11 Consider the system 3 in Example 1 of the text Recall that this system has an asymptot ically 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 8a and b for the species x and y respectively In this case Eqs 3 are replaced by dx dt x 1 x y a i dy dt y 0 75 y 0 5x Sb 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 da 0 y 0 75 y 0 5x 8b 0 One way to proceed is to assume that x and y are given by power series in the parameter 5 thus iii 11 x x0 x 8 y yo y o Substitute Eqs iii into Eqs ii and collect terms according to powers of d b From the constant terms the terms not involving 8 show that xo 0 5 and yo 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 8 show that x 4a 4b y 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