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Population Ecology

AP Biology· difficulty 4/5

A population follows fracdNdt=rNleft(1−fracNKright)\\frac{dN}{dt} = rN\\left(1 - \\frac{N}{K}\\right) with r=0.5r = 0.5/year and K=1000K = 1000. The growth rate dN/dtdN/dt is maximum at what population size?

  • A

    N=1000N = 1000 (at carrying capacity)

  • B

    N=500N = 500 (half of KK)

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  • C

    N=100N = 100 (when growth is exponential)

  • D

    N=0N = 0 (the rate-limiting starting point)

Explanation

Take derivative of rN(1−N/K)rN(1-N/K) with respect to NN and set to 0: r−2rN/K=0RightarrowN=K/2r - 2rN/K = 0 \\Rightarrow N = K/2. Logistic growth peaks at half of carrying capacity.

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