TY - JOUR

T1 - On the populations of competing species

AU - Brauer, Fred

N1 - Funding Information:
A standard mathematical model for the populations of two species competing for the same food supply consists of a system of two ordinary differential equations. It is usually assumed that the growth rate of each species depends linearly on the two population sizes. The resulting system can be analyzed qualitatively to study whether there is an equilibrium solution in which the species co-exist (see, for example, [4], pp. 53365, or [5], pp. 46-50). It is also possible to give a criterion for the asymptotic stability of such an equilibrium solution [2]. In particular, it is easy to choose the coefficients in such a mathematical model so as to violate the controversial biological principle of competitive exclusion. In this note, we give conditions under which the principle of competitive exclusion is valid mathematically for this particular model. Since experimental results have been reported under which two species do co-exist even though these conditions are satisfied, we are led to the conclusion that the “linear” mathematical *This research was supported by the National Science Foundation, Grant No. GP-28267.
Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1974/4

Y1 - 1974/4

N2 - The population sizes of two species competing for the same food supply or living space are often modelled by a pair of ordinary differential equations. If the growth rates of the two population sizes are linear in the population sizes, then coexistence in stable equilibrium implies qualified competition at equilibrium, in the sense that the effect of the competition is to increase the total population. Since experiments indicate that a stable equilibrium with unqualified competition is possible, this suggests that non-linear growth rates would give a more accurate model. An example is given of a model with non-linear growth rates which exhibits a stable equilibrium with unqualified competition.

AB - The population sizes of two species competing for the same food supply or living space are often modelled by a pair of ordinary differential equations. If the growth rates of the two population sizes are linear in the population sizes, then coexistence in stable equilibrium implies qualified competition at equilibrium, in the sense that the effect of the competition is to increase the total population. Since experiments indicate that a stable equilibrium with unqualified competition is possible, this suggests that non-linear growth rates would give a more accurate model. An example is given of a model with non-linear growth rates which exhibits a stable equilibrium with unqualified competition.

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U2 - 10.1016/0025-5564(74)90045-5

DO - 10.1016/0025-5564(74)90045-5

M3 - Article

AN - SCOPUS:0016184387

VL - 19

SP - 299

EP - 306

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 3-4

ER -