8. Consider the Lotka-Volterra equations for the joint dynamics of rabbits r(t) and coyotes c(t): dr...

 

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8. Consider the Lotka-Volterra equations for the joint dynamics of rabbits r(t) and coyotes c(t): dr | | dt dr dt dc dt de dt = 5r - 2rc = c+rc (a) Solve for the equilibrium solutions of this system of equations. [Hint: there are two equilibrium solutions.] = Now, while we can't solve analytically for r(t), c(t), we can study what happens near each equilibrium point. To do this we follow a procedure called linearization: we set r(t) = ro+ eri(t), c(t) = co + ec(t), where (ro, co) is an equilibrium solution, < 1 is assumed to be a small number, and r, C represent perturbations from the equilibrium solution. Then, we substitute these expressions into the system above, expand in powers of e, and keep only terms involving the first power of e. This gives us the following, linear system of ODEs, for the perturbation (r, C): = = 5r12(cori + roc) = -c + (cor + roc) (b) Solve this linear system of ODEs, for each of the equilibrium solutions you found above. For each, sketch some trajectories in the (r, c)-plane. (c) Interpret your solutions. What does this seem to imply, about how the populations of rabbits and coyotes evolve, near each of the equilibrium solutions? 8. Consider the Lotka-Volterra equations for the joint dynamics of rabbits r(t) and coyotes c(t): dr | | dt dr dt dc dt de dt = 5r - 2rc = c+rc (a) Solve for the equilibrium solutions of this system of equations. [Hint: there are two equilibrium solutions.] = Now, while we can't solve analytically for r(t), c(t), we can study what happens near each equilibrium point. To do this we follow a procedure called linearization: we set r(t) = ro+ eri(t), c(t) = co + ec(t), where (ro, co) is an equilibrium solution, < 1 is assumed to be a small number, and r, C represent perturbations from the equilibrium solution. Then, we substitute these expressions into the system above, expand in powers of e, and keep only terms involving the first power of e. This gives us the following, linear system of ODEs, for the perturbation (r, C): = = 5r12(cori + roc) = -c + (cor + roc) (b) Solve this linear system of ODEs, for each of the equilibrium solutions you found above. For each, sketch some trajectories in the (r, c)-plane. (c) Interpret your solutions. What does this seem to imply, about how the populations of rabbits and coyotes evolve, near each of the equilibrium solutions?