In Exercise 5, we considered the problem of predicting the population in a predator-prey model. Another problem of this type is concerned with two species competing for the same food supply. If the numbers ofspecies alive attime t are denoted by X| (/) and *2(0, it is often assumed that, although the birthrate of each of the species is simply proportional to the number ofspecies alive at that time, the death rate of each species depends on the population of both species. We will assume that the population of a particular pair of species is described by the equations d ^ll = Xl (f )[4 - 0.0003xi(t) - 0.0004x2(0] and dt dX2(t) = X2(Ot2 - 0.0002xi(0 - 0.0001x2(0]. dt If it is known that the initial population of each species is 10,000, find the solution to this system for 0 < t < 4. Is there a stable solution to this population model? If so, for what values of x\ and X2 is the solution stable?

th Math 340 Lecture – Introduction to Ordinary Differential Equations – April 18 , 2016 What We Covered: 1. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.9: Inhomogeneous Linear Systems i. Definition: You’re given the linear equation = + () where f(t) is the inhomogeneous term because it’s not dependent on y ii. Theorem: Suppose that y is p particular solution to the inhomogeneous equation and that 1 2..., frm a fundamental set of solutions to the associated ′ homogeneous equation =