HW4: Additional Problem Suppose we have a species of sh in a pond. Let x(t) be the population of the species at time t. In the simplest model of population growth the rate of growth is assumed to be proportional to the size of the population i.e. dx = kx, where k is called the growth rate. However, the growth of the dt population must eventually be limited by the amount of resources. This implies the growth rate should be a decreasing function of the population size x i.e. dx = xk(x). If we now taylor expand dt k(x) and drop the error term we have k(x) = a bx, where a, b are positive constants. (The negative insures that as the population increases the growth rate decreases.) We then have the nonlinear model dx = x(a bx) dt Now, consider two species of sh in a pond with populations x(t), y(t). If we assume the two sh share a common food supply, then we can expect the growth rate of one sh species to depend on the population of the other sh species i.e. x = x k(x, y). Doing taylor expansions, etc... we get a reasonable nonlinear model x = x(a bx cy) y = y(d ey f x) 1. Suppose we went to the pond and measured the actual paramaeters a, b, c, d, e, f above; we found x = x(1 x y) 1 3 y = y( 2 1 y 4 x) 4 (a) Show that (x , y ) = (0, 0), (1, 0), (0, 2), ( 1 , 1 ) are critical points. (Notice that at the critical 2 2 points the populations aren't changing i.e. x = y = 0) (b) Suppose we measure the populations and nd that they are close to the critical point (0, 0). Should we be worried that both sh populations will die out? (c) Suppose that they are close to the critical point (1, 0). Should we be worried that species y will die out? (d) Suppose that they are close to the critical point (0, 1). Should we be worried that species x will die out? 1 (e) Suppose that they are close to the critical point ( 2 , 1 ). What should we expect about the sh 2 in this case? 1