we constantly see a great fluctuation in numbers, the parasite rapidly increasing immediately after the increase of the host species, overtaking it numerically, and reducing it to the bottom of another ascending period of development. (L. O. HOWARD 1897)
Living organisms exist within webs of interactions with other living creatures, the most important of which involve eating or being eaten (trophic interactions). Complex interactions among several species are called food webs while simpler linear ones within a particular food web are called food chains. In this course we will develop 3 approaches to modeling the interaction between members of a food chain. For convenience we will think about a simple food chain consisting of a plant, an herbivore (insect pest), and a carnivore (e.g., an insect parasitoid) (see Fig. 1). Each species can be affected by its own density through intraspecific competition for fixed resources (the negative link from the population to itself), and by other species directly below or above it in the trophic chain. Notice also that the interaction between trophic levels gives rise to a negative feedback loop (the product of + and - interactions between species). Finally, we can see that the most general model for a member of a food chain is that for the central species, in our case the herbivore (H), for it is this population that is affected by species both below (plant, P) and above it in the chain (consumer, C). Hence, our main objective will be to obtain a model for the herbivore population H in Figure 1.
Fig. 1. A food chain consisting of three trophic levels, plant (P), herbivore (H) and carnivore (C), and their interactions and feedbacks.
The model for a population in a food chain is defined by the quantitative relationships indicated by the arrows in Figure 1. For example, the dynamics of the herbivore population are defined, in general terms, by a mathematical function describing how the densities of plant, herbivore and carnivore populations affect changes in herbivore density. For example,
is a continuous-time differential equation defining a change in herbivore density (dH) over a small unit of time (dt) as a function (f) of the densities of plants, herbivores and carnivores.
There are three basic ways in which the modeling of trophic dynamics has been approached: Some have appealed to the laws of physics as a metaphor for population interactions and have built their models around the law of "mass action". Others have taken an energy-flow approach by modeling the general process of consumption, or the flow of biomass through the food chain. Yet others model the effects of plant and carnivore densities on the reproduction and survival of the herbivore population. We will examine each of these approaches to modeling predator-prey systems in this course.
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©1998 Alan A. Berryman