1.3: A Model of Species Competition

A model of species competition

Suppose that two species compete for the same resources. To build a model, we can start with logistic equations for both species. Different species would have different growth rates and different carrying capacities. If we let \(N_{1}\) and \(N_{2}\) be the number of individuals of species one and species two, then

\[\begin{aligned} &\frac{d N_{1}}{d t}=r_{1} N_{1}\left(1-N_{1} / K_{1}\right) \\[4pt] &\frac{d N_{2}}{d t}=r_{2} N_{2}\left(1-N_{2} / K_{2}\right) \end{aligned} \nonumber \]

These are uncoupled equations so that asymptotically, \(N_{1} \rightarrow K_{1}\) and \(N_{2} \rightarrow K_{2}\). How do we model the competition between species? If \(N_{1}\) is much smaller than \(K_{1}\), and \(N_{2}\) much smaller than \(K_{2}\), then resources are plentiful and populations grow exponentially with growth rates \(r_{1}\) and \(r_{2}\). If species one and two compete, then the growth of species one reduces resources available to species two, and viceversa. Since we do not know the impact species one and two have on each other, we introduce two additional parameters to model the competition. A reasonable modification that couples the two logistic equations is

\[\begin{align} \frac{d N_{1}}{d t} &=r_{1} N_{1}\left(1-\frac{N_{1}+\alpha_{12} N_{2}}{K_{1}}\right), \\[4pt] \frac{d N_{2}}{d t} &=r_{2} N_{2}\left(1-\frac{\alpha_{21} N_{1}+N_{2}}{K_{2}}\right), \end{align} \nonumber \]

where \(\alpha_{12}\) and \(\alpha_{21}\) are dimensionless parameters that model the consumption of species one’s resources by species two, and vice-versa. For example, suppose that both species eat exactly the same food, but species two consumes twice as much as species one. Since one individual of species two consumes the equivalent of two individuals of species one, the correct model is \(\alpha_{12}=2\) and \(\alpha_{21}=1 / 2\). Another example supposes that species one and two occupy the same niche, consume resources at the same rate, but may have different growth rates and carrying capacities. Can the species coexist, or does one species eventually drive the other to extinction? It is possible to answer this question without actually solving the differential equations. With \(\alpha_{12}=\alpha_{21}=1\) as appropriate for this example, the coupled logistic equations (1.3.1 and 1.3.2) become

\[\frac{d N_{1}}{d t}=r_{1} N_{1}\left(1-\frac{N_{1}+N_{2}}{K_{1}}\right), \quad \frac{d N_{2}}{d t}=r_{2} N_{2}\left(1-\frac{N_{1}+N_{2}}{K_{2}}\right) \nonumber \]

For sake of argument, we assume that \(K_{1}>K_{2}\). The only fixed points other than the trivial one \(\left(N_{1}, N_{2}\right)=(0,0)\) are \(\left(N_{1}, N_{2}\right)=\left(K_{1}, 0\right)\) and \(\left(N_{1}, N_{2}\right)=\left(0, K_{2}\right) .\) Stability can be computed analytically by a two-dimensional Taylor-series expansion, but here a simpler argument can suffice. We first consider \(\left(N_{1}, N_{2}\right)=\left(K_{1}, \epsilon\right)\), with \(\epsilon\) small. Since \(K_{1}>K_{2}\), observe from (1.3.3) that \(\dot{N}_{2}<0\) so that species two goes extinct. Therefore \(\left(N_{1}, N_{2}\right)=\left(K_{1}, 0\right)\) is a stable fixed point. Now consider \(\left(N_{1}, N_{2}\right)=\) \(\left(\epsilon, K_{2}\right)\), with \(\epsilon\) small. Again, since \(K_{1}>K_{2}\), observe from (1.3.3) that \(\dot{N}_{1}>0\) and species one increases in number. Therefore, \(\left(N_{1}, N_{2}\right)=\left(0, K_{2}\right)\) is an unstable fixed point. We have thus found that, within our coupled logistic model, species that occupy the same niche and consume resources at the same rate cannot coexist and that the species with the largest carrying capacity will survive and drive the other species to extinction. This is the so-called principle of competitive exclusion, also called \(K\)-selection since the species with the largest carrying capacity wins. In fact, ecologists also talk about \(r\)-selection; that is, the species with the largest growth rate wins. Our coupled logistic model does not model \(r\)-selection, demonstrating the potential limitations of a too simple mathematical model.

For some values of \(\alpha_{12}\) and \(\alpha_{21}\), our model admits a stable equilibrium solution where two species coexist. The calculation of the fixed points and their stability is more complicated than the calculation just done, and I present only the results. The stable coexistence of two species within our model is possible only if \(\alpha_{12} K_{2}<K_{1}\) and \(\alpha_{21} K_{1}<K_{2}\).