then B b N and D d N and the

population change in this instant of time is

dN/dt b N d N, which can be

expressed as dN/dt (b d) N.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). - Then B b N and D d N and the

population change in this instant of time

is dN/dt b N d N, which can be

expressed as dN/dt (b d) N. - Let b d r (instantaneous, intrinsic,

or per capita growth rate). Then the

growth is expressed as dN/dt rN which

is the exponential growth model for continuously

breeding populations.

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). - Then B b N and D d N and the

population change in this instant of time

is dN/dt b N d N, which can be

expressed as dN/dt (b d) N. - Let b d r (instantaneous, intrinsic,

or per capita growth rate). Then the

growth is expressed as dN/dt rN which

is the exponential growth model for continuously

breeding populations. Note that r 0

means no growth in the population and r lt

0 means the population is declining! - 2. Discrete population growth model (FIG. 4)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations (isolated from other

populations of the species) - 1. Continuous population growth model (FIG.

3). - Then B b N and D d N and the

population change in this instant of time

is dN/dt b N d N, which can be

expressed as dN/dt (b d) N. - Let b d r (instantaneous, intrinsic,

or per capita growth rate). Then the

growth is expressed as dN/dt rN which

is the exponential growth model for continuously

breeding populations. Note that r 0

means no growth in the population and r lt

0 means the population is declining! - 2. Discrete population growth model (FIG. 4)

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

Most mammals, birds, and other animals and

plants dont breed continuously throughout

the year. For these populations, well use a

different model.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

Most mammals, birds, and other animals and

plants dont breed continuously throughout

the year. For these populations, well use a

different model. Let rd a constant

proportional change in the population each

year. Example rd 0.10 means a 10

increase in the population each year and rd

0.20 means a 20 decrease in the

population each year.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

Most mammals, birds, and other animals and

plants dont breed continuously throughout

the year. For these populations, well use a

different model. Let rd a constant

proportional change in the population each

year. Example rd 0.10 means a 10

increase in the population each year and rd

0.20 means a 20 decrease in the

population each year. We can now write our

original population growth equation slightly

differently Nt1 Nt rd Nt or

Nt1 (1 rd) Nt. - Now well let 1 rd ?. This ? is

called the finite rate of increase or

decrease from one year to the next.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

Most mammals, birds, and other animals and

plants dont breed continuously throughout

the year. For these populations, well use a

different model. Let rd a constant

proportional change in the population each

year. Example rd 0.10 means a 10

increase in the population each year and rd

0.20 means a 20 decrease in the

population each year. We can now write our

original population growth equation slightly

differently Nt1 Nt rd Nt or

Nt1 (1 rd) Nt. - Now well let 1 rd ?. This ? is

called the finite rate of increase or

decrease from one year to the next. - Then Nt1 ? Nt and ? Nt1/ Nt.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Now well let 1 rd ?. This ? is

called the finite rate of increase or

decrease from one year to the next. - Then Nt1 ? Nt and ? Nt1/ Nt.
- The present time is indicated as t 0.

If t 0, our equation becomes N1 ?N0

(which says the population next year is ?

times the current population).

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Now well let 1 rd ?. This ? is

called the finite rate of increase or

decrease from one year to the next. - Then Nt1 ? Nt and ? Nt1/ Nt.
- The present time is indicated as t 0.

If t 0, our equation becomes N1 ?N0

(which says the population next year is ?

times the current population). Two years from

now, the population is predicted to be N2

?N1 which is the same as N2 ??N0

or N2 ?2 N0 .

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- The present time is indicated as t 0.

If t 0, our equation becomes N1 ?N0

(which says the population next year is ?

times the current population). Two years from

now, the population is predicted to be N2

?N1 which is the same as N2 ??N0

or N2 ?2 N0 . - Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any number

of years, t, in the future is Nt ?t N0

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any number

of years, t, in the future is Nt ?t N0 - 3. Assumptions of exponential growth models
- a.
- b.
- c.
- d.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any

number of years, t, in the future is Nt ?t

N0 - 3. Assumptions of exponential growth models
- a. No element of chance. The model is

predictable (deterministic). - b.
- c.
- d.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any

number of years, t, in the future is Nt ?t

N0 - 3. Assumptions of exponential growth models
- a. No element of chance. The model is

predictable (deterministic). - b. No resource limitation. The growth

rate stays the same

forever. - c.
- d.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any

number of years, t, in the future is Nt ?t

N0 - 3. Assumptions of exponential growth models
- a. No element of chance. The model is

predictable (deterministic). - b. No resource limitation. The growth

rate stays the same

forever. - c. No I or E. The population is

closed. - d.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 2. Discrete population growth model (FIG. 4).

- Continuing in the same way N3 ?3 N0

estimates the population in three years

and the estimated population for any

number of years, t, in the future is Nt ?t

N0 - 3. Assumptions of exponential growth models
- a. No element of chance. The model is

predictable (deterministic). - b. No resource limitation. The growth

rate stays the same

forever. - c. No I or E. The population is

closed. - d. No population structure. All

individuals have same b d

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- C. Mathematical models of exponential growth in

closed populations - 3. Assumptions of exponential growth models
- a. No element of chance. The model is

predictable (deterministic). - b. No resource limitation. The growth

rate stays the same forever. - c. No I or E. The population is closed.
- d. No population structure. All

individuals have same b d. - D. Projecting future population size with the

models - 1. Discrete model example (FIG. 4)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 1. Discrete model example (FIG. 4). Remember

that Nt ?t N0

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 1. Discrete model example (FIG. 4). Remember

that Nt ?t N0 If the current

population is N0 2000 and the finite growth

rate is ? 1.2, then the projected population

in 4 years is N4

1.24 2000 2.0736 2000 4147.

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 1. Discrete model example (FIG. 4). Remember

that Nt ?t N0 If the current

population is N0 2000 and the finite growth

rate is ? 1.2, then the projected

population in 4 years is

N4 1.24 2000 2.0736 2000 4147. - 2. How do we project future population size

with the continuous model? - a. The link between discrete and continuous

models - b. Example of projecting future population

size with continuous model

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - a. The link between discrete and continuous

models - As time interval ?t gets smaller, the

discrete rd approaches the

continuous growth rate r and er ?. - b. Example of projecting future population

size with continuous model

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - a. The link between discrete and continuous

models - As time interval ?t gets smaller, the

discrete rd approaches the

continuous growth rate r and er ?. Thus,

the model for discrete breeding

populations Nt ?t N0 becomes

Nt (er)tN0 which equals Nt

ert N0 for continuously breeding

populations. - b. Example of projecting future population

size with continuous model

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - Suppose you bought a herd of N0

20 dairy cows to put on your 160 acres

of grassland and pasture. You plan to breed

your cows regularly and know that the

intrinsic growth rate is r 0.2

female calves/cow/yr.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - Suppose you bought a herd of N0

20 dairy cows to put on your 160 acres

of grassland and pasture. You plan to breed

your cows regularly and know that the

intrinsic growth rate is r 0.2

female calves/cow/yr. You want to predict how

many cows youll have in the future and

decide to use the continuous model of

population growth. How many cows will you have?

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - Suppose you bought a herd of N0

20 dairy cows to put on your 160 acres

of grassland and pasture. You plan to breed

your cows regularly and know that the

intrinsic growth rate is r 0.2

female calves/cow/yr. You want to predict how

many cows youll have in the future and

decide to use the continuous model of

population growth. How many cows will you have? - Youll use the equation Nt ert N0
- In two years you should have N2 e

0.22 20 30 cows.

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30 54

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30 54

148

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30 54

148 1092

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30 54

148 1092 59620

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- D. Projecting future population size with the

models - 2. How do we project future population size

with the continuous model? - b. Example of projecting future population

size with continuous model. - In two years you should have N2

e 0.22 20 30 cows. - t 0 2 5 10

20 40 - N 20 30 54

148 1092 59620 - Whats wrong with these predictions?

What about our assumptions?

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 1. Add stochasticity (FIG. 5)
- 2. Add effects of resource limitation (FIG.

6)(See Part II below) - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - 4. Add effects of genetic structure

(Population Genetics)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 1. Add stochasticity (FIG. 5)
- Instead of using a constant r at each time

step (e.g. year), randomly select r values

from a normal distribution of values

to introduce chance elements into population

projections. - 2. Add effects of resource limitation (FIG.

6)(See Part II below) - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - 4. Add effects of genetic structure

(Population Genetics)

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 1. Add stochasticity (FIG. 5)
- Instead of using a constant r at each time

step (e.g. year), randomly select r values

from a normal distribution of values

to introduce chance elements into population

projections. - 2. Add effects of resource limitation (FIG.

6)(See Part II below) - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - 4. Add effects of genetic structure

(Population Genetics)

(No Transcript)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 1. Add stochasticity (FIG. 5)
- Instead of using a constant r at each time

step (e.g. year), randomly select r values

from a normal distribution of values to

introduce chance elements into population

projections. - 2. Add effects of resource limitation (FIG.

6)(See Part II below) - Why doesnt the actual population growth in

FIG. 6 match the projected growth? - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - 4. Add effects of genetic structure

(Population Genetics)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 1. Add stochasticity (FIG. 5)
- Instead of using a constant r at each time

step (e.g. year), randomly select r values

from a normal distribution of values

to introduce chance elements into population

projections. - 2. Add effects of resource limitation (FIG.

6)(See Part II below) - Why doesnt the actual population growth in

FIG. 6 match the projected growth?

The main reason is that food, habitat, and

other resources the pheasants need will become

limiting on the island. - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - 4. Add effects of genetic structure

(Population Genetics)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 2. Add effects of resource limitation (FIG.

6)(See Part II below) - Why doesnt the actual population growth in

FIG. 6 match the projected growth?

The main reason is that food, habitat, and

other resources the pheasants need will become

limiting on the island. - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - Its not very realistic to assume that all

individuals are the same! Well develop

more realistic matrix models next lecture. - 4. Add effects of genetic structure

(Population Genetics)

Lecture 9 Population Growth Models

- I. Exponential Population Growth
- E. Things we can do with population growth

models - 3. Add effects of age, size, gender, stage of

development (Lec. 10) - Its not very realistic to assume that all

individuals are the same! Well develop

more realistic matrix models next lecture. - 4. Add effects of genetic structure

(Population Genetics) - Its also interesting to examine the

distribution of genotypes and attempt to

determine genetic relationships among

individuals. This is in the realm

of population genetics.

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors
- 2. Density-independent factors
- B. Mathematical models of density-dependent

growth

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors - increase in

importance as N increases. - 2. Density-independent factors

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors - increase in

importance as N increases. - 2. Density-independent factors - chance events

that have about the same effect on b

and d in small populations as they have

in large populations.

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors - increase in

importance as N increases. Resource

availability (food, habitat, etc.), disease,

predation rates, etc. - 2. Density-independent factors - chance events

that have about the same effect on b

and d in small populations as they have

in large populations.

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors - increase in

importance as N increases. Resource

availability (food, habitat, etc.), disease,

predation rates, etc. - 2. Density-independent factors - chance events

that have about the same effect on b

and d in small populations as they have

in large populations. Disturbances, changes in

weather or climate, and other chance

occurrences. We can use stochastic models

to imitate chance events (FIG. 5).

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- A. What are the limitations on exponential

growth? - 1. Density-dependent factors - increase in

importance as N increases. Resource

availability (food, habitat, etc.), disease,

predation rates, etc. - 2. Density-independent factors - chance events

that have about the same effect on b

and d in small populations as they have

in large populations. Disturbances, changes in

weather or climate, and other chance

occurrences. We can use stochastic models

to imitate chance events (FIG. 5). - B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- 2. Discrete logistic growth model
- 3. Modifications of the logistic growth model

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7)
- b. Population size and growth rate (FIG.

8) - c. Continuous logistic growth model (FIG.

9)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. - b. Population size and growth rate (FIG.

8) - c. Continuous logistic growth model (FIG.

9)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. The population stabilizes

when b d. b. Population size and

growth rate (FIG. 8) - c. Continuous logistic growth model (FIG.

9)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. The population stabilizes

when b d. This stable population

size is called K, the carrying capacity of the

environment or habitat. - b. Population size and growth rate (FIG.

8) - c. Continuous logistic growth model (FIG.

9)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. The population stabilizes

when b d. This stable population

size is called K, the carrying capacity of the

environment or habitat. This pattern is

called logistic growth. - b. Population size and growth rate (FIG.

8) - c. Continuous logistic growth model (FIG.

9)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. The population stabilizes

when b d. This stable population

size is called K, the carrying capacity of the

environment or habitat. This pattern is

called logistic growth. - b. Population size and growth rate (FIG.

8). - c. Continuous logistic growth model (FIG.

9)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- a. Birth and death rates (FIG. 7). As N

increases, the birth rate, b,

usually decreases and the death rate, d, usually

increases. The population stabilizes

when b d. This stable population

size is called K, the carrying capacity of the

environment or habitat. This pattern is

called logistic growth. - b. Population size and growth rate (FIG.

8). In logistic growth, dN/dt

(the rate of change in N) first increases but

then levels off and gradually declines

as N gets larger (FIG. 8a). - c. Continuous logistic growth model (FIG.

9)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- b. Population size and growth rate (FIG.

8). In logistic growth dN/dt

(the rate of change in N) first increases but

then levels off and gradually declines

as N gets larger (FIG. 8a).

In exponential growth, dN/dt keeps

increasing (FIG. 8b)! - c. Continuous logistic growth model (FIG.

9)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- b. Population size and growth rate (FIG.

8). In logistic growth dN/dt

(the rate of change in N) first increases but

then levels off and gradually declines

as N gets larger (FIG. 8a).

In exponential growth, dN/dt keeps

increasing (FIG. 8b)! - c. Continuous logistic growth model (FIG.

9)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- c. Continuous logistic growth model (FIG.

9). - dN/dt rN(1N/K) is the continuous

logistic growth model where r is the

intrinsic growth rate (as in the exponential

model), N is population size, and K is the

carrying capacity.

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- c. Continuous logistic growth model (FIG.

9). - dN/dt rN(1N/K) is the continuous

logistic growth model where r is the

intrinsic growth rate (as in the exponential

model), N is population size, and K is the

carrying capacity. When N is small,

the population growth is exponential, but

as N increases, growth slows and eventually

stops at K.

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- c. Continuous logistic growth model (FIG.

9). - dN/dt rN(1N/K) is the continuous

logistic growth model where r is the

intrinsic growth rate (as in the exponential

model), N is population size, and K is the

carrying capacity. When N is small,

the population growth is exponential, but

as N increases, growth slows and

eventually stops at K. The rate of

growth starts to decline at the inflection point,

halfway between N 0 and N K.

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 1. Continuous population growth model
- c. Continuous logistic growth model (FIG.

9). - dN/dt rN(1N/K) is the continuous

logistic growth model where r is the

intrinsic growth rate (as in the exponential

model), N is population size, and K is the

carrying capacity. When N is small,

the population growth is exponential, but

as N increases, growth slows and

eventually stops at K. The rate of

growth starts to decline at the inflection point,

halfway between N 0 and N K. - 2. Discrete logistic growth model (FIG. 10)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 2. Discrete logistic growth model (FIG. 10).

Remember the discrete model for

exponential growth Nt1 Nt rd Nt

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 2. Discrete logistic growth model (FIG. 10).

Remember the discrete model for

exponential growth Nt1 Nt rd Nt

Well add the same term for resource limitation

that we used in the continuous

logistic growth model to get the discrete

model Nt1 Nt rd Nt (1Nt/K)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 2. Discrete logistic growth model (FIG. 10).

Remember the discrete model for

exponential growth Nt1 Nt rd Nt

Well add the same term for resource limitation

that we used in the continuous

logistic growth model to get the discrete

model Nt1 Nt rd Nt (1Nt/K) - 3. Modifications of the logistic growth model
- a. Time lag in response to resource

limitation (FIGS. 10,11) - b. Stochastic variation (FIG. 12)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 3. Modifications of the logistic growth model
- a. Time lag in response to resource

limitation (FIGS. 10,11) - Birth rates and death rates dont

respond immediately to changes in

resource availability. - b. Stochastic variation (FIG. 12)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 3. Modifications of the logistic growth model
- a. Time lag in response to resource

limitation (FIGS. 10,11) - Birth rates and death rates dont

respond immediately to changes in

resource availability. This time lag causes a

fluctuation in N as the habitat cant

support all the new offspring and high

mortality occurs. This is seen clearly in

the discrete logistic model (FIG. 10). - b. Stochastic variation (FIG. 12)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 3. Modifications of the logistic growth model
- a. Time lag in response to resource

limitation (FIGS. 10,11) - Birth rates and death rates dont

respond immediately to changes in

resource availability. This time lag causes a

fluctuation in N as the habitat cant

support all the new offspring and high

mortality occurs. This is seen clearly in

the discrete logistic model (FIG. 10).

Organisms with higher reproductive rate

(higher rd) have greater fluctuation in

population size (N). - b. Stochastic variation (FIG. 12)

(No Transcript)

Lecture 9 Population Growth Models

- II. Logistic Population Growth
- B. Mathematical models of density-dependent

growth - 3. Modifications of the logistic growth model
- a. Time lag in response to resource

limitation (FIGS. 10,11) - Birth rates and death rates dont

respond immediately to changes in

resource availability. This time lag causes a

fluctuation in N as the habitat cant

support all the new offspring and high

mortality occurs. This is seen clearly in

the discrete logistic model (FIG. 10).

Organisms with higher reproductive rate

(higher rd) have greater fluctuation in

population size (N). A time lag, called tau

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