159 Lecture 7

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159 Lecture 7

• Population Models in Excel

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Toads Again!

• Lets 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 n0.
• We can think of this function as a recurrence
  relation with
• x(0) 36449
• x(n) f(x(n-1)) for n1,
• for some function f(x)!

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Toads Again! (cont.)

• Lets 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 populations 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.)
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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 doesnt make sense in
  the real world for a population, try x0 -1 and
  x0 -10.
• What happens?

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Example 1 (cont.)
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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, lets 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)aR(n-1) bR(n-1)F(n-1)
• F(n) F(n-1)-cF(n-1) dR(n-1)F(n-1) for
  n1,
• where a, b, c, and d are all greater than zero.

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Example 2

• As an example, lets 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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Example 2 (cont.)
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Example 2 (cont.)
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Example 2 (cont.)
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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)aR(n-1) bR(n-1)F(n-1)
  eR(n-1)R(n-1)
• F(n) F(n-1)-cF(n-1) dR(n-1)F(n-1)
  fF(n-1)F(n-1) for n1,
• 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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Example 3 (cont.)
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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