(a) To find the new critical point, we must solve the equations x(1 x y) + a = 0, (ii)

Chapter 9, Problem 12

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(a) To find the new critical point, we must solve the equations x(1 x y) + a = 0, (ii) y(0.75 y 0.5x) + b = 0. One way to proceed is to assume that x and y are given by power series in the parameter ; thus x = x0 + x1 + , y = y0 + y1 + . (iii) Substitute Eqs. (iii) into Eqs. (ii) and collect terms according to powers of . (b) From the constant terms (the terms not involving ), show that x0 = 0.5 and y0 = 0.5, thus confirming that, in the absence of immigration or emigration, the critical point is (0.5, 0.5). (c) From the terms that are linear in , show that x1 = 4a 4b, y1 = 2a + 4b. (iv) (d) Suppose that a > 0 and b > 0 so that immigration occurs for both species. Show that the resulting equilibrium solution may represent an increase in both populations, or an increase in one but a decrease in the other. Explain intuitively why this is a reasonable result