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7 March, 11:07

There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1-m/P). Thus the modified logistic model is given by the differential equation dP/dt=kP (1-P/M) (1-m/P) Use the differential equation to show that any solution is increasing if m

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  1. 7 March, 12:39
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    a) If m < P 0, P is increasing

    b) If 0 < P < m, then all factors < 0, P is decreasing

    Step-by-step explanation:

    Given:

    - the modified Logistics Equation is:

    dP/dt = kP (1 - P/M) * (1-m/P)

    Find:

    Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m

    Solution:

    - If m < P < M, then:

    P/M 0

    similarly m/P 0

    - Since all factors are positive then dP/dt > 0, so P is increasing.

    - If 0 < P < m, then:

    m/P > 1, then (1 - P/M) < 0

    similarly P is still < M, so

    - Since all factors are positive then dP/dt < 0, so P is decreasing.
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