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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

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159 lecture 7 l.jpg

159 Lecture 7

Population Models in Excel


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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)!


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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!!


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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…)


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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.


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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.


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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).


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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.



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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?



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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.


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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.


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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:


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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.


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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.





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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:


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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.


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Example 3

  • Revise our model from Example 2 with e = 0.00015 and f = 0.00001.

  • Keep all other parameters the same.



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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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