Consider the system (3) in Example 1 of the text. Recall
Question and Solution
Consider the system (3) in Example 1 of the text. Recall that this system has an asymptotically stable critical point at
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Consider the system (3) in Example 1 of the text. Recall that this system has an asymptotically stable critical point at (0.5, 0.5), corresponding to the stable coexistence of the two population species. Now suppose that immigration or emigration occurs at the constant rates of ?a and ?b for the species x and y, respectively. In this case, Eqs. (3) are replaced by dx dt = x(1 ? x ? y) + ?a, dy (i) dt = y(0.75 ? y ? 0.5x) + ?b. The question is what effect this has on the location of the stable equilibrium point. (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.
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