Biologists introduce the same fish species into two isolated alpine lakes, Lake A and…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Biologists introduce the same fish species into two isolated alpine lakes, Lake A and Lake B. In each lake the population follows a logistic model with a common carrying capacity of 50005000 fish. However, differing nutrient levels produce distinct growth parameters: in Lake A the population yA(t)y_{A}(t) satisfies dyAdt=0.02 yA(5000−yA)\dfrac{dy_{A}}{dt}=0.02\,y_{A}(5000-y_{A}), while in Lake B the population yB(t)y_{B}(t) satisfies dyBdt=0.05 yB(5000−yB)\dfrac{dy_{B}}{dt}=0.05\,y_{B}(5000-y_{B}), where tt is measured in years. At time t=0t=0, both lakes contain 10001000 fish of the introduced species. (a) Determine the instantaneous growth rate of the fish population in each lake at t=0t=0. State your answers in fish per year. (b) Show analytically that there exists a population size y∗y^{\ast} (with 0<y∗<50000<y^{\ast}<5000) for which the instantaneous growth rate in Lake B exceeds that in Lake A by dyBdt−dyAdt>200 fish per year.\frac{dy_{B}}{dt}-\frac{dy_{A}}{dt}>200\text{ fish per year}. Derive an inequality that y∗y^{\ast} must satisfy. (c) Solve the inequality from part (b) to find the range of population sizes for which Lake B’s instantaneous growth rate exceeds Lake A’s by more than 200200 fish per year. (d) Discuss how the difference in the growth parameters (0.050.05 versus 0.020.02) influences long-term population dynamics, even though both lakes share the same carrying capacity. Support your discussion with reasoning based on the logistic model.

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