In the last three sessions three basic approaches to modeling predator-prey interactions have been presented. These approaches are summarized and compared below:

The Lotka-Volterra approach is based on the assumption that the predation process is similar to the reaction of molecules in a homogeneously mixed gas or liquid, the so-called "law of *mass action*". The L-V model is usually modified by adding a logistic self-limitation term to the prey equation and a cyrtoid functional response. This results in the familiar prey-dependent L-V model

The main characteristics of this model are a humped prey isocline and a vertical predator isocline (see figure). Models with this general isocline structure were widely used by Robert MacArthur and Michael Rosenzweig in the 60's and 70's, and are frequently referred model as Rosenzweig-MacArthur models (e.g., see Rosenzweig and MacArthur1969, MacArthur and Connell 1966 ).

The stability of the L-V model with logistic prey growth and cyrtoid functional response depends on the position of the community equilibrium (the intersection of the isoclines) on the prey isocline. The model becomes more stable as the community equilibrium moves down the right-hand arm of the prey isocline (to the right of the hump), and becomes less stable as the equilibrium point moves down the left-hand arm (to the left of the hump). These stability properties give rise to a couple of paradoxes:

**The paradox of enrichment**. Rosenzweig (1971) showed that L-V models with consumer satiation are destabilized if the carrying capacity for the resource (the place where the prey population intersects its own axis) is increased. He called this the "paradox of enrichment" because enriching the system, say by increasing the level of nutrients entering the system, will tend to destabilize the interaction between resource and consumer.**The paradox of biological control**. Biological control is the regulation of a pest population at very sparse and stable densities by predators and/or parasitoids. There are many examples of this in the history of insect pest management, yet the Rosenzweig-MacArthur model (above) predicts that such a condition, with the consumer isocline far to the left, will be very unstable.

Apart from these paradoxes, L-V models with the above characteristics fail to satisfy condition 3 for a plausible model because consumers do not compete with each other for prey. To deal with this problem, Arditi and Ginzberg (1989) suggest using a ratio-dependent behavioral response to produce a ratio-dependent L-V model

which satisfies all conditions for a biologically plausible model. Notice that, in this version of the L-V model, the attack rate declines with the density of the consumer and so it describes the process of intraspecific competition amongst consumers for resources. The only difference between this ratio-dependent L-V model and the prey-dependent form (above) is that the predator isocline now slopes to the right because of reduced consumer efficiency with increasing consumer density. As this isocline structure is identical to all other ratio-dependent models, they will all be discussed together at a later time.

The consumption approach builds upon the hunger models of Ivlev and Watt to arrive at a general model for the transfer of biomass or energy through a food chain. This approach culminated in the *metabolic pool* model of Gutierrez and his colleagues (e.g., Gutierrez and Baumgärtner 1984, Gutierrez et al. 1994), which has the basic structure

Here *X _{i}* is the population density of any organism in a food chain. In this truly general model, each trophic level acquires biomass or energy from the next lower trophic level (

The Ivlev-Watt-Gutierrez model fulfills all the conditions for a credible food chain model. However, it is really an energy flow model and difficulties may be encountered in applying it to systems where organisms are identified as individuals. This problem can be visualized by placing a hypothetical population in an environment without food or predators. With *X _{i}*-1 =

which means that the population dies out exponentially, like radioactive decay. If *X* is total biomass, then this type of decay seems reasonable. However, if *X* is the number of individuals, then we would expect them to die more or less simultaneously; i.e., they should get thinner and thinner as they starve and lose biomass, but then suddenly die when their reserves are no longer sufficient to support basal metabolism.

It is also important to remember that this is an *instantaneous* equation and should be integrated before it can be applied to real data. Also remember that it is a ratio-dependent model and, therefore, has similar dynamics to other ratio-dependent formulations (see later).

Finally, there is the individual survival approach which grew out of the theoretical work of Verhulst, Thompson and Royama, and the field studies of Morris and Varley in the 60's and 70's. This approach has culminated in the *logistic food chain model* of Berryman and his colleagues (Berryman 1992, Berryman et al. 1995)

which can be written more conveniently as a per-capita *R*-function

As in the biomass consumption model, this is a truly general model because *X _{i}* represents the density of any species in a food chain of any length. Remember that the survival terms (all but the first term in the left-hand side of the equation) should be interpreted broadly to include the effects of competition and predation on the reproductive capacity of individuals as well as on their death rates. This model is also a general ratio-dependent equation because the survival terms have the consumer in the numerator and the total amount of resources in the denominator. Remember that these demand/supply ratios are the inverse of the supply/demand ratios in other ratio-dependent models.

The individual survival model also meets all the conditions for a plausible food chain model. Because it has been developed in discrete time steps, it is an *overall* model and is most suitable for modeling populations of *individual* organisms observed at discrete time intervals. Notice that when a population is placed in an environment without resources or predators, the quantity in the exponent becomes minus infinity and *X _{i,t}* = 0. In other words, a starved population dies out within one time step. For this reason, the time step of observation should be greater than the longevity of starved individuals.

Because the logistic food chain model is derived from assumptions about the survival of individual organisms, it is inappropriate to apply it to the dynamics of biomass or energy flow. When used in this way, the individual survival model does not adhere to the laws of thermodynamics because consumed energy is not directly converted into consumer biomass.

Arditi, R. and L. R. Ginzburg. 1989. Coupling in predator-prey dynamics: ratio-dependence. *Journal of Theoretical Biology* 139:311-326.

Berryman, A. A. 1992. The origins and evolution of predator-prey theory. *Ecology* 73: 1530-1535.

Berryman, A. A., Michalski, J., Gutierrez, A. P. and Artiditi, R. 1995b. Logistic theory of food dynamics. *Ecology ***76**: 336-343.

Gutierrez, A. P. and J. U. Baumgärtner. 1984. Multitrophic level models of predator-prey energetics. 1. Age-specific energetic models - pea aphid,* Acyrthosiphon pisum* (Homoptera: Aphididae) as an example. *Canadian Entomologist* 116: 924-932.

Gutierrez, A. P., N. J. Mills, S. J. Schreiber and C. K. Ellis. 1994. A physiologically based tritrophic perspective on bottom-up top-down regulation of populations. *Ecology* 75: 2227-2242.

MacArthur, R. H. and J. H. Connell. 1966. *The biology of populations*. John Wiley & Sons, New York.

Rosenzweig, M. P. and R. H. MacArthur. 1969. Graphic representation and stability conditions of predator-prey interaction. *American Naturalist* 97: 209-223.

©1998 Alan A. Berryman