Cons the predator-prey model with b = 0, equation 50.2. Suppose that the value o

Question

Cons the predator-prey model with b = 0, equation 50.2. Suppose that the value of k was increased. (a) What does increasing the value of k correspond to ecologically? (b) How does this affect the nonzero equilibrium populations of fish and sharks? (c) Briefly explain this effect ecologically. Consider the predator-prey model with b notequalto 0, equation 50.1. Calculate all possible equilibrium solutions. Compare these populations to the ones which occur if b = 0. Briefly explain the qualitative and quantitative differences between the two cases, b = 0 and b notequalto 0. Reconsider exercise 50.2. In the phase plane, sketch the isocline (s) along which dF/dS = 0. Also sketch the isocline (s) along which dF/dS = infinity. Indicate arrows as in Fbig. 50 - 3. Explain on your diagram where all possible equilibrium populations are. (If needed, sketch all possible cases.) Reconsider the predator-prey model of exercise 50.2 (b notequalto 0). Determine the constant coefficient linear system of differential equations which governs the small displacements from the equilibrium populations, F = k/lambda, S = a/c - bk/c lambda > 0. Do this in two ways: Using the Taylor series for a function of two variables (equation 44.5). Using perturbation methods as described in Sec. 50. Show that the two are equivalent. Do not attempt to solve the resulting system. Show that your answer to exercise 50.4 can be put in the form: dF_1/dt = k/lambda (- bF_1 - cS_1) dS_1/dt = (a/c - bk/c lambda) lambda F_1, where F_1 and S_1 are the displacements from the equilibrium populations of the fish and sharks respectively. Eliminate F_1 (from the second equation) to derive a second-order constant coefficient differential equation for S_1. Analyze that equation and determine the conditions under which the equilibrium population is stable. Give an ecological interpretation of the inequality in exercise 50.4, a/c > bk/c lambda

Explanation / Answer

a/b > k/

Means that equilibrium number of prey without predators a/b is smaller than the prey necessary to sustain

the predators k/. This means that there is an imbalance and it will divert from equilibrium.

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