Suppose that a certain population obeys the logistic equation dy/dt = ry(1 − (y/K)). a.If y0 = K/3, find the time τ at which the initial population has doubled. Find the value of τ corresponding to r = 0.025 per year. b.If y0/K = α, find the time T at which y(T)/K = β, where 0 < α, β < 1. Observe that T → ∞ as α → 0 or as β → 1. Find the value of T for r = 0.025 per year, α = 0.1, and β = 0.9.

Question

Suppose that a certain population obeys the logistic equation dy/dt = ry(1 − (y/K)).

a.If y0 = K/3, find the time τ at which the initial population has doubled. Find the value of τ corresponding to r = 0.025 per year.

b.If y0/K = α, find the time T at which y(T)/K = β, where 0 < αβ < 1. Observe that T → ∞ as α → 0 or as β → 1. Find the value of T for r = 0.025 per year, α = 0.1, and β = 0.9.

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