Lecture 5

Lecture 5Density-dependent population growth

WILD3810 (Spring 2021)1 / 47

Readings

Mills 126-141

2 / 47

Density-independence vs density-dependence

In lecture 3, we learned about population growth models that assume demographic rates are unrelated to population size

3 / 47

Density-independence vs density-dependence

We also learned that this assumption leads to exponential population growth

4 / 47

Limitless population growth?

No population can grow exponentially forever (or even for relatively short periods of time)

5 / 47

Limitless population growth?

No population can grow exponentially forever (or even for relatively short periods of time)
Thomas Malthus was the first to propose that no population could grow without bound forever (1798)

At some point, resources will be become limited and populations must either stop growing or decline

5 / 47

Limitless population growth?

No population can grow exponentially forever (or even for relatively short periods of time)
Thomas Malthus was the first to propose that no population could grow without bound forever (1798)

At some point, resources will be become limited and populations must either stop growing or decline

Malthus' work inspired Darwin (1859) to suggest that limitation of resources is what drives evolution by natural selection

5 / 47

Stochasticity and extinction risk over time

We also learned that, given enough time, populations that experience stochasticity will eventually go extinct

6 / 47

Stochasticity and extinction risk over time

We also learned that, given enough time, populations that experience stochasticity will eventually go extinct

Why isn't extinction more common?

6 / 47

Density-dependence

The tendency of population vital rates, and therefore population growth rate, to change (increase or decrease) as a function of population size

7 / 47

Density-dependence

The tendency of population vital rates, and therefore population growth rate, to change (increase or decrease) as a function of population size

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

7 / 47

Density-dependence

The tendency of population vital rates, and therefore population growth rate, to change (increase or decrease) as a function of population size

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

As the population grows, competition, disease, and predation increase

7 / 47

Competition

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

As $N$ increases, the availability of resources per organism will decrease, leading to increased competition

8 / 47

Competition

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

As $N$ increases, the availability of resources per organism will decrease, leading to increased competition
Intra-specific competition:

interaction between individuals of a single species brought about by the need for a shared resource

8 / 47

Competition

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

As $N$ increases, the availability of resources per organism will decrease, leading to increased competition
Intra-specific competition:

interaction between individuals of a single species brought about by the need for a shared resource

Intra-specific competition can arise in multiple ways:

Animals

food
shelter
breeding sites
mates

Plants

space
light
water
nutrients

8 / 47

Competition

Ecologists generally distinguish between two types of competition:

9 / 47

Competition

Ecologists generally distinguish between two types of competition:
1) Exploitation competition

consumption of limited resource by individuals depletes the amount available for others
also known as: depletion, consumption, or scramble competition
indirect

9 / 47

Image courtesy of U.S. Fish and Wildlife Service Headquarters, via Wikimedia Commons

Competition

Ecologists generally distinguish between two types of competition:

1) Exploitation competition

2) Interference competition

individuals actively prevent others from attaining a resource in a given area or territory
also known as: encounter or contest competition
direct

10 / 47

Image courtesy of Muhammad Mahdi Karim, via Wikimedia Commons

Competition

As population size increases, the resources available to each individual will eventually shrink to the point where demographic parameters are negatively effected

11 / 47

Competition

As population size increases, the resources available to each individual will eventually shrink to the point where demographic parameters are negatively effected

Increased density can also increase rates of disease transmission or predation

12 / 47

Population regulation

13 / 47

Population regulation

14 / 47

Left of the intersection, population growth rate is positive so the population will grow

Population regulation

15 / 47

Right of the intersection, population growth is negative so the population will shrink

Population regulation

Carrying capacity $K$ :

the population size that the environment can maintain $^1$

16 / 47

$^1$ At the intersection, population growth is 0 so the population will be stable

Population regulation vs limitation

The density-dependent processes we just learned about are called regulating factors

Regulating factors keep population size from going too far above or below $K$ $^2$

Limiting factors determine the actual value of $K$

Limiting factors can be density-dependent (competition) or density independent (disturbance or extreme weather)

17 / 47

$^2$ Think of regulating factors like a thermostat - if the temperature goes below the desired temperature ($N < K$), the heater kicks on ($b$ increases). If once the temperature increases above the set temperature ($N > K$), the heater turns off ($d$ increases)

Modeling density-dependent population growth18 / 47

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

$\Large \frac{dN}{dt} = N \times r$

19 / 47

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

$\Large \frac{dN}{dt} = N \times r$

How can we modify this equation to include density-dependence?

19 / 47

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

$\Large \frac{dN}{dt} = N \times r$

How can we modify this equation to include density-dependence?

To start, remember what density-dependence means:

the rate of population growth changes as population size increases

19 / 47

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

$\Large \frac{1}{N}\frac{dN}{dt} = r$

20 / 47

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

$\Large \frac{1}{N}\frac{dN}{dt} = r$ (Note that all we did was divide both sides by $N$ )

20 / 47

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

$\Large \frac{1}{N}\frac{dN}{dt} = r$ (Note that all we did was divide both sides by $N$ )

How do we interpret this equation?

20 / 47

Modeling D-D population growth

$\Large \frac{1}{N}\frac{\color{red}{dN}}{dt} = \color{red}{r}$

There are $r$ new individuals added to the population

21 / 47

Modeling D-D population growth

$\Large \frac{1}{N}\frac{dN}{\color{red}{dt}} = r$

There are $r$ new individuals added to the population

every time step (e.g., hour, day, week, etc.)

22 / 47

Modeling D-D population growth

$\Large \color{red}{\frac{1}{N}}\frac{dN}{dt} = r$

There are $r$ new individuals added to the population

every time step (e.g., hour, day, week, etc.)

per individual currently in the population

23 / 47

Modeling D-D population growth

$\Large \color{red}{\frac{1}{N}}\frac{dN}{dt} = r$

There are $r$ new individuals added to the population

every time step (e.g., hour, day, week, etc.)

per individual currently in the population

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

23 / 47

Modeling D-D population growth

$\Large \color{red}{\frac{1}{N}}\frac{dN}{dt} = r$

There are $r$ new individuals added to the population

every time step (e.g., hour, day, week, etc.)

per individual currently in the population

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

For example, if $r = 1$ , each individual replaces itself at each time step

23 / 47

Modeling D-D population growth

$\Large \color{red}{\frac{1}{N}}\frac{dN}{dt} = r$

There are $r$ new individuals added to the population

every time step (e.g., hour, day, week, etc.)

per individual currently in the population

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

For example, if $r = 1$ , each individual replaces itself at each time step
Note that the growth rate will always be $r$ , no matter how many individuals are in the population

23 / 47

Modeling D-D population growth

It's also helpful to visualize this equation:

24 / 47

Modeling D-D population growthHow do we add density-dependence to our model?25 / 47

Modeling D-D population growth

How do we add density-dependence to our model?

Remember the equation for a line: $^3$

$\Large y = a + bx$

25 / 47

Modeling D-D population growth

How do we add density-dependence to our model?

Remember the equation for a line: $^3$

$\Large y = a + bx$

So one way to include density-dependence is: $^4$

$\Large \frac{1}{N} \frac{dN}{dt} = r + cN$ .

25 / 47

$^3$ $a$ is the y-intercept (the value of $y$ when $x=0$ ) and $b$ is the slope (the change in $y$ per unit change is $x$ ).

$^4$ I changed the slope to $c$ to ensure that it is not confused with the birth rate $b$

Modeling D-D population growth

Again, it might help to visualize this equation:

26 / 47

Modeling D-D population growth

Again, it might help to visualize this equation:

So now, the per capita growth rate $\Big(\frac{1}{N} \frac{dN}{dt}\Big)$ decreases as $N$ increases

26 / 47

Modeling D-D population growthIn the new population model, r represents the rate of increase when the population size is 0 (i.e., the y-intercept)27 / 47

Modeling D-D population growthIn the new population model, r represents the rate of increase when the population size is 0 (i.e., the y-intercept)We can see in the figure that this is the largest value of r the population can experience27 / 47

Modeling D-D population growthIn the new population model, r represents the rate of increase when the population size is 0 (i.e., the y-intercept)We can see in the figure that this is the largest value of r the population can experienceCall that r0  
27 / 47

Modeling D-D population growth

In the new population model, $\large r$ represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of $\large r$ the population can experience

Call that $r_0$
Because $r_0$ the maximum rate of increase (nothing limiting population growth), it is equivalent to $r$ in the D-I model

27 / 47

Modeling D-D population growthWhat is c?28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population

28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population
In this case, it has to be negative (growth rate decreases as $N$ increases)

28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population
In this case, it has to be negative (growth rate decreases as $N$ increases)

Notice also that there is a point where the line crosses the x-axis (growth rate = 0):

28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population
In this case, it has to be negative (growth rate decreases as $N$ increases)

Notice also that there is a point where the line crosses the x-axis (growth rate = 0):

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + c N^* = 0$

28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population
In this case, it has to be negative (growth rate decreases as $N$ increases)

Notice also that there is a point where the line crosses the x-axis (growth rate = 0):

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + c N^* = 0$ solving for $N^*$ , we can see that:

$\Large N^* = -\frac{r_0}{c}$

28 / 47

Modeling D-D population growth

What is $\large c$ ?

The slope of the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$
i.e., the amount by which population growth changes for every individual added to the population

In this case, it has to be negative (growth rate decreases as $N$ increases)

Notice also that there is a point where the line crosses the x-axis (growth rate = 0):

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + c N^* = 0$

solving for $N^*$ , we can see that:

$\Large N^* = -\frac{r_0}{c} = \color{red}K$

29 / 47

Modeling D-D population growth

30 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 100 =$ 0.9

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 100 =$ 0.9

What about when $N=500$ ?

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 100 =$ 0.9

What about when $N=500$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 500=$ 0.5

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 100 =$ 0.9

What about when $N=500$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 500=$ 0.5

When $N=1000$ ?

31 / 47

Models of D-D population growth

We now have a full equation for the relationship between $\large \frac{1}{N} \frac{dN}{dt}$ and $\large N$ :

$\Large \frac{1}{N} \frac{dN}{dt} = r_0 + cN$ How does this work?

If $r_0 = 1$ and $c = -0.001$ , what is the population growth rate when $N=100$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 100 =$ 0.9

What about when $N=500$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 500=$ 0.5

When $N=1000$ ?

$\large \frac{1}{N} \frac{dN}{dt} = 1 - 0.001 \times 1000=$ 0

31 / 47

Models of D-D population growthHow do we use the density-dependent growth rate model to project population size?32 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \frac{dN}{dt} = N \times \Big(r_0 + cN\Big)$

32 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \frac{dN}{dt} = N \times \Big(r_0 + cN\Big)$

Now read this as:

32 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \color{red}{\frac{dN}{dt}} = N \times \Big(r_0 + c N\Big)$

Now read this as:

The change in population size per time step

33 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \frac{dN}{dt} = \color{red}{N} \times \Big(r_0 + c N\Big)$

Now read this as:

The change in population size per time step

equals the current population size

34 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \frac{dN}{dt} = N \times \color{red}{\Big(r_0 + c N\Big)}$

Now read this as:

The change in population size per time step

equals the current population size

times the growth rate when population size equals N

35 / 47

Models of D-D population growth

How do we use the density-dependent growth rate model to project population size?

Multiply both sides by $N$

$\Large \frac{dN}{dt} = N \times \color{red}{\Big(r_0 + c N\Big)}$

Now read this as:

The change in population size per time step

equals the current population size

times the growth rate when population size equals N

This is called the logistic growth model

35 / 47

Models of D-D population growthHow does this work? Again, r0=1 and c=−0.00136 / 47

Models of D-D population growth

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

What is $\frac{dN}{dt}$ when $N=100$ ?

$\large \frac{dN}{dt} = 100 \times \bigg(1 - 0.001 \times 100\bigg)=$ 90

36 / 47

Models of D-D population growth

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

What is $\frac{dN}{dt}$ when $N=100$ ?

$\large \frac{dN}{dt} = 100 \times \bigg(1 - 0.001 \times 100\bigg)=$ 90

What about when $N=500$ ?

$\large \frac{dN}{dt} = 500 \times \bigg(1 - 0.001 \times 500\bigg)=$ 250

36 / 47

Models of D-D population growth

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

What is $\frac{dN}{dt}$ when $N=100$ ?

$\large \frac{dN}{dt} = 100 \times \bigg(1 - 0.001 \times 100\bigg)=$ 90

What about when $N=500$ ?

$\large \frac{dN}{dt} = 500 \times \bigg(1 - 0.001 \times 500\bigg)=$ 250

When $N=1000$ ?

$\large \frac{dN}{dt} = 1000 \times \bigg(1 - 0.001 \times 1000\bigg)=$ 0

36 / 47

What happens to $\frac{dN}{dt}$ when $N > K$ ?

Models of D-D population growth

How does this work? $^6$

37 / 47

$^6$ Thickness and color of line correspond to magnitude of population change (thicker and darker red correspond to larger rate of change)

Models of D-D population growth

How does this work? $^6$

38 / 47

$^6$ Thickness and color of line correspond to magnitude of population change (thicker and darker red correspond to larger rate of change)

Rate of change is largest when abundance is at half of the carrying capacity $\bigg(N = \frac{K}{2}\bigg)$

Above $K$ , the rate of change becomes negative

Models of D-D population growth

39 / 47

Allee effects40 / 47

Allee effects

So far, we have assumed that $\Large b$ and $\Large d$ (and therefore $\Large r$ ) decrease as population size increases

This is called negative density dependence

41 / 47

Allee effects

So far, we have assumed that $\Large b$ and $\Large d$ (and therefore $\Large r$ ) decrease as population size increases

This is called negative density dependence

In some cases, the slope could also be positive

Positive relationships between $\large N$ and $\large d$ or $\large b$ generally occur at small population sizes

42 / 47

Allee effectsWhy might the death rate be high at small N?43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
Why might the birth rate be low at small N?43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
Why might the birth rate be low at small N?Pollination failure  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
Why might the birth rate be low at small N?Pollination failure  
Unable to find mates because of rarity  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
Why might the birth rate be low at small N?Pollination failure  
Unable to find mates because of rarity  
Unable to find mates because of skewed sex ratio  
43 / 47

Allee effectsWhy might the death rate be high at small N?Group signaling breaks down, predation increases  
Cooperative foraging becomes less efficient  
Inbreeding depression  
Why might the birth rate be low at small N?Pollination failure  
Unable to find mates because of rarity  
Unable to find mates because of skewed sex ratio  
Inbreeding depression
43 / 47

Allee effects

When abundance drops below the minimum viable population (MVP), the population will likely approach extinction without help!

44 / 47

Discrete dynamics45 / 47

Discrete dynamics

Remember the discrete-time model of density-independent growth

$\Large N_{t+1} = N_t \lambda$

46 / 47

Discrete dynamics

Remember the discrete-time model of density-independent growth

$\Large N_{t+1} = N_t \lambda$

As before, we need to account for possible changes in $\lambda$ caused by changes in population density

46 / 47

Discrete dynamics

Remember the discrete-time model of density-independent growth

$\Large N_{t+1} = N_t \lambda$

As before, we need to account for possible changes in $\lambda$ caused by changes in population density

Remember that:

$\Large \lambda = e^r$

and:

$\Large r = \frac{1}{N} \frac{dN}{dt} = r_0 + cN$

46 / 47

Discrete dynamics

Therefore, one discrete-time density-dependent growth model is:

$\Large N_{t+1}=N_t e^{\Big[r_0 + cN\Big]}$

This known as the Ricker model

47 / 47

↑, ←, Pg Up, k	Go to previous slide
↓, →, Pg Dn, Space, j	Go to next slide
Home	Go to first slide
End	Go to last slide
Number + Return	Go to specific slide
b / m / f	Toggle blackout / mirrored / fullscreen mode
c	Clone slideshow
p	Toggle presenter mode
t	Restart the presentation timer
?, h	Toggle this help

Lecture 5

Density-dependent population growth

WILD3810 (Spring 2021)

Readings

Density-independence vs density-dependence

Density-independence vs density-dependence

Limitless population growth?

Limitless population growth?

Limitless population growth?

Stochasticity and extinction risk over time

Stochasticity and extinction risk over time

Why isn't extinction more common?

Density-dependence

Density-dependence

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

Density-dependence

At small population sizes, individual organisms may be able to acquire all of the resources they need to survive and reproduce

As the population grows, competition, disease, and predation increase

Competition

Competition

Competition

Competition

Competition

Competition

Competition

Competition

Population regulation

Population regulation

Population regulation

Population regulation

Population regulation vs limitation

Modeling density-dependent population growth

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

How can we modify this equation to include density-dependence?

Modeling D-D population growth

Remember the (continuous time) density-independent model of population growth:

How can we modify this equation to include density-dependence?

To start, remember what density-dependence means:

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

Modeling D-D population growth

First, let's modify the density-independent model is a small but useful way:

How do we interpret this equation?

Modeling D-D population growth

Modeling D-D population growth

Modeling D-D population growth

Modeling D-D population growth

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are N\large N individuals currently in the population

Modeling D-D population growth

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are N\large N individuals currently in the population

Modeling D-D population growth

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are N\large N individuals currently in the population

Modeling D-D population growth

Modeling D-D population growth

How do we add density-dependence to our model?

Modeling D-D population growth

How do we add density-dependence to our model?

Remember the equation for a line: 3^3

Modeling D-D population growth

How do we add density-dependence to our model?

Remember the equation for a line: 3^3

So one way to include density-dependence is: 4^4

Modeling D-D population growth

Again, it might help to visualize this equation:

Modeling D-D population growth

Again, it might help to visualize this equation:

Modeling D-D population growth

In the new population model, r\large r represents the rate of increase when the population size is 0 (i.e., the y-intercept)

Modeling D-D population growth

In the new population model, r\large r represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of r\large r the population can experience

Modeling D-D population growth

In the new population model, r\large r represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of r\large r the population can experience

Modeling D-D population growth

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

So this formulation tells us the per capita (i.e., per individual) rate of growth when there are $\large N$ individuals currently in the population

Remember the equation for a line: $^3$

Remember the equation for a line: $^3$

So one way to include density-dependence is: $^4$

In the new population model, $\large r$ represents the rate of increase when the population size is 0 (i.e., the y-intercept)

In the new population model, $\large r$ represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of $\large r$ the population can experience

In the new population model, $\large r$ represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of $\large r$ the population can experience

In the new population model, $\large r$ represents the rate of increase when the population size is 0 (i.e., the y-intercept)

We can see in the figure that this is the largest value of $\large r$ the population can experience

What is $\large c$ ?

What is $\large c$ ?

What is $\large c$ ?

What is $\large c$ ?

What is $\large c$ ?

What is $\large c$ ?

What is $\large c$ ?

Multiply both sides by $N$

Multiply both sides by $N$

Multiply both sides by $N$

Multiply both sides by $N$

Multiply both sides by $N$

Multiply both sides by $N$

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

How does this work? Again, $\large r_0 = 1$ and $\large c = -0.001$

How does this work? $^6$

How does this work? $^6$

So far, we have assumed that $\Large b$ and $\Large d$ (and therefore $\Large r$ ) decrease as population size increases