[DEFINITIONS] Modelling with first-order equations

Logistic model

The logistic model combines proportional growth with a reduction in the per-capita growth rate as the population approaches a carrying capacity $K$ :

P'=\alpha P\left(1-\frac{P}{K}\right).

Here $\alpha>0$ , $K>0$ and $P\ge0$ ; if $t$ is measured in a time unit, $\alpha$ has inverse-time units. The equilibria are $P=0$ and $P=K$ . For $0<P<K$ , the population grows; for $P>K$ , it decreases. Its separated solution has the form

P(t)=\frac{K}{1+Ae^{-\alpha t}}.

This expression is interpreted only on intervals where its denominator is nonzero; for a physical population one usually takes the forward-time branch with $P(0)>0$ ; the zero equilibrium is listed separately below.

To see where the formula comes from, separate on a branch with $P>0$ and $P\ne K$ . Since

\frac{1}{P(1-P/K)}=\frac1P+\frac1{K-P},

integration gives $\ln|P|-\ln|K-P|=\alpha t+C$ . Solving this ratio for $P$ gives the displayed expression, with $A=K/P(0)-1$ when $P(0)>0$ . The zero equilibrium $P=0$ is not obtained from a finite $A$ and must be included separately; $P=K$ corresponds to $A=0$ . The bounded relaxation model $P'=k(P_c-P)$ also approaches a target, but logistic growth is proportional to both the current population and the remaining fraction of carrying capacity.

[1]:A city with a carrying capacity

Definitions from Modelling with first-order equations

Logistic model

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