Question

Often scientists use rate of change equations in their study of population growth for one or more species. In this problem we study systems of rate of change equations designed to inform us about the future populations for two species that are either competitive (that is both species are harmed by interaction) or cooperative (that is both species benefit from interaction. Which system of rate of change equations below describes a situation where the two species compete and which system describes cooperative species? Explain your reasoning.

a. dx/dt= -5x + 2xy
b. dx/dt= 4x-2xy
c. dy/dt= -4y + 3xy
d. dy/dt= 2y-xy

Answer 1

a: cooperative.

b: competitive.

c: cooperative.

d: competitive.

Explanation:

The rate of change:

dx/dt

gives information about how the quantity x changes as a function of time.

If dx/dt > 0, the population x increases as the time passes.

if dx/dt < 0, the population x decreases as the time passes.

Now, let's analyze the options:

a) dx/dt= -5x + 2xy

in this case, x represents one population and y represents the other population. Where x and y should only be positive numbers because we can not have a negative population.

Notice that in this case, if y increases, then dx/dt increases.

Then we have cooperative species because the population y helps the growth of population x.

b) dx/dt = 4*x - 2*x*y

Similar to the previous case, but now as the value of y increases, the value of dx/dt decreases, then we have competitive species.

c) dy/dy = -4*y + 3*x*y

Notice that now we have the rate of change of y, in the right side, we can see that as x increases, also the rate of change dy/dt.

Then the population x helps to increase the population y, which means that the species are cooperative.

d) dy/dt = 2*y - x*y

Now as x increases, the value of dy/dt decreases, then these species are competitive.