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Part 1.2: Construction of the Model
As noted in the Introduction, in order to to track the time evolution of both predators and prey, we will need two differential equations -- one for the rate of growth of the prey population and another for the rate of growth of the predator population. Suppose we let x(t) represent the population of the prey and let y(t) represent the population of the predators in appropriate units. For the hare/lynx case, we will measure the populations in thousands of members.
We repeat our (admittedly simplistic) assumptions from Part 1a:
If there were no predators, the second assumption would imply that the prey species grows exponentially, i.e., we would have
dx/dt = ax.
But there are predators, which should account for a negative term in the prey growth rate.
We'll investigate what this term should look like.
We'll assume that a fixed fraction of encounters between a predator and a prey result in the death of the prey. Say one out of five or one out of ten encounters between a lynx and a hare is fatal to the hare. We don't need to settle on a particular fraction now.
Now suppose we suddenly double the number of hares. Since there are twice as many hares, there should be twice as many encounters and twice as many fatal ones. Similarly, if we triple the number of hares, there should be three times as many encounters fatal to the hares. In mathematical terms, we are assuming that the death rate caused by the predators is proportional to the number of prey.
Suppose we keep the number of hares fixed but double the number of lynx. Again there should be twice as many encounters and twice as many encounters fatal to the hare. This time we are assuming that the death rate caused by the predators is proportional to the number of predators. When one quantity is proportional to two different variables, we speak of "joint proportionality." In our case, the negative term in the hare growth rate is jointly proportional to x and y.We will model the effect of the predator-prey interactions on the growth rate of the prey by a term of the form -bxy, where b is a constant.
(b) Explain why a term of the form -b(x + y) would not model joint proportionality.
(c) Can you think of any other term that would model joint proportionality?
dx/dt = ax - bxy.
Now we consider the predator population. If there were no food supply, the population would die out at a rate proportional to its size, i.e. we would find
dy/dt = -cy.
(Keep in mind that the "natural growth rate" is a composite of birth and death rates, both presumably proportional to population size. In the absence of food, there is no energy supply to support the birth rate.) But there is a food supply: the prey. And what's bad for hares is good for lynx. That is, the energy to support growth of the predator population is proportional to deaths of prey, so
dy/dt = -cy + pxy.
We now have our first model for the variations in the populations of the predator and prey -- called the Lotka-Volterra Predator-Prey Model:
dx/dt = ax - bxy,
dy/dt = -cy + pxy,
where a, b, c, and p are positive constants.
Recall that for a single first-order differential equation, we need to give an initial value for the dependent variable in order to specify a particular solution. In the case of a system of two first-order differential equations, we need to give initial values for both of the dependent variables in order to specify a particular pair of functions x(t) and y(t). In other words, a particular pair of solutions is determined by the Lotka-Volterra equations together with values of x(0) and y(0).
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