Question

Consider the following predator-prey systems of differential equations dx/dt = -x/2 + xy/2 + y, dy/dt = y(1 - y) - xy/2 + y.

Answer 1

Answer:

a) Prey: $\frac{dy}{dt}=y(1-y)-\frac{xy}{2+y}$

Predator: $\frac{dx}{dt}=\frac{xy}{2+y}-\frac{x}{2}$

b) they grow as a function y(1-y).

c) It will grow faster and probably exponentially,

Step-by-step explanation:

a)

Prey

For prey population the rate of change is given by its own growth rate, it would be a constant times the variable, minus the rate at which it is preyed upon, usually it is represent a parameter times x and y.

So the differential equation could be:

$\frac{dy}{dt}=y(1-y)-\frac{xy}{2+y}$

Predator

By the other hand, the rate of change of the predator's population depends only the rate at which the predators eat preys, minus predator's intrinsic death rate.

$\frac{dx}{dt}=\frac{xy}{2+y}-\frac{x}{2}$

b)  If there are no predators present the differential equation that models prey population will be:

$\frac{dy}{dt}=y(1-y)$

So they grow as a function y(1-y).

c)  In the case of the population of prey is much grader than predator, it will grow faster and exponentially, because it we assume that preys have unlimited food supply.

I hope it helps you!