Differential Offensive-Defensive Games between Plants and Herbivores

Abstract

A mathematical theory of chemically mediated plant-herbivore interaction dynamics is developed in the context of the assumptions basic to Rhoades' verbal optimal-defence theory. Evidently, plants produce some important defensive chemicals in allometric proportion to biomass, and are also capable of spontaneous response to damage caused by physical stress and/or herbivore attack, by actually increasing percentages of these chemicals in plant tissues over as short a time as a few hours. In addition, herbivores have evolved offensive strategies to counter the plants' defences. Our mathematical theory centres on a differential zero-sum game in production-consumption space. The associated discrete game with noise and particular sequences of play is also considered. The concept of a zero-sum production-consumption game as a representation of an interactive coevolved two-species predator-prey system is basic to our approach. The dynamics of the game are developed in a series of biological arguments ending finally with a pair of second-order nonlinear coupled symmetric Duffing oscillators with variable second-order damping. Plant vigour or resistance may be studied in terms of properties of this damping which depends upon chemical response levels in plant tissues. Both plant and herbivore players use bang-bang controls and automatic feed-back programs. To be explicit, die plant controls are die Antonelli-Rhoades allometric response parameter v and the coefficient of palatability u₁ δ. The herbivore's controls are its coefficient of aggregation γy and its coefficient of foraging efficiency u₂ . The parameter δ has been proved identifiable by us, elsewhere. The theory is applied to the population dynamics of locusts and larch bud-moths in the context of stealthy (γ δ 0) and opportunistic strategies (γ » 0).