Resource competition is an intensely discussed topic in ecological modeling. Based on theoretical reasoning and early modeling suggestions, it has been assumed that the interaction of two species competing for the same resource will necessarily result in the extinction of one of the species. A two-predator-one-prey-system cannot be stable.

However, natural species hardly ever compete on just one resource in reality. The assumption thus is purely theoretic and driven by the abstract concept of stability. In the attempt to show that competition can lead to instability if resources are limited, a range of models has been suggested. One simple example depicts competition among two consumers, \(N_1\) and \(N_2\), without regarding the resource itself.

\[\frac{dN_1}{dt}=r_1*N_1*(1 - \frac{N_1 + \alpha * N_2}{K_1})\] \[\frac{dN_2}{dt}=r_2*N_2*(1 - \frac{N_2 + \beta * N_1}{K_2})\]Parameter \(r_i\) indicates the growth rates, the term \(\frac{N_i}{K_i}\) indicates the carrying capacity of the two consumer populations, and \(\alpha\) and \(\beta\) indicate the competition coefficient with which the two populations act upon each other. An example could be a system with two kinds of squirrels feeding on the same kind of nuts, with the dynamics of the nuts (the reproduction of this resource) not explicitly considered. The plots below show that one of the squirrel-species eventually will disappear.

Volterra, V. (1928). Variations and fluctuations of the number of individuals in animal species living together. Journal de Conseil Int. Explor. Mer. 3, p. 3-51.

A sensitivity analysis of this model - here, as an example, by starting the model repeatedly with different growth rates for predator 2 - supports this assumption.

The following somehow different model subjects the dynamics of the resource to investigation. The resource regrows in this case.

\[\frac{dR}{dt}=\alpha * R*(1-R) - b_1*R*N_1 - b_2*R*N_2\] \[\frac{dN_1}{dt}=c_1*R*N_1 - d_1*N_1\] \[\frac{dN_2}{dt}=c_2*R*N_2 - d_2*N_2\]\(\alpha=\) 0.1 indicates the reproduction rate of the resource; \(b_1 = b_2 =\) 0.03 indicates the death rates of the resource in respect to consumers \(N_1\) and \(N_2\); \(c_1 = c_2 =\) 0.1 indicate the reproduction rates of consumers \(N_1\) and \(N_2\) in respect to their consumption of the resource; and finally \(d_1= \) 0.01 and \(d_2=\) 0.011 indicate the death rates of the consumers \(N_1\) and \(N_2\). As can be seen from the plots below, the small disadvantage of predator 2, due to its slightly larger death rate, makes it prone to extinction.

However, alternative models have been proposed to demonstrate the possibility of coexistence. A broadly discussed model for instance, is the one of Armstrong and McGehee (1980) that shows a coexistence of two species \(X_1\) and \(X_2\) competing on one biotic resource \(R\).

\[\frac{dR}{dt}= R*[r*(1-\frac{R}{K})-\frac{\mu_1*N_1}{R+\Gamma}-\mu_2*N_2]\] \[\frac{dN_1}{dt}= N_1*(-m_1+\frac{c_1*\mu_1*R}{R+\Gamma})\] \[\frac{dN_2}{dt}= N_2*(-m_2+c_2*\mu_2*R)\]where \(m_1\) and \(m_2\) are constant death rates, \(\mu_1\) and \(\mu_2\) are parameters for resource consumption per unit competitor; \(c_1\) and \(c_2\) represent the conversion efficiency of resource biomass into competitor biomass; \(r\) and \(K\) are the resource growth rate and the carrying capacity of the resource respectively; and \(\Gamma\) is a half-saturation constant in the functional response of the first predator.

Parameters as suggested by Armstrong and Mcgehee (1980: 155) are as follows: \(m_1\) = 0.1, \(c_1\) = 0.3, \(\mu_1\)= 0.5, \(\Gamma\) = 50, \(r\) = 0.1, \(K\) = 300, \(m_2\) = 0.11, \(c_2\) = 0.33, \(\mu_2\)= 0.003. The initial values are: \(N1\) = 1, \(R\) = 400 , \(N_2\) = 0, which marks a point near to the two-species periodic orbit of \(N_1\) and \(R\). A small fraction (\(N_2\) = 0.01) of predator 2 is added only at \(t\) = 194, which readily invades the limit cycle, as can be seen in the plots below, particularly in the enlargement of the predator 2 dynamics to the right.

Armstrong R.A. and McGehee R. (1980). Competitive exclusion. American naturalist 115/2. p. 151-170.