**Introduction:** Interspecific competition refers to the competition
between two or more species for some limiting resource. This limiting resource
can be food or nutrients, space, mates, nesting sites-- anything for which
demand is greater than supply. When one species is a better competitor,
interspecific competition negatively influences the other species by reducing
population sizes and/or growth rates, which in turn affects the population
dynamics of the competitor. The Lotka-Volterra model of interspecific competition
is a simple mathematical model that can be used to understand how different
factors affect the outcomes of competitive interactions.

**Importance: **Competitive interactions between organisms can have
a great deal of influence on species evolution, the structuring of communities
(which species coexist, which don't, relative abundances, etc.), and the
distributions of species (where they occur). Modeling these interactions
provides a useful framework for predicting outcomes.

**Question: **Under what circumstances can two species coexist? Under
what circumstances does one species outcompete another?

**Variables:**

N | population size |

t | time |

K | carrying capacity |

r | intrinsic rate of increase |

a | competition coefficient |

**Methods:** The logistic equation below models
a rate of population increase that is limited by __intraspecific__ competition
(i.e., members of the same species competing with one another).

The first term on the right side of the equation
(*rN*, the intrinsic rate of increase [*r*] times the population
size [*N*]) describes a population's growth in the absence of competition.
The second term ([*K*-*N*] / *K*) incorporates intraspecific
competition, or density-dependence, into the model, and takes a value between
0 and 1. As population size (*N*) approaches carrying capacity (*K*),
the numerator (*K*-*N*) becomes smaller but the denominator (*K*)
stays the same and the second term decreases. The addition of this term
describes a rate of population growth that slows down as population size
increases, until the population reaches its carrying capacity. In other
words, the growth curve described by the logistic equation is sigmoidal,
and the rate of growth depends on the density of the population.

The logistic equation can be modified to include
the effects of __interspecific__ competition as well as intraspecific
competition. The Lotka-Volterra model of interspecific competition is comprised
of the following equations for population 1 and population 2, respectively:

The big difference (other than the subscripts denoting
populations 1 and 2) is the addition of a term involving the competition
coefficient, a. The
competition coefficient represents the effect that one species has on the
other: a_{12}represents
the effect of species 2 on species 1, and a_{21}represents
the effect of species 1 on species 2 (the first number of the subscript
always refers to the species being affected). In the first equation of
the Lotka-Volterra model of interspecific competition, the effect that
species 2 has on species 1 (a_{12})
is multiplied by the population size of species 2 (*N*_{2}).
When a_{12 }is
< 1 the effect of species 2 on species 1 is less than the effect of
species 1 on its own members. Conversely, when a_{12}is
> 1 the effect of species 2 on species 1 is greater than the effect of
species 1 on its own members. The product of the competition coefficient,
a_{12},
and the population size of species 2, *N*_{2}, therefore represents
the effect of an equivalent number of individuals of species 1, and is
included in the intraspecific competition, or density-dependence, term.
The a_{21}*N*_{1
}term
in the second equation is interpreted in the same way.

To understand the predictions of the model it is helpful to look at
graphs that show how the size of each population increases or decreases
when we start with different combinations of species abundances (i.e.,
low *N*1 low *N*2, high *N*1 low *N*2, etc.). These
graphs are called state-space graphs, in which the abundance of species
1 is plotted on the *x*-axis and the abundance of species 2 is plotted
on the *y*-axis. Each point in a state-space graph represents a combination
of abundances of the two species. For each species there is a straight
line on the graph called a zero isocline. Any given point along, for example,
species 1's zero isocline represents a combination of abundances of the
two species where the species 1 population does not increase or decrease
(the zero isocline for a species is calculated by setting *dN/dt*,
the growth rate, equal to zero and solving for *N*). The two graphs
below show the zero isoclines for species 1 (left, solid yellow line) and
species 2 (right, dashed pink line). (All graphs adapted from Begon et
al. [1996] and Gotelli [1998])

Note that the zero isoclines divide each graph into two parts. Below
and to the left of the isocline the population size increases because the
combined abundances of both species are less than the carrying capacity
of the one, while above and to the right the population size decreases
because the combined abundances are greater than the carrying capacity.
For the graph of the isocline of species 1, the isocline intersects the
graph on the *x*-axis when *N*1 reaches its carrying capacity
(*K*_{1}) and no individuals of species 2 are present. The
isocline intersects the graph on the *y*-axis at *K*_{1}/a_{12},
when the carrying capacity of species 1 is filled by the equivalent number
of individuals of species 2 and no individuals of species 1 are present.
The intersections of the isocline for species 2 are essentially the same,
but on different axes.

These two graphs illustrate what happens to a population when it is below or above its isocline, but they only account for one isocline at a time. The following four graphs include both species' isoclines, and illustrate the possible outcomes of interspecific competition depending on where each species' isocline lies in relation to the other. In each graph, the solid yellow line represents the isocline of species 1, and the dashed pink line represents the isocline of species 2. The thick black arrows represent the joint trajectory of the two populations, and the thinner colored arrows indicate the trajectories of the individual populations.

**Interpretation:** The first scenario is one in which the isocline
for species 1 is above and to the right of the isocline for species two.
For any point in the lower left corner of the graph (i.e., any combination
of species abundances), both populations are below their respective isoclines
and both increase. For any point in the upper right corner of the graph,
both species are above their respective isoclines and both decrease. For
any point in between the two isoclines, species 1 is still below its isocline
and increases, while species 2 is above its isocline and decreases. The
joint movement of the two populations (thick black arrows) is down and
to the right, so species 2 is driven to extinction and species 1 increases
until it reaches carrying capacity (*K*_{1}). The open circle
at this point represents a stable equilibrium. In this scenario, species
1 always outcompetes species 2, and is referred to as the competitive exclusion
of species 2 by species 1.

The second scenario is the opposite of the first; the isocline of species
2 is above and to the right of the isocline for species 1. This graph can
be interpreted in much the same way as the previous one, except that the
joint trajectory of the two populations when starting in between the isoclines
is up and to the left. In this case species 2 always outcompetes species
1, and species 1 is competitively excluded by species 2.

In the third scenario, the isoclines of the two species cross one another.
Here, the carrying capacity of species 1 (*K*_{1}) is higher
than the carrying capacity of species 2 divided by the competition coefficient
(*K*_{2}/a_{21}),
and the carrying capacity of species 2 (*K*_{2}) is higher
than the carrying capacity of species 1 divided by the competition coefficient
(*K*_{1}/a_{12}).
Below both isoclines and above both isoclines the populations increase
or decrease as in the first two scenarios, and there is an unstable equilibrium
point (closed circle) where the isoclines intersect. For points above the
dashed pink line (species 2 isocline) and below the solid yellow line (species
1 isocline), the outcome is the same as in the first scenario: competitive
exclusion of species 2 by species 1. On the other hand, for points above
the solid yellow line (species 1 isocline) and below the dashed pink line
(species 2 isocline), the outcome is the same as in the second scenario:
competitive exclusion of species 1 by species 2. The two stable equilibrium
points are again represented by open circles. In this scenario, the outcome
depends on the initial abundances of the two species.

Finally, in the fourth scenario we can see that the isoclines cross one another, but in this case both species' carrying capacities are lower than the other's carrying capacity divided by the competition coefficient. Again, below both isoclines the populations increase and above both isoclines the populations decrease. In this case, however, when the populations of the two species are between the isoclines their joint trajectories always head toward the intersection of the isoclines. Rather than outcompeting one another, the two species are able to coexist at this stable equilibrium point (open circle). This is the outcome regardless of the initial abundances.

**Conclusions:** The Lotka-Volterra model of interspecific competition
has been a useful starting point for biologists thinking about the outcomes
of competitive interactions between species. The assumptions of the model
(e.g., there can be no migration and the carrying capacities and competition
coefficients for both species are constants) may not be very realistic,
but are necessary simplifications. A variety of factors not included in
the model can affect the outcome of competitive interactions by affecting
the dynamics of one or both populations. Environmental change, disease,
and chance are just a few of these factors.

**Additional Question:**

1. The Lotka-Volterra model predicts that stable coexistence of two
species is possible only when __intraspecific__ competition has a greater
effect than __interspecific__ competition. Why would this be the case?

**Sources:** Begon, M., J. L. Harper, and C. R. Townsend. 1996.
__Ecology:
Individuals, Populations, and Communities__, 3rd edition. Blackwell Science
Ltd. Cambridge, MA.

Gotelli, N. J. 1998. __A Primer of Ecology__, 2nd edition. Sinauer
Associates, Inc. Sunderland, MA.

*copyright 1999, M. Beals, L. Gross, S. Harrell*