The broader ecological setting

Like other ecological interactions, host-parasite dynamics take place in a broader ecological setting. Grenfell (1988, 1992) constructed a scenario that connects parasite dynamics and grazing systems, with some interesting and remarkable conclusions. Here I describe the key features of a model that is similar, but not identical, to those used by Grenfell and, as before, you are encouraged to consult the original papers after this development.

We imbed the host-parasite dynamics in an herbivore-plant interaction. We allow V(t) to denote the biomass of vegetation at time t and assume that, in the absence of the herbivore, the vegetation grows logistically. We assume that the herbivores consume vegetation according to a type II functional response (Grenfell (1988, 1992) assumed a type III functional response, which makes the analysis much more complicated - and interesting - so I encourage you to read his papers) and that herbivore per capita reproduction is proportional to this consumption. We assume that the free living stage of the parasite can be ignored (Grenfell includes it), so that the simpler hostparasite dynamics apply. When the plant-herbivore dynamics are coupled to the host-parasite dynamics, we end up with the system of equations dV — rV (l- V-Y eHV

In these equations, the new terms should be easily interpretable by you. The question, of course, is what does one do with them now that one has them?

Let us start by thinking about the steady states; we do so by holding the herbivore population size fixed. Then the equation for vegetative biomass is uncoupled from all of the others and the steady state of the vegetation is determined by the solution of rV(1 -(V/K)) = eHV/(Vo + V). The graphical solution of this equation is shown in Figure 5.15. If we think that e (the maximum per capita consumption rate of the herbivores) and V0 (the level of vegetative biomass at which 50% of this maximum is reached) are fixed by the biology of the system, then steady states of the vegetation are set by the steady state level of the herbivores, HT. The origin, V= 0, is always a steady state but depending upon the values of herbivores, there may be another steady state.

Figure 5.15. The graphical solution of rV(1 — (V/K)) — eHV/( V0 + V) for three values of H.

H increasing

Determine the stability of the steady states of V at the origin and at its positive value, when appropriate.

However, the herbivore population level cannot be manipulated at will, as I have just done. Rather it comes out of the simultaneous solution of the three nonlinear equations obtained by setting the left hand side of Eq. (5.42) equal to 0. We are not going to do that (once again, look at Grenfell's papers), but make the following observation: host populations will decline as parasite populations increase. Thus, the stability of the plant-herbivore system depends upon the parasites: the parasites function as top predator in this system! This means, for example, that one wants to think carefully about actions that will reduce parasite populations. For example, the indiscriminate application of antihelminth agents might have exactly the undesired effect of causing the entire system to crash.

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