Differential offensive-defensive games between plants and herbivores

Abstract

A mathematical model of chemically mediated plant/herbivore interaction dynamics is developed in the context of the assumptions basic to Optimal Foraging and Optimal Defense Theories (Rhoades). 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' defenses. Our mathematical theory centers on a differential zero-sum game in production /consumption space. However, 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 feedback programs. To be explicit, the plant controls are the Rhoades' Response Parameter (ν) and the coefficients of palatability (u ⌝δ ). The herbivore's controls are its coefficient of aggregation (γ) and its coefficient of foraging efficiency (u 2 δ). The parameter δ has been proved identifiable by us, elsewhere. As an outcome of this theory locusts and larch budmoths, can be successfully modelled as alternating herbivore strategies, first stealthy γ = 0 and then Opportunistic γ ⪢ 0, in a single zero-sum (production/consumption)-game . Thus, the (Larch/Larch budmoth)-system is quite different from the “cooperative behaviour” of limit cycle models, when viewed from the zero-sum game perspective.