you are studying a community in a lake that contains two species of fish competeing with eachother for aquatic insects
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Problem 7: Two species of fish that compete with each other for food, but do not prey on each other, are bluegill and redear. Suppose that a pond is stocked with bluegill and redear, and let x(t) and y(t) be the populations of bluegill and redear, respectively, at time t. Let B ≥ 1 be the carrying capacity of the pond for bluegill (in the absence of redear) and R ≥ 1 the carrying capacity for redear (in the absence of bluegill). Suppose that the competition is modeled by the equations: dx/dt = ε₁x(1 - (1/B)x - (γ₁/B)y) dy/dt = ε₂y(1 - (1/R)y - (γ₂/R)x) where ε₁, ε₂, γ₁, γ₂ > 0 are constants. a) Suppose that γ₁ > B/R and R/B > γ₂. Show that the only equilibrium populations in the pond are no fish, no redear, or no bluegill. b) Suppose that B/R > γ₁ and R/B > γ₂. Show that there is an equilibrium point (x*, y*) at which both species can coexist (that is x* > 0 and y* > 0). Find the critical point (x*, y*) in terms of B, R, γ₁, γ₂ and compute the Jacobian matrix at (x*, y*). Classify (x*, y*) as sink, source, or saddle and determine whether it is asymptotically stable, stable, or unstable. We remark that, by fishing, it is possible to reduce the population of bluegill to such a level that they will die out. Fishing only for bluegill has the effect of reducing B at such a level that B/R < γ₁. We are then in the situation of part a) where we don't have an equilibrium at which both species can coexist.
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