159 Lecture 7

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# 159 Lecture 7 - PowerPoint PPT Presentation

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

Population Models in Excel

• 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)!

• 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!!

• 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…)
• 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