Competition-colonization models In Exercise 7.Review.23 a metapopulation model for two

Chapter 10, Problem 27

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Competition-colonization models In Exercise 7.Review.23 a metapopulation model for two species was introduced. The equations were dp1 dt c1 p1s1 2 p1d 2 m1 p1 dp2 dt c2 p2s1 2 p1 2 p2d 2 m2 p2 2 c1 p1 p2 where pi is the fraction of patches occupied by species i, and ci and mi are the species-specific rates of colonization and extinction of patches, respectively. These equations assume that any patch has at most one species, and species 2 patches can be taken over by species 1, but not vice versa. (a) Suppose that m1 m2 3, c1 5, and c2 30. Find all equilibria. (b) Calculate the Jacobian matrix. (c) Determine the local stability properties of each equilibrium found in part (a) using the Jacobian from part (b). (d) Are the species predicted to be able to coexist at a stable equilibrium?