# Differential Equations: Populations

The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations

ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r).

In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution for x at time t if x(0) = x0, where x0 > 0. Is this model realistic? Why?

How would I determine all the equilibrium points for this system of diﬀerential equations expressing my answer in terms of p, q, r and s.

How would I classify each equilibrium point using the method of matrices with two real distinct eigenvalues?

© BrainMass Inc. brainmass.com March 5, 2021, 12:44 am ad1c9bdddfhttps://brainmass.com/math/calculus-and-analysis/differential-equations-populations-531030

#### Solution Preview

Please see attachment for complete solution.

The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations

ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r).

In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution for x at time t if x(0) = x0, where x0 > 0. Is this ...

#### Solution Summary

The expert determines all the equilibrium points for a system.