A Dynamic Game Approach to Uninvadable Strategies for Biotrophic Pathogens

This paper studies a zero-sum state-feedback game for a system of nonlinear ordinary differential equations describing one-seasonal dynamics of two biotrophic fungal cohorts within a common host plant. From the perspective of adaptive dynamics, the cohorts can be interpreted as resident and mutant populations. The invasion functional takes the form of the difference between the two marginal fitness criteria and represents the cost in the definition of the value of the differential game. The presence of a specific competition term in both equations and marginal fitnesses substantially complicates the reduction in the game to a two-step problem that can be solved by using optimal control theory. Therefore, a general game-theoretic formulation involving uninvadable strategies has to be considered. First, the related Cauchy problem for the Hamilton–Jacobi–Isaacs equation is investigated analytically by the method of characteristics. A number of important properties are rigorously derived. However, the complete theoretical analysis still remains an open challenging problem due to the high complexity of the differential game. That is why an ad hoc conjecture is additionally proposed. An informal but rather convincing and practical justification for the latter relies on numerical simulation results. We also establish some asymptotic properties and provide biological interpretations.