Joining or opting out of a Lotka–Volterra game between predators and prey: does the best strategy depend on modelling energy lost and gained?

Abstract

One contribution of 11 to a Theme Issue 'Modelling biological evolution: recent progress, current challenges and future direction'. Apart from interacting, prey and predators may also avoid each other by moving into refuges where they lack food, yet survive by switching to an energy-saving physiological state. Lotka -Volterra models of predatorprey interactions ignore this option. Therefore, we have modelled this game of 'joining versus opting out' by extending Lotka-Volterra models to include portions of populations not in interaction and with different energy dynamics. Given this setting, the prey's decisions to join or to opt out influence those of the predator and vice versa, causing the set of possible strategies to be complex and large. However, using game theory, we analysed and published two models showing (i) which strategies are best for the prey population given the predator's strategy, and (ii) which are best for prey and predator populations simultaneously. The predicted best strategies appear to match empirical observations on plant-inhabiting predator and prey mites. Here, we consider a plausible third model that does not take energy dynamics into account, but appears to yield contrasting predictions. This supports our assumption to extend Lotka -Volterra models with 'interaction-dependent' energy dynamics, but more work is required to prove that it is essential and that what is best for the population is also best for the individual.

Game theory as a conceptual framework for managing insect pests

“…Aside from the evolution of resistance, pests have other ways to escape control. They may undertake otherwise risky migrations to establish a population elsewhere [55] or they may move into a refuge, give up reproduction and enter a state of physiological dormancy [56,57]. Resistance may simply involve avoiding contact with or ingestion of the chemical agent.…”

Model of neo-Malthusian population anticipating future changes in resources

“…The foundation of the mathematical modeling approach is based on the Lotka-Volterra equations [36] of coupled predator-prey (or consumer-resource) dynamics. Moreover, in some works (for instance [21] p277, [31,32]), the authors have introduced the optimization of a gain functional, to optimize the number of prey and predators at, or during, a given time. This approach leads to adding an adjoint equation to the Lotka-Volterra system of equations (Bellman's equation [5]).…”

Modelling biological evolution: recent progress, current challenges and future direction

“…Beckett & Williams [2] elegantly demonstrate that a very simple model of bacteriaphage interaction can result in patterns which are in very good agreement with those observed in the real ocean. On the contrary, Staň ková et al [16] show that simplifying a description of a predator-prey system by neglecting the energy level of the individuals within populations can result in erroneous conclusions regarding the optimal strategy of organisms within a population. Thus, their insightful study emphasizes that we need to be very careful about oversimplifying biological systems when constructing our models.…”

“…The authors construct a set of models of varying complexity and compare the predicted strategies with empirical observations. They conclude that the model which does not take into account the energy dynamics of the organisms yields contradictory predictions, because it shows multiple entering/exiting the diapause state by the mites, which is not observed in reality [16].…”

“…The standard theoretical approach to determine the best strategy for population persistence is implementation of game theory and optimal control theory: the optimal strategy is determined by maximizing a certain functional related to fitness. Staň ková et al [16] use these approaches (based on the HamiltonJacobi-Bellman equations of dynamic programming) to find the optimal strategy for allocating time between the active state and the diapause state in a generic predator-prey model, together with a case study of the interaction between predator and prey mites on fruit trees. The authors construct a set of models of varying complexity and compare the predicted strategies with empirical observations.…”