The population dynamics ofthree competing species can be described by dxiit) = riXid) dt

Chapter 10, Problem 10

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The population dynamics ofthree competing species can be described by dxiit) = riXid) dt 1 -^CtijXjil) /=i for each i = I, 2, 3 where the population ofthe (th species attimer isx,(r). The growth rate ofthe/th species isr,-, and , measuresthe extentto which speciesj affectsthe growth rateofspeciesi. Assume that the three growth rates equal r. By scaling time by the factor r, we can effectively make r = 1. Also, we assume species 2 affects 1 the same as 3 affects 2 and as I affects 3. Thus, an = asi, which we let equal a, and, similarly, a2i = 32 = i3 = /* The populations can be scaled so that all an 1. This yields the system of differential equations X|(r) =X|(0[l -xK?) -ax2(r) - ^(r)], x'2{t) = X2{t)[\ -X2(?)-/Ixi(?)-ffX3(?)], and x'3(t) = X3(0[l - Aid) - axi(t) - 0X2(0]. If a = 0.3 and = 0.6, find a stable solution (xj(r) = x^it) = x3(t) = 0) of the scaled populations xi(r), X2(r), X3(r) in the set described by 0.5 < X|(r) < 1, 0 < X2(r) < 1, and 0.5 < X3(r) < 1