Department of Biological Sciences

University of South Carolina

- Competition occurs when organisms consume a
**limiting resource**. **Intraspecific**competition - among individuals of the**same species**. This sort of competition results in**logistic growth**.**Interspecific**competition - among individuals of**different species**.- Lotka (1925) and Volterra (1926) proposed a model of competition derived
from the logistic equation. Consider two species (N
_{1},N_{2}) with growth rates (r_{1},r_{2}), carrying capacities (K_{1},K_{2}), and interaction strengths (a,b): - dN
_{1}/dt = r_{1}*N_{1}*(1 - N_{1}/K_{1}- a*N_{2}/K_{1})

dN_{2}/dt = r_{2}*N_{2}*(1 - N_{2}/K_{2}- b*N_{1}/K_{2}) - In this
model, each species reduces the carrying capacity of the
environment for the other. Species 2 reduces the carrying capacity of species 1
by a*N
_{2}. This has the effect of reducing the growth rate of species 1 (dN_{1}/dt). Species 1 has a similar effect on the carrying capacity and growth rate of species 2. - Under what conditions is coexistence possible? In a habitat with 2 species, presumably there is a mosaic of habitat patches, some empty, some with one species, some with the other, and some with both. It is unlikely that both species simultaneously colonize empty patches, so the patches with two species in them resulted from the invasion of occupied patches. This invasion could occur as species 1 invades habitat occupied by species 2 or species 2 invades habitat occupied by species 1.
- Invasion can occur if the invader has a positive growth rate when the
habitat patch is occupied fully by the other species (the occupier population
is equal to carrying capacity). We assume that the invader is rare, so it does
not affect its own growth: We also assume that r
_{1}and N_{1}are positive, so they have no effect on our calculations. The only factors that determine whether growth is positive or negative are in the expression (1 - N_{1}/K_{1}- a*N_{2}/K_{1}). We assume N_{1}= 0 (it is invading and is so rare as to be negligible) and N_{2}= K_{2}(species 2 is at its carrying capacity because it already occupies the patch) :

dN_{1}/dt > 0 (species 1 growth rate is positive during invasion)

therefore (1 - a*K_{2}/K_{1}) > 0

therefore 1/a > K_{2}/K_{1} - similar logic can be applied to the invasion by species 2:

dN_{2}/dt > 0 (species 2 growth rate is positive during invasion)

therefore (1 - b*K_{1}/K_{2}) > 0

therefore K_{2}/K_{1}> b - The conditions for coexistence are therefore (each species can invade the
other's habitat):

1/a > K_{2}/K_{1}> b

Consider two cases:**Strong competition**(a and b large (example: a=b=0.9)):

1/0.9 > K_{2}/K_{1}> 0.9

1.1 > K_{2}/K_{1}> 0.9

This is very restrictive. If competition is strong, coexistence can only occur if the carrying capacities of the species are very close to one another. Given that carrying capacities fluctuate from year to year,**coexistence is very unlikely in cases of strong competition**.

**Weak competition**(a and b small (example: a = b = 0.1))

1/0.1 > K_{2}/K_{1}> 0.1

10 > K_{2}/K_{1}> 0.1

This is a very loose constraint. If competition is weak, coexistence can occur over a wide range of carrying capacities. Even with year to year variation in carrying capacities, this constraint is so broad that**coexistence is very likely in cases of weak competition**.

**Graphical Analysis of Competition**- We can simultaneously examine the populations of both species in a
**phase plot**in which we graph the density of species 2 against the density of species 1. On this graph we can plot the conditions of equilibrium for each species. - Equilibrium for species 1:
dN This is a straight line with endpoints K_{1}= 0 = 1 - N_{1}/K_{1}- a* N_{2}/K_{1}._{1}and K_{1}/ a. It is called a**nullcline**or**zero growth isocline**It is plotted as the red line on the graphs. This line is the carrying capacity of the environment as a function of the densities of species 1 and 2. When N_{1}lies is above and to the right of this line, species 1 will decline (yellow region on graph). When N_{1}lies below and to the left of this line (blue region on graph), species 1 will grow. The carrying capacities of both species are at a density of 1, and a and b are both less than 1 (a = 0.5, b=0.4).

The black line is the equivalent carrying capacity for species 2. When N_{2}is above and to the right of this line, species 2 will decline (yellow region on graph). In the blue region below and to the left of this line, species 2 will grow because it is below carrying capacity.

We model the invasion of species 1 into species 2 habitat and species 2 into species 1 habitat. The blue line represents species 1 invading species 2. It starts at (N_{1}=0.01, N_{2}=1 (K)). The black line represents species 2 invading species 1. It starts at (N_{1}=1 (K), N_{2}=0.01). In this case, the intensity of competition is weak, and each species can invade the other successfully. On the graph, the endpoint of the trajectories is at the crossing point of the carrying capacity nullclines. The red arrows on the graph represent the trajectories that would be followed by populations starting with other initial abundances.

Plotted as numbers versus time, the invasion of species 1 into species 2 habitat can be seen as an increase in species 1 (blue line) and a decrease in species 2 (black line), and both eventually level off at intermediate densities, indicating coexistence. Note that under conditions of coexistence, both species are reduced to densities below their original carrying capacities.

- We can simultaneously examine the populations of both species in a

- Lotka (1925) and Volterra (1926) proposed a model of competition derived
from the logistic equation. Consider two species (N