Write down the system of differential equations for the population of both predators and prey by using the assumptions above.

Solution

We will start off through letting that the x and y present the population of the predators and the population of the prey.

Here, the first assumption tells us that, in the lacking of predators, the prey will develop at a rate of ay where a > 0. Similarly the second assumption tells us as, in the absence of prey, the predators will reduce at a rate of -xy where b > 0.

Subsequently, the third and fourth assumptions tell us how the population is influenced by encounters among predators and prey. Therefore, with each encounter the population of the predators will raise at a rate of axy and the population of the prey will reduce at a rate of -b xy, here a> 0and b < 0.

Putting all of this together we arrive at the following system.

x′ = -bx + axy = x (ay - b)

y′ = ay - b xy = y (a - b x)

Note that it is a nonlinear system and we not have (nor will we here) discuss how to determine this type of system. We simply wanted to provide a "better" model for several population problems and to point out this not all systems will be simple and nice linear systems.