Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification

In many host-parasite relations, the parasite has numerous variants, antigenic strains, or types. The host also has many types of reactions and defenses. A simple phenomenological model proposed here shows how evolution by natural selection could explain this diversity, and why the diversity of a host roughly corresponds to the diversity of a parasite. At the level of species diversity of hosts and parasites, the model provides a theoretical basis for Eichler's rule. The model is based on a two-player (host and parasite), zero-sum game. To focus attention on the number of strategies of each player rather than on the details of payoffs, the model assumes that the elements of the payoff matrix are chosen at random, once and for all. For concreteness, the model supposes that the host and parasite contend over changes in the parasite's net rate of reproduction. The model implies that it is to each player's advantage to diversify its strategies if the cost of additional strategies can be neglected. This elementary and unremarkable conclusion does not depend on any assumptions about the details of payoffs or the numbers of strategies that the host and parasite currently use. Analysis of the model gives bounds on how quickly each player must diversify, relative to its opponent, to avoid any change in average net rate of reproduction (NRR). Within these bounds, the probability of a substantial change in average NRR for either player tends to zero as both players diversify. The model suggests that, when the change in NRR for different strategies is generated by a random mechanism, which on the average does not favor either player, an antagonistic host-parasite relation will either evolve large numbers of parasite and host strategies or else become evolutionarily unstable. The model shows that it is not necessary to invoke selective effects of multiple species of parasites and multiple species of hosts to explain this diversity of strategies. It shows that even directly opposed interests can be stabilized by sufficient diversification of strategies on both sides.