Competition-colonization models The metapopulation model from Exercise 15 can be extended to include two species, where one is a superior competitor. The equations are dp1 dt

− c1p1s1 2 p1d 2 m1p1 dp2 dt

− c2p2s1 2 p1 2 p2d 2 m2p2 2 c1p1p2 where p1 and p2 are the fractions of patches occupied by species 1 and 2, respectively. These equations model a process in which any patch has at most one species, and where species 2 patches can be “taken over” by species 1, but not vice versa.

(a) Explain how the terms in the equations reflect the assumption that species 1 is the superior competitor.

(b) Suppose that m1 − m2 − 3, c1 − 5, and c2 − 30.

Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. Show that, despite species 1 being a better competitor, the two species are predicted to coexist.