Lecture 5

class: center, middle, inverse, title-slide

# Lecture 5
## Density-dependent population growth
### <br/><br/><br/>WILD3810 (Spring 2021)

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## Readings

> Mills 126-141

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## Density-independence vs density-dependence

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

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## Density-independence vs density-dependence

We also learned that this assumption leads to exponential population growth

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## Limitless population growth?

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

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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

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

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###  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?

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class: inverse

## Density-dependence

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

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

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

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## 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

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**Intra-specific competition:**

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

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Intra-specific competition can arise in multiple ways:

.pull-left[
**Animals**
 * food
 * shelter
 * breeding sites
 * mates
 ]
 
 .pull-right[
 **Plants**
 * space
 * light
 * water
 * nutrients
]

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## Competition

Ecologists generally distinguish between two types of competition:

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1) **Exploitation competition**

+ consumption of limited resource by individuals depletes the amount available for others

+ also known as: depletion, consumption, or scramble competition

+ *indirect*

???

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

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## 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*

???

Image courtesy of Muhammad Mahdi Karim, via Wikimedia Commons

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## Competition

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

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## 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

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## Population regulation

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## Population regulation

???

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

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## Population regulation

???

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

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## Population regulation

Carrying capacity `$K$`:

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

???

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

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## 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)

???

`$^2$` 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)

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class: inverse, middle, center

# Modeling density-dependent population growth

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## Modeling D-D population growth

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

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

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

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### To start, remember what density-dependence means:

> the rate of population growth changes as population size increases

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## Modeling D-D population growth

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

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

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(Note that all we did was divide both sides by `$N$`)

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### How do we interpret this equation?

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## 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

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## 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.)

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## 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

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### 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

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

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+ Note that the growth rate will **always** be `$r$`, no matter how many individuals are in the population

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## Modeling D-D population growth

It's also helpful to visualize this equation:

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## Modeling D-D population growth

#### How do we add density-dependence to our model?

#### Remember the equation for a line: `$^3$`

`$$\Large y = a + bx$$`  
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#### So one way to include density-dependence is: `$^4$`

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

???

`$^3$` `$a$` 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 `$b$`

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## 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

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## 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)

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

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+ Call that `$r_0$`

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

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## Modeling D-D population growth

#### What is `$\large c$`?

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+ 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):

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

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

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## 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:

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

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## Modeling D-D population growth

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## Models of D-D population growth

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

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

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

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

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What about when `$N=500$`?

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

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When `$N=1000$`?

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

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## Models of D-D population growth

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

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#### Multiply both sides by `$N$`

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

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#### Now read this as:

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## 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

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## 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

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## 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**

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## Models of D-D population growth

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

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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

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When `$N=1000$`?

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

???

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

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## Models of D-D population growth

#### How does this work? `$^6$`

???

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

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## Models of D-D population growth

#### How does this work? `$^6$`

???

`$^6$` 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

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## Models of D-D population growth

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# Allee effects

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## 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

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## 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

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## Allee effects

#### Why might the death rate be high at small `$\large N$`?

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* Group signaling breaks down, predation increases

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* Cooperative foraging becomes less efficient

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* Inbreeding depression

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#### Why might the birth rate be low at small `$\large N$`?

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* Pollination failure

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* Unable to find mates because of rarity

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* Unable to find mates because of skewed sex ratio

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* Inbreeding depression

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## Allee effects

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

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class: inverse, center, middle

# Discrete dynamics

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## Discrete dynamics

#### Remember the discrete-time model of density-independent growth

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

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

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#### Remember that:

`$$\Large \lambda = e^r$$`

and:

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

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## 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**