An Economic Model of Population Growth and Competition in Natural Communities

Abstract

A fundamental problem of ecology is to relate the dynamics of the growth and competition of biological populations to the underlying foraging process and to the detailed structure of the environment. The logistic growth model

$$ \frac{{{\text{dn}}}} {{{\text{dt}}}}\, = \,{\text{rn}}\left( {{\text{1 - }}\frac{{\text{n}}} {{\text{k}}}} \right) $$

(1)

is the best known descriptive model of population growth in a limited environment. It describes a population coming into balance with its resources at the carrying capacity K of the environment. The Lotka-Volterra competition model (Slobodkin, 1961)

$$ \begin{array}{*{20}c} {\frac{{{\text{dn}}_1 }} {{{\text{dt}}}}\, = \,{\text{r}}_1 {\text{n}}_1 \left( {1 - \frac{{{\text{n}}_1 }} {{{\text{K}}_1 }} - \frac{{{\text{an}}_2 }} {{{\text{k}}_1 }}} \right)} \\ {\frac{{{\text{dn}}_2 }} {{{\text{dt}}}}\, = \,{\text{r}}_2 {\text{n}}_2 \left( {1 - \frac{{{\text{bn}}_1 }} {{{\text{K}}_2 }} - \frac{{{\text{n}}_2 }} {{{\text{k}}_2 }}} \right)} \\ \end{array} $$

(2)

generalizes the logistic growth model to describe competition of two (or more) populations for limited resources. These models can be derived from varying assumptions concerning the dependence of birth and death rates on population densities (Pielou, 1969).