Modeling Fish Population

Calculus 2 · Axiom Academy

EXAMPLE Modeling Fish Population Using the logistic growth model to predict population dynamics in a lake ecosystem. A lake can sustain at most 10 , 000 fish. Biologists stock it with 1 , 000 fish, and the population grows logistically with rate constant k = 0.5 per year. Find the population model P(t) , then determine when the lake reaches half its carrying capacity ( 5 , 000 fish). Nice work. You built a logistic model from scratch and used it to answer a real question about the lake. Logistic model: describes growth that levels off at a carrying capacity K . Finding A : the initial condition fixes the constant — , here . Carrying capacity: K = 10 , 000 is the largest population the lake can support; P(t) rises toward it but never exceeds it. Solving for time: to find when a target population is reached, set P(t) equal to it and undo the exponential with a natural log. Half-capacity time: P = 5 , 000 at years — and K/2 is exactly where logistic growth is fastest. The same logistic model governs constrained growth far beyond fish — epidemics, product adoption, and resource use all bend the same way once a limit comes into view.

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