Competition
Competition occurs when organisms reduce one another's access to something that limits population growth. The limiting factor may be food, water, light, nutrients, territory, nesting sites or another resource.
Start with one species
Without another species, logistic growth is
\[\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{K_1}\right).\]The term \(N_1/K_1\) represents the density-dependent pressure created by species 1 on itself. As \(N_1\) approaches \(K_1\), its growth slows.
Now introduce a competitor
Suppose species 2 uses some of the same limiting resource. From the viewpoint of species 1, an individual of species 2 may not have exactly the same competitive effect as an individual of species 1.
We therefore convert species 2 into an equivalent amount of species-1 competition using a coefficient \(\alpha_{12}\). The effective crowding experienced by species 1 becomes
\[N_1+\alpha_{12}N_2.\]This gives
\[\boxed{\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1+\alpha_{12}N_2}{K_1}\right)}.\]What exactly does \(\alpha_{12}\) mean?
The coefficient \(\alpha_{12}\) measures the effect of one individual of species 2 on species 1, relative to the effect of one additional individual of species 1 on itself.
| Value | Interpretation for species 1 |
|---|---|
| \(\alpha_{12}=1\) | one species-2 individual has the same competitive effect as one species-1 individual |
| \(\alpha_{12}=0.5\) | two species-2 individuals have the same effect as approximately one species-1 individual |
| \(\alpha_{12}=2\) | one species-2 individual has the effect of approximately two species-1 individuals |
| \(\alpha_{12}=0\) | species 2 has no competitive effect on species 1 in this model |
The effect need not be symmetrical
Species 1 may affect species 2 differently. We therefore use a second coefficient \(\alpha_{21}\).
\[\boxed{\frac{dN_2}{dt}=r_2N_2\left(1-\frac{N_2+\alpha_{21}N_1}{K_2}\right)}.\]The complete Lotka–Volterra competition model is the pair of coupled equations above.
Meaning of every parameter
| Symbol | Meaning |
|---|---|
| \(N_1,N_2\) | population sizes of species 1 and species 2 |
| \(r_1,r_2\) | intrinsic growth-rate parameters |
| \(K_1,K_2\) | carrying capacities each species would have when alone |
| \(\alpha_{12}\) | competitive effect of species 2 on species 1, measured in species-1 equivalents |
| \(\alpha_{21}\) | competitive effect of species 1 on species 2, measured in species-2 equivalents |
Competition changes per-capita growth
Dividing the first equation by \(N_1\) gives
\[\frac{1}{N_1}\frac{dN_1}{dt}=r_1\left(1-\frac{N_1+\alpha_{12}N_2}{K_1}\right).\]Increasing either \(N_1\) or \(N_2\) lowers the per-capita growth rate of species 1. This is the mathematical signature of competition in this model.
What is a nullcline?
A nullcline is a set of population combinations for which one species is momentarily neither increasing nor decreasing.
For species 1, setting its growth rate to zero gives the non-zero nullcline
\[N_1+\alpha_{12}N_2=K_1,\]or
\[N_2=\frac{K_1-N_1}{\alpha_{12}}.\]Its intercepts are \(K_1\) on the \(N_1\)-axis and \(K_1/\alpha_{12}\) on the \(N_2\)-axis.
For species 2,
\[N_2+\alpha_{21}N_1=K_2,\]with intercepts \(K_2/\alpha_{21}\) on the \(N_1\)-axis and \(K_2\) on the \(N_2\)-axis.
How to read a nullcline
Below the species-1 nullcline, effective competition is less than \(K_1\), so species 1 increases. Above it, competition is greater than \(K_1\), so species 1 decreases.
The same reasoning applies to species 2 using its own nullcline.
The four classical outcomes
The relative positions of the two nullclines determine the qualitative outcome.
| Outcome | Biological interpretation |
|---|---|
| species 1 excludes species 2 | species 1 can persist under the competitive pressure created by species 2, but species 2 cannot persist under species 1 |
| species 2 excludes species 1 | the reverse situation |
| stable coexistence | each species limits itself more strongly than it limits the other species, in the relevant scaled sense |
| unstable coexistence | both single-species states can be stable; the eventual winner depends on starting populations |
Stable coexistence
Stable coexistence occurs when each species can increase while rare in a population dominated by the other species. For this model, the conditions can be written as
\[K_1\gt\alpha_{12}K_2,\qquad K_2\gt\alpha_{21}K_1.\]Equivalently,
\[\frac{K_1}{\alpha_{12}}\gt K_2,\qquad \frac{K_2}{\alpha_{21}}\gt K_1.\]Coexistence nullclines
Finding the coexistence equilibrium
At a positive coexistence equilibrium, both species have zero growth simultaneously:
\[N_1+\alpha_{12}N_2=K_1,\qquad \alpha_{21}N_1+N_2=K_2.\]Solving these equations gives
\[\boxed{N_1^*=\frac{K_1-\alpha_{12}K_2}{1-\alpha_{12}\alpha_{21}}},\qquad \boxed{N_2^*=\frac{K_2-\alpha_{21}K_1}{1-\alpha_{12}\alpha_{21}}}.\]These values are biologically meaningful as a coexistence equilibrium only when they lie in the positive population region and have the appropriate stability.
Competitive exclusion
If one species' nullcline lies outside the other's in the relevant way, that species can exclude its competitor.
For species 1 to exclude species 2 in the classical model, a typical condition is
\[K_1\gt\alpha_{12}K_2,\qquad K_2\lt\alpha_{21}K_1.\]Species 1 can invade the species-2 equilibrium, while species 2 cannot invade the species-1 equilibrium.
Why invasion when rare is so useful
Suppose species 2 is alone at its carrying capacity \(K_2\). Introduce a very small amount of species 1. Because \(N_1\) is nearly zero, species 1 initially has positive per-capita growth when
\[1-\frac{\alpha_{12}K_2}{K_1}\gt0,\]which is exactly
\[K_1\gt\alpha_{12}K_2.\]This gives an intuitive test: can species 1 increase when rare?
Unstable coexistence and priority effects
Another possibility is that neither species can invade when rare. Then the coexistence intersection is unstable, while either single-species equilibrium can be stable.
The outcome depends on initial population sizes. Whichever species begins with a sufficient numerical advantage may drive the other to exclusion. This is called a priority effect.
Population trajectories under stable coexistence
Competition does not require direct fighting
Competition can be indirect. Two plant species may never physically interact but can reduce each other's growth by drawing water or nutrients from the same soil.
This is called exploitative competition. Direct interference, such as territorial aggression or chemical inhibition, is another mechanism.
Intraspecific and interspecific competition
Intraspecific competition occurs within the same species and is represented by the species' own density term. Interspecific competition occurs between species and is represented by the terms containing the competition coefficients.
The balance between these two forms of competition is central to whether coexistence is possible.
Resource partitioning and coexistence
Species that use resources differently may exert weaker competitive effects on one another. For example, plants with roots concentrated at different soil depths may compete less strongly for water.
In the model this can correspond to smaller competition coefficients, which can make stable coexistence more likely.
Connection with ecological niches
Competition theory helps formalise the idea of niche differences. If species depend on exactly the same limiting resource in exactly the same way, persistent coexistence can be difficult. Differences in resource use, habitat, timing or other ecological traits can reduce interspecific competition.
Assumptions and limitations
The basic Lotka–Volterra competition model assumes constant parameters, logistic self-limitation, fixed competition coefficients and homogeneous populations. It does not explicitly represent the resource being competed for.
Real competitive effects can change with density, environment, age, stage, space or time. More detailed models may therefore represent resources explicitly or allow interaction strengths to vary.
Stochastic competition
In small populations, random births and deaths can change competitive outcomes. A species predicted to coexist deterministically may disappear by chance. Environmental variation can also alter carrying capacities and competition strengths through time.
Stochastic models therefore replace a single guaranteed outcome with probabilities of coexistence, exclusion or extinction.