Lecture 9 Population Growth Models

1
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

2
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

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4
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

5
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

6
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

7
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

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9
European rabbit (Oryctolagus cuniculus)
Map of rabbit-proof Fences in Australia
Archive.amol.org.au
library.thinkquest.org
10
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

11
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

12
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

13
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

14
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.

15
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.

16
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.

17
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.

18
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.

19
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.

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21
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)

22
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)

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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.

25
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.

26
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.

27
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.

28
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).

29
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 .

30
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

31
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.

32
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.

33
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.

34
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.

35
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

36
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)

37
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

38
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.

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40
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

41
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

42
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

43
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.

44
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?

45
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.

46
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

47
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

48
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

49
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

50
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

51
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?

52
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)

53
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)

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55
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)

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57
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)

58
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)

59
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)

60
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.

61
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

62
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

63
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.

64
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.

65
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).

66
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67
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
  

68
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)

69
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)

70
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71
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)

72
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)

73
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)

74
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75
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)

76
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)

77
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78
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)

79
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80
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)

81
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.

82
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83
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.

84
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85
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.

86
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87
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)

88
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

89
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)

90
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91
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)

92
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)

93
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)

94
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95
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)

96
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97
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