15.3: Interspecific (Two Species) Competition
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Interspecific competition may occur when individuals of two separate species share a limiting resource in the same area. If the resource cannot support both populations, then lowered fecundity, growth, or survival may result in at least one species. Interspecific competition has the potential to alter populations, communities, and the evolution of interacting species. An example among animals could be the case of cheetahs and lions; since both species feed on similar prey, they are negatively impacted by the presence of the other because they will have less food, however, they still persist together, despite the prediction that under competition one will displace the other. In fact, lions sometimes steal prey items killed by cheetahs. Potential competitors can also kill each other, in so-called 'intraguild predation'. For example, in southern California coyotes often kill and eat gray foxes and bobcats, all three carnivores sharing the same stable prey (small mammals).
Interspecific Competition: The Lotka-Volterra Model
Given two populations, N1 and N2, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions. Thus the competitive Lotka–Volterra equations are:
Population 1:
\[ \frac{d N_{1}}{d_{t}}=r_{1} N_{1}\left(\frac{K_{1}-N_{1}-a_{12} N_{2}}{K_{1}}\right) \nonumber\]
Population 2:
\[ \frac{d N_{2}}{d_{t}}=r_{2} N_{2}\left(\frac{K_{2}-N_{2}-a_{21} N_{1}}{K_{2}}\right) \nonumber\]
Here, α12 represents the effect species 2 has on the population of species 1 and α21 represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all α-values are positive. Note that each species can have its own growth rate and carrying capacity. The Lotka-Volterra model for competition and how it is used to predict competitive outcomes is described in more detail in the Quantifying Competition section of this chapter.
Resource-Ratio Hypothesis (R* rule)
The R* rule (also called the resource-ratio hypothesis) is a hypothesis that attempts to predict which species will become dominant as the result of competition for resources. It predicts that if multiple species are competing for a single limiting resource, then whichever species can survive at the lowest equilibrium resource level (i.e., the R*) can outcompete all other species . If two species are competing for two resources, then coexistence is only possible if each species has a lower R* on one of the resources. For example, two phytoplankton species may be able to coexist if one is more limited by nitrogen, and the other is more limited by phosphorus.
Consider a community with multiple species. We will assume that each species competes for a single resource, and ignore the effects of interference or apparent competition. Each population increases by consuming resources, and declines when resources are too scarce. For example, we could model their population dynamics as
\[ \frac{d N_{j}}{d t}=N_{j}\left(a_{j} R-d\right) \nonumber\]
\[ \frac{d R}{d t}=r-R \sum_{j} a_{j} N_{j} \nonumber\]
where Nj is the density of species j, R is the density of the resource, a is the rate at which species j eats the resource, d is species j's death rate, and r is the rate at which resources grow when not consumed. It is easy to show that when species j is at equilibrium by itself (i.e., dNj/dt = 0), that the equilibrium resource density, R*j, is
\[ R_{j}^{*}=d / a_{j} \nonumber\]
When R > R*j, species j's population will increase; when R is less than R*j, species j's population will decline. Because of this, the species with the lowest R* will eventually dominate. Consider the two species case, where R*1 < R*2. When species 2 is at equilibrium, R = R*2, and species 1's population will be increasing. When species 1 is at equilibrium, R = R*1, and species 2's population will be decreasing.