Ch15_predators and prey

Ch15_predators and prey - Chapter 15 Predators and Prey...

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Chapter 15 Predators and Prey Models of population growth. The simplest model for the growth, or decay, of a population says that the growth rate, or the decay rate, is proportional to the size of the population itself. Increasing or decreasing the size of a the population results a proportional increase or decrease in the number of births and deaths. Mathematically, this is described by the differential equation ˙ y = ky The proportionality constant k relates the size of the population, y ( t ), to its rate of growth, ˙ y ( t ). If k is positive, the population increases; if k is negative, the population decreases. As we know, the solution to this equation is a function y ( t ) that is proportional to the exponential function y ( t ) = ηe kt where η = y (0). This simple model is appropriate in the initial stages of growth when there are no restrictions or constraints on the population. A small sample of bacteria in a large Petri dish, for example. But in more realistic situations there are limits to growth, such as finite space or food supply. A more realistic model says that the population competes with itself. As the population increases, its growth rate decreases linearly. The differential equation is sometimes called the logistic equation. ˙ y = k (1 - y μ ) y Copyright c ± 2009 Cleve Moler Matlab R ± is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1
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Chapter 15. Predators and Prey 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 20 22 Figure 15.1. Exponential growth and logistic growth. The new parameter μ is the carrying capacity . As y ( t ) approaches μ the growth rate approaches zero and the growth ultimately stops. It turns out that the solution is y ( t ) = μηe kt ηe kt + μ - η You can easily verify for yourself that as t approaches zero, y ( t ) approaches η and that as t approaches infinity, y ( t ) approaches μ . If you know calculus, then with quite a bit more effort, you can verify that y ( t ) actually satisfies the logistic equation. Figure 15.1 shows the two solutions when both η and k are equal to one. The exponential function y ( t ) = e t gives the rapidly growing green curve. With carrying capacity μ = 20, the logistic function y ( t ) = 20 e t e t + 19 gives the more slowly growing blue curve. Both curves have the same initial value and initial slope. The exponential function grows exponentially, while the logistic function approaches, but never exceeds, its carrying capacity. Figure 15.1 was generated with the following code.
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This note was uploaded on 10/11/2011 for the course MTHSC 365 taught by Professor Adams during the Spring '11 term at Clemson.

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Ch15_predators and prey - Chapter 15 Predators and Prey...

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