Species Competition and Predation - Ecological Systems - page 228

Lotka-Volterra equations are in principle most accurate, given that they do not
explicitly represent the dynamics of resources or natural enemies outside the guild
in question. In these cases, they are valid for short-term as well as long-term
predictions. In these cases of direct interference, conditions ( Eq. 13.2 ) and
( Eq. 13.3 ) then apply, respectively, to exclusion and coexistence, showing in
particular that coexistence requires each species to interfere more strongly intraspe-
cifically than interspecifically.
When the Lotka-Volterra equations are used to represent trophic interactions
such as those of the food web Fig. 13.1 , the coefficients of density dependence a can
be defined in terms of quantities that summarize the interactions in the food web:
niche overlap, r , and species-level average fitness, k [ 17 ]. The niche overlap, r ,
between any pair of species is a measure of the relative strength of the density-
dependent feedback between versus within species through resources and through
predators. This quantity takes the value 1 when there is complete overlap, and zero
when then there is no overlap. No overlap occurs when the members of the pair of
species under consideration do not share resources and do not share predators.
Thus, they do not have arrows to or from any of the same resources or predators in
Fig. 13.1 . With complete overlap, they have arrows to and from all of the same
resources and predators, and these predators and resources are of the same relative
importance for each species in the guild. If the resources and predators vary in
importance for different species, but nevertheless both species under consideration
are affected by them, then r will be between 0 and 1. Figure 13.2 gives various
scenarios for different strengths of niche overlap.
The average fitness measure, k , for any given species measures its ability to meet
its energy needs and avoid predation when all species in the guild are at low density
and thus not providing any feedback through density changes. This measurement is
also expressed in special units, namely, in units of average sensitivity of the per
capita growth rate of the species to changes in resources and predators. These
quantities r and k now relate to the coefficients of density dependence according to
the relationship

a ij
a jj ¼

k j
k i r:

(13.4)

Thus, the ratio of interspecific to intraspecific density dependence for species j' s
impact on species i is equal to the ratio of the fitness of species j to species i ,
multiplied by the overlap measure. This relationship is correct regardless of how
many species are present in the guild in question. However, when there are just two
species, the condition that the ratio ( Eq. 13.4 ) be greater than 1 is the condition
( Eq. 13.2 ) for species j to exclude species i . The ratio being less than 1 means that
exclusion does not occur. These conditions in terms of the fitness ratio, multiplied
by the overlap measure, are also strongly intuitive and instructive in terms of how
stable coexistence occurs.
Consider the case of complete niche overlap, r = 1. Then the formula ( Eq. 13.4 )
implies that whichever species has the larger fitness will exclude the other species.