Populations and Simple Food Webs as Energy Filters. II. Two-Species Systems

In these two papers I have restricted myself to linear dynamic models of populations and simple food webs as energy filters. This paper deals exclusively with two-species systems. Three cases are briefly examined: simple predator-prey, simple interspecific competition, and combined predator-prey and competition. Several novel emergent dynamic properties of these two-species systems are revealed. In simple predator-prey two-species food webs, the study predicts that: 1. The system is most oscillatory (smallest damping ratio) when the intrinsic rates of predator and prey are equal, other parameters remaining unchanged. 2. The smaller the intrinsic rate parameters of both species, the smaller is the amplitude of the predator-prey oscillations, but the longer they take to damp away, relative to the period of the oscillation. 3. A parameter-space study suggests that most natural predator-prey systems are only moderately oscillatory to nonoscillatory. 4. If herbivores are represented collectively by the "prey species," and carnivores collectively by the "predator species," the study demonstrates that the steady-state abundance of herbivores is controlled simultaneously by predation intensity and by the rate of food supply to the herbivores. It appears, however, that in most natural systems herbivore abundance is more responsive to changes in predation intensity than to changes in food level. In simple two-species competition the study suggests that: 1. All such systems are effectively nonoscillatory for all possible values of the intrinsic rates of natural increase of both species. This is true even when the values of the intrinsic rates are less than four times the product of the predation rate and efficiency parameters, a condition which in the single-species case always produces oscillatory populations (Hubbell 1973). 2. The values of the competition coefficients do not influence the oscillatory state of the system significantly, but they are very important in setting the steady-state abundance of each species. 3. The sensitivity of the combined abundance of the competing species to a change in the level of shared food source is principally a function of the larger of the two competition coefficients. Finally, in the rarely studied two-species system combining both predatorprey and competitive interactions, the study predicts that: 1. The two-species interaction is unstable if the intrinsic rate of natural increase of the prey-competitor is too low. 2. The level at which the intrinsic rate of the prey-competitor is "too low" is set by all the other parameters of the system except one: the predator-competitor's intrinsic rate of natural increase. 3. Changes in the parameters of the system vary greatly in their relative stabilizing or destabilizing effects on the two-species interaction. For example, increases in the feeding rates of the predator-competitor on the prey-competitor, and of the prey-competitor on the shared food supply, have large destabilizing effects relative to the other parameters, whereas an increase in the production efficiency of the predator-competitor feeding on the prey-competitor has a large stabilizing effect. 4. If either the trophic link from the prey-competitor to the predator-competitor or from the shared food to the predator-competitor is broken, all of the potentially unstable systems can be stabilized. This fact is significant because it indicates that the trophic structure of a food web is indeed critical for its stability. This simple example also, however, illustrates that the trophic complexity of a food web need bear no necessary relationship to its dynamic stability-a relationship that has often been asserted to exist (e.g., MacArthur 1955). The theory developed in these two papers is, at best, primitive when measured against the mathematical needs of community ecology. Nevertheless its principal function will be served if it directs more attention to the value and necessity of adopting operational approaches to the study of population and community dynamics. If and when further research develops more realistic, nonlinear operational descriptions of ecological systems, we can happily give the linear dynamic approach its deserved, unceremonial burial.