Question

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

Answer 1

a) If m < P < M , then all factors > 0, P is increasing

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

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

similarly m/P < 1, then (1-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.