FAQ: Why is the parameter "a" in the predator-prey problem used as both death of rabbit and birth of foxes?
Zhipu Jin, 02-10-07
Note: this question refers to the discrete time predator prey model used in Chapter 2 - System Modeling.
We repeat our (admittedly simplistic) assumptions: 1. The predator species is totally dependent on the prey species as its only food supply; 2. The prey species has an food supply and no threat to its growth other than the specific predator.
The rate at which predators encounter prey is jointly proportional to the sizes of the two populations. A fixed proportion of encounters leads to the death of the prey. These assumptions lead to the conclusion that the negative component of the prey growth rate is proportional to the product "R[k]*F[k]" of the population sizes.
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 F[k+1] = F[k]-d*F[k].(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 for the predator: the prey. And what's bad for rabbit is good for foxes. That is, the energy to support growth of the predator population is proportional to deaths of prey, so F[k+1] = F[k]-d*F[k]+a*R[k]*F[k].
Richard selected "a" as the parameters for the product "R[k]*F[k]" in both difference equations just in order to simplify the problem. Of course we can use different parameters in those equations.