Variable-rate harvesting Suppose that the logistic equation dx/dt = kx(M - x) models a population x...

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Variable-rate harvesting Suppose that the logistic equation dx/dt = kx(M - x) models a population x (t) of fish in a lake after t months during which no fishing occurs. Now suppose that, because of fishing, fish are re- moved from the lake at the rate of hx fish per month (with h a positive constant). Thus fish are "harvested" at a rate the proportional to the existing fish population, rather than at (a) If 0 <h < KM, show the constant rate of that the population is still logistic. What is the new limit- ing population? (b) If h≥kM, show that x(t) → 0 are t → +∞, so the lake is eventually fished out. Variable-rate harvesting Suppose that the logistic equation dx/dt = kx(M - x) models a population x (t) of fish in a lake after t months during which no fishing occurs. Now suppose that, because of fishing, fish are re- moved from the lake at the rate of hx fish per month (with h a positive constant). Thus fish are "harvested" at a rate the proportional to the existing fish population, rather than at (a) If 0 <h < KM, show the constant rate of that the population is still logistic. What is the new limit- ing population? (b) If h≥kM, show that x(t) → 0 are t → +∞, so the lake is eventually fished out.