## Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit

Question

Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance). Decide whether each system describes competition or cooperation and explain why it is a reasonable model. (Ask yourself what effect an increase in one species has on the growth rate of the other.)
dx/dt = 0.12x – 0.0006x^2 + 0.00001xy
dy/dt = 0.08x + 0.00004xy

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3 weeks 2021-09-25T23:52:24+00:00 2 Answers 0

The systems describe cooperation.

Step-by-step explanation:

Given the following systems:

dx/dt = 0.12x – 0.0006x² + 0.00001xy

dy/dt = 0.08x + 0.00004xy

Consider the following equations:

dx/dt = A1x – B1x² + C1xy

dy/dt = A2x – B2x² + C2xy

The system is

COMPETITIVE if the existence of one specie hurts both species, that is, when C1 < 0 and C2 < 0.

COOPERATIVE if the existence of one specie benefits both species, that is, when C1 > 0 and C2 > 0

Comparing this with the equations given,

C1 = 0.00001

C2 = 0.00004

Because C1 > 0 and C2 > 0, we say that the systems are cooperative.

Increasing the y population causes an increase in the rate of growth of x, that is dx/dt.

Similarly, increase in the x population causes an increase in the rate of growth of y, that is dy/dt. Therefore, this system of equations describes the populations of two species which are cooperating with each other for mutual benefit.