In Subsection 8.3.3, we introduced a hierarchical competition model. We will use this

Chapter 8, Problem 8

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In Subsection 8.3.3, we introduced a hierarchical competition model. We will use this model to investigate the effects of habitat destruction on coexistence. We assume that a fraction D of the sites is permanently destroyed. Furthermore, we restrict our discussion to two species and assume that species 1 is the superior and species 2 the inferior competitor. In the case in which both species have the same mortality (m1 = m2), which we set equal to 1, the dynamics are described by dp1 dt = c1 p1(1 p1 D) p1 (8.98) dp2 dt = c2 p2(1 p1 p2 D) p2 c1 p1 p2 (8.99) where pi , i = 1, 2, is the fraction of sites occupied by species i . (a) Explain in words the meanings of the different terms in (8.98) and (8.99). (b) Show that p1 = 1 1 c1 D is an equilibrium for species 1, which is in (0, 1), and is stable if D < 1 1/c1 and c1 > 1. (c) Assume that c1 > 1 and D < 11/c1. Show that species 2 can invade the nontrivial equilibrium of species 1 [computed in (b)] if c2 > c2 1(1 D) (d) Assume that c1 = 2 and c2 = 5. Then species 1 can survive as long as D < 1/2. Show that the fraction of sites that are occupied by species 1 is then p1 = 1 2 D for 0 D 1 2 0 for 1 2 D 1 Show also that p2 = 1 10 + 2 5 D for 0 D 1 2 For D > 1/2, species 1 can no longer persist. Explain why the dynamics for species 2 reduce to dp2 dt = 5p2(1 p2 D) p2 in this case. Show, in addition, that the nontrivial equilibrium is of the form p2 = 1 1 5 D for 1 2 D 1 1 5 Plot p1 and p2 as functions of D in the same coordinate system. What happens for D > 1 1/5? Use the plot to explain in words how each species is affected by habitat destruction. (e) Repeat (d) for c1 = 2 and c2 = 3.