Variations on the Exponential Model

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The general exponential model is dN/dt = (b-c)N. Interesting situations can be noted by making the birth rate (b) and the death rate (c) functions of t.

Periodic Growth Rate; dN/dt = -acos(wt)N.

This model could be used to describe seasonal growth rate in a population. The minimum growth rate is -a and the maximum growth rate is a. The differential equation is separable and has solution N = nexp[(-a/w)sinwt]. The minimum population size is nexp(-a/w) and the maximum population size is nexp(a/w).

a(N, t) = A(1-acoswt)

This growth rate could be used to represent a seasonally modulated growth rate. This is separable, with solution N = nexp[At - (aA/w)sinwt].