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### 159 Lecture 7

Population Models in Excel

Toads Again!

- Let’s look at the toad data again, but this time let n be the number of years after 1939 and x(n) be the area covered by toads at year n.
- Using Excel, we find that the best-fit exponential function for this data is
- x(n) = 36449e0.0779n for n≥0.
- We can think of this function as a recurrence relation with

x(0) = 36449

x(n) = f(x(n-1)) for n≥1,

for some function f(x)!

Toads Again! (cont.)

- Let’s find f(x).
- To do so, look at x(n) – x(n-1):
- x(n) – x(n-1)

= 36449e0.0779n - 36449e0.0779(n-1)

= 36449e0.0779(n-1)(e0.0779 – 1)

= (e0.0779 – 1)*x(n-1)

- Solving for x(n), we see that
- x(n) = x(n-1)+(e0.0779 -1)*x(n-1) =

= e0.0779 *x(n-1), so our function is

f(x) = e0.0779 *x!!

Toads Again! (cont.)

- Thus, the toad growth can be modeled with the recurrence relation

x(0) = 36449

x(n) = e0.0779 *x(n-1) for n ≥ 1.

- The closed form solution is given by our original model!
- For this model, the growth of the toad population is exponential (no surprise…)

Toads Again! (cont.)

- So how realistic is an exponential growth model for the toad population?
- For such a model, the population grows without bound, with no limitations built in.
- Realistically, there should some way to limit the growth of a population due to available space, food, or other factors.

The Logistic Model

- As a population increases, available resources must be shared between more and more members of the population.
- Assuming these resources are limited, here are some “reasonable” assumptions one can make how a population should grow:
- The population’s growth rate should eventually decrease as the population levels increase beyond some point.
- There should be a maximum allowed population level, which we will call a carrying capacity.
- For population levels near the carrying capacity, the growth rate is near zero.
- For population levels near zero, the growth rate should be the greatest.

The Logistic Model (cont.)

- The simplest model that takes these assumptions into account is the logistic model:
- x(0) = x0

x(n) = x(n-1)*(R(1-x(n-1)/K)+1) for n ≥ 1

- Here, x0 is the initial population size,

R is the intrinsic growth rate (i.e. growth rate without any limitations on growth),

and K is the carrying capacity.

- Notice that when x(n-1) is close to zero, the growth is exponential.
- Also, when x(n-1) is close to K, the population stays near the constant value of K (so growth rate is close to zero).

Example 1

- Use Excel to study at the long-term behavior of a population the grows logistically, with carrying capacity K = 100 and growth rate R = 0.5 (members/year).
- Use x0 = 0, 25, 50, 75, 100, 125, and 150.

Example 1 (cont.)

- Notice that X = 100 and X = 0 are fixed points of the logistic recurrence relation.
- X = 100 is stable.
- What about X = 0?
- For fun, even though this doesn’t make sense in the real world for a population, try x0 = -1 and x0 = -10.
- What happens?

Example 1 (cont.)

- Fixed point X = 0 is unstable!
- In general, for the logistic equation, the fixed points turn out to be X = 0 and X = K.
- This can be shown by solving the equation X = X*(R(1-X/K)+1) for X.

Two or More Populations

- If two or more populations interact, we can use a system of recurrence equations to model the population growth!
- Typical examples include predator-prey, host-parasite, competitive hunters and arms races.

Predator-Prey Model

- As an example, let’s consider two populations that interact – foxes (predator) and rabbits (prey). Assume no other species interact with the foxes or rabbits.
- Assume the following:
- There is always enough food and space for the rabbits.
- In the absence of foxes the rabbit population grows exponentially.
- In the absence of rabbits, the fox population decays exponentially.
- The number of rabbits killed by foxes is proportional to the number of encounters between the two species.
- This in turn is proportional to the product of the two populations (this assumption implies fewer kills when the number of foxes or rabbits is small).
- These assumptions can be modeled with the following system:

Predator-Prey Model (cont.)

- Let R(n) be the number of rabbits at time n and F(n) be the number of foxes at time n.
- R(0) = R0

F(0) = F0

R(n) = R(n-1)+a*R(n-1) – b*R(n-1)*F(n-1)

F(n) = F(n-1)-c*F(n-1) + d*R(n-1)*F(n-1) for n≥1,

where a, b, c, and d are all greater than zero.

Example 2

- As an example, let’s try the Rabbit-Fox Population model with a = 0.15, b = 0.004, c = 0.1, and d = 0.001.
- Assume that initially there are 200 rabbits and 50 foxes, i.e. R0 = 200 and F0 = 50.
- Plot R(n) and F(n) vs. n, for 200 years.
- Repeat with F(n) vs. R(n), for 200 years.

Revised Predator-Prey Model (cont.)

- A more realistic model takes into account the fact that there may be limits to the space available for the foxes and rabbits.
- This can be modeled via a logistic growth model, in the absence of the other species!
- This amounts to the following:

Revised Predator-Prey Model

- Let R(n) be the number of rabbits at time n and F(n) be the number of foxes at time n.
- R(0) = R0

F(0) = F0

R(n) = R(n-1)+a*R(n-1) – b*R(n-1)*F(n-1) – e*R(n-1)*R(n-1)

F(n) = F(n-1)-c*F(n-1) + d*R(n-1)*F(n-1) – f*F(n-1)*F(n-1) for n≥1,

where a, b, c, d, e, and f are all greater than zero.

Example 3

- Revise our model from Example 2 with e = 0.00015 and f = 0.00001.
- Keep all other parameters the same.

References

- A Course in Mathematical Modeling by Douglas Mooney and Randall Swift
- An Introduction to Mathematical Models in the Social and Life Sciences by Michael Olinick

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