One way to approach the question posed in the title of this Web Module is to assume that in the absence of competition, a species grows according to the logistic equation. Then, to model the effect of a competitor species, the logistic equation can be altered in a way that mimics the effects of competition. This was the approach taken in the papers by A. J. Lotka (1932) and Vito Volterra (1926) mentioned in Chapter 11.
Recall from Chapter 9 that in the logistic equation, the rate at which a population changes in size (dN/dt) is:
where N is population density, r is the intrinsic rate of increase (the maximum possible per capita growth rate for the species, achieved only under ideal conditions), and K is the density at which the population stops increasing in size (which can be interpreted as the carrying capacity of the population).
As we see in Chapter 11, competition deprives species of resources and hence reduces per capita population growth rates. Thus, the presence of a competitor should reduce the per capita growth rate of the original species.
To add the effects of competition to the logistic equation, we note that in the absence of a competitor species, the per capita growth rate can be found by dividing both sides of Equation 1 by N:
This growth rate is close to its maximum value of r when N is small, and it equals zero when N equals K.
To account for effects of a competitor species, the per capita growth rate of the logistic equation can be reduced from:
to:
where N2 is the density of the competitor species (species 2) and α is a constant (greater than zero) that indicates how strong a competitive effect species 2 has on the original species, called a competition coefficient.
Putting it all together, we can use Equation 2 to modify the logistic equation so that it includes the effects of competition:
Finally, to set the stage for modeling the effects of two competing species on each other, we indicate the density, growth rate, and carrying capacity of the original species (species 1) by N1, r1, and K1, resulting in:
This equation is the first of the two Lotka–Volterra competition equations (see Equation 11.1 in the textbook); the second Lotka–Volterra competition equation is derived in a similar way.
