The Lotka-Volterra predator-prey model assumes that in the absence of predators the

Chapter 10, Problem 20

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The Lotka-Volterra predator-prey model assumes that in the absence of predators the number of prey grows exponentially. If we make the alternative assumption that the prey population grows logistically, the new system is , where a, b, c, r, and K are positive and K ab. (a) Show that the system has critical points at (0, 0), (0, K), and , where and . (b) Show that the critical points at (0, 0) and (0, K) are saddle points, whereas the critical point at is either a stable node or a stable spiral point. (c) Show that is a stable spiral point if . Explain why this case will occur when the carrying capacity K of the prey is large.