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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

159 Lecture 7

Population Models in Excel

toads again
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
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 cont4
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 cont5
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
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 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
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 cont10
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 cont12
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
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
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
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
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
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
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
Example 3
  • Revise our model from Example 2 with e = 0.00015 and f = 0.00001.
  • Keep all other parameters the same.
references
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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