The Logistic Model

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The exponential model only works when the population has unlimited resources. If there is a finite amount of resources then there will be a maximum population size that can be supported. This maximum is called the carrying capacity (M). In these circumstances, the growth rate must depend on the population size; in particular it must approach zero as the population approaches the carry capacity. The simplest way to model this is taking growth rate a(N) = a (1 - N/M), where a is a positive constant called the linear growth rate.

This brings us to the logistic model for population growth; dN/dt = a(1 - N/M)N.

This is a separable first order ODE which may be rewritten as M dN/dt = a(M - N)N, which has two particular solutions N(t) = 0 and N(t) = M for all t. So, for N(t) not equal to 0, M, we have N(t) = Mn / ((M - n)exp(-at) + n).

We can draw logistic curves from the logistic model.