Exploitation is the final type of interaction that can occur between two species. In this case, however, the feedback structure is not symmetrical because one species benefits while the other suffers from the association (see figure). This produces an overall negative feedback structure which has a stable equilibrium under most conditions. Exploitation of one species by another usually involves trophic or feeding interactions; e.g., herbivore-plant or predator-prey interactions. However, some forms of mimicry, where one species assumes the protective display of another and, in so doing, increases the other's likelihood of being eaten by predators, can also be considered a form of exploitation. Once again, the logistic equation can be extended to describe exploitative situations.
The simplest exploitation model is derived from what are usually called the Lotka-Volterra (or LV) equations. Here the per-capita rate of change of the prey is negatively related to the density of consumers, while the consumer per-capita rate of change is positively related to prey density. These effects are incorporated in the logistic model to give
The zero growth isocline for the exploited species is given by
which is identical to the isocline for a competitor. In a way, the interaction is "competitive" because prey are competing with predators for the energy resources held in their own bodies (see Berryman 1981).
The zero growth isocline for the exploiter, on the other hand, is
which is identical to that for an organism receiving benefits from a cooperator; i.e., from the standpoint of the predator, the prey is cooperating by being there for it to eat.
When the isoclines for exploited and exploiting populations are superimposed, a community equilibrium (an intersection) exists over a wide range of conditions; i.e., except when K2, the carrying capacity for the exploiter, has unrealistically large positive or negative values. This is quite unlike competition and cooperation, where community equilibria only exist under a narrow set of conditions. If we plot the dynamic trajectory of a exploitative interaction in phase space, we see that it takes a circular path which may eventually converge on a stable equilibrium (see the gray line in the figure). In a variable environment, however, this circular motion can be sustained indefinitely (figure) (see Berryman 1986, 1991). Thus, exploitative interactions are often characterized by cycles of abundance of exploiting and exploited populations.
Of course, we can also generalize this model to nonlinear intraspecific interactions by including a coefficient of curvature, Q, in the model.
There are some problems with the LV interpretation of two-species interactions: First, the interspecific effect is assumed to be dependent only on the density of the other species. In fact, the likelihood of an individual receiving benefits or being harmed by another species can also depend on the density of its own population. This is most clearly visualized in an exploitative relationship where the per-capita risk of being attacked by predators can be diluted by large prey populations, a phenomenon that has been called the "hiding in a crowd effect", and could well be the reason why schooling, herding and flocking behaviors have evolved in many prey species (Berryman 1981, Berryman et al. 1987). In these cases the per-capita rate of change of the prey would be expected to vary in inverse relationship to the exploiter/exploited (or predator/prey) ratio. A similar argument can be made for an exploiting population. In the LV model, the benefit received by an individual predator is assumed to depend only on the density of the prey population. In many cases, however, we would expect prey to be more difficult to catch when other predators are present, so that the per-capita rate of change of the predator should be related to the ratio of predators to their prey. This relationship is actually present in our original logistic equation because, if you remember, the rate of increase of the population is assumed to be inversely related to the ratio of population density to its carrying capacity, K, which is often considered to be a measure of the exploitable resources available to the population.
The second problem with the LV exploitation equation is that the observed per-capita rate of change can exceed its maximum value, A, when prey density is high. This also means that the equilibrium population density can exceed K, which violates our original definition of carrying capacity.
Let us then return to the elemental logistic R-function and write it as follows
where Ni is the density of species i, Si
is a measure of the space available to that species, and Nj
is the amount of food available in the form of species j;
i.e., Nj is the density of the population in the lower
trophic level. In this variation of the logistic model, we recognize
that the carrying capacity can be determined by both space and
food. Therefore, the parameters 
and 
specify the demands
of the average individual for these resources, respectively. This
equation reduces to the logistic when space and food are constant
because we can then set
It is easy to extend this equation to two species because a second exploitable species is already present in the form of Nj. Writing the equation in terms of the exploiting population, N2, we have
Because space is not a variable in our two-species system, we
can set C22 = 
and, with C21 = 
,
we obtain
The only difference between this model and the LV equation is that the interspecific effect is defined by the ratio of consumer to consumed. We call this the theory of ratio-dependent or RD predator/prey interactions.
An RD equation can also be rewritten for an exploited population
In this case, however, both space and the base trophic level, N0, are not part of the two-species model and must be assumed constant. Accordingly, the equation reduces to the single-species logistic model
At this point, the model for an exploited species does not contain an interspecific effect caused by its exploiter. As discussed earlier, this term should also consist of the predator/prey ratio and should have a negative effect on the per-capita rate of change of the exploited population. Thus, we can write
Notice that the interspecific effect on the exploited population is determined by the same exploiter/exploited ratio as in the exploiter equation. These equations represent, more generally, species that are harmed by the interaction (competing with or being exploited by another species). Hence, we can write the general equation for a harmed species as
and for those that benefit from the interaction (cooperating with or exploiting another species)
Notice that the only difference between the two equations is in the interspecific effect; i.e., beneficial effects are signified by the ratio of the species to its benefactor, Ni / Nj, while harmful effects are represented by the ratio of the harmful species to the one that is harmed, Nj / Ni. Thus, we can model RD competition with two harmful equations, cooperation with two beneficial equations, and exploitation with one harmful equation for the exploited population and one beneficial equation for the exploiting population. It should be emphasized, however, that RD models arise from thinking about trophic relationships and so may not be appropriate for modeling competition or cooperation.
Equilibrium isoclines for RD models are a little more complicated than those for LV models. In particular, the isocline for a population which is harmed by the association has two roots which gives it a humped (parabolic) shape (see figure, blue line). The isocline can be written most easily in terms of the densities of the other species, N2, that produce zero growth in the harmed population, N1,
which is a parabola with roots at zero and A1 / C11 = K1. The isocline for a species receiving benefits is not so complicated, being a convex monotonic curve that reaches an asymptote at A2 / C22 = K2 (figure, red line),
Of course, we can also write these equations in terms of their carrying capacities and can add coefficients of curvature, Q. In addition, we can introduce the notion of alternative food for generalist exploiters in RD exploitation models, which gives us the two-species model
where F is the availability of alternate foods for exploitation.
NOTICE that the same ratio of consumers to their resources appears in both interspecific terms.
Ratio-dependent logistic equations are frequently superior to LV models at describing trophic (predator/prey) interactions. They may also be useful for describing cooperation, or even competition, but we have not been able to test these kinds of interactions with real field data. You should note that a pair of harmful RD interactions can produce very interesting isocline configurations with up to five equilibrium points.
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