József Garay

Ideal Free Distributions, Evolutionary Games, and Population Dynamics in Multiple‐Species Environments

In this article, we develop population game theory, a theory that combines the dynamics of animal behavior with population dynamics. In particular, we study interaction and distribution of two species in a two‐patch environment assuming that individuals behave adaptively (i.e., they maximize Darwinian fitness). Either the two species are competing for resources or they are in a predator‐prey relationship. Using some recent advances in evolutionary game theory, we extend the classical ideal free distribution (IFD) concept for single species to two interacting species. We study population dynamical consequences of two‐species IFD by comparing two systems: one where individuals cannot migrate between habitats and one where migration is possible. For single species, predator‐prey interactions, and competing species, we show that these two types of behavior lead to the same population equilibria and corresponding species spatial distributions, provided interspecific competition is patch independent. However, if differences between patches are such that competition is patch dependent, then our predictions strongly depend on whether animals can migrate or not. In particular, we show that when species are settled at their equilibrium population densities in both habitats in the environment where migration between habitats is blocked, then the corresponding species spatial distribution need not be an IFD. Thus, when species are given the opportunity to migrate, they will redistribute to reach an IFD (e.g., under which the two species can completely segregate), and this redistribution will also influence species population equilibrial densities. Alternatively, we also show that when two species are distributed according to the IFD, the corresponding population equilibrium can be unstable.

A predator-prey refuge system: Evolutionary stability in ecological systems.

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individuals fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator-prey systems are discussed from this perspective.

The effects of opportunistic and intentional predators on the herding behavior of prey

In this article, we study how predator behavior influences the aggregation of prey into herds. Game-theoretic models of herd formation are developed based on different survival probabilities of solitary prey and prey that join the herd and on the predators preference of what type of prey to search for. For an intentional predator that will only pursue its preferred type of prey, a single herd with no solitaries cannot emerge unless the herd acts as a prey refuge. If neither prey choice provides a refuge, it is shown that an equilibrium always exists where there are both types of prey and the predator does not always search for the same type of prey (i.e., a mixed equilibrium exists). On the other hand, if the predator is opportunistic in that it sometimes shifts to pursue the type of prey that is observed first, there may be a single herd equilibrium that does not act as a prey refuge when there is a high level of opportunistic behavior. For low opportunistic levels, a mixed equilibrium is again the only outcome. The evolutionary stability of each equilibrium is tested to see if it predicts the eventual herding behavior of prey in its corresponding model. Our analysis confirms that both predator and prey preferences (for herd or solitary) have strong effects on why prey aggregate. In particular, in our models, only the opportunistic predator can maintain all prey in a single herd that is under predation risk.

Observation and control in a model of a cell population affected by radiation

The effect of radiation on a cell population is described by a two-dimensional nonlinear system of differential equations. If the radiation rate is not too high, the system is known to have an asymptotically stable equilibrium. First, for the monitoring of this effect, the concept of observability is applied. For the case when the total number of cells is observed, without distinction between healthy and affected cells, a so-called observer system is constructed, which, at least near the equilibrium state, makes it possible to recover the dynamics of both the healthy and the affected cells, from the observation of the total number of cells without distinction. Results of simulations with illustrative data are also presented. If we want to control the system into a required new equilibrium state, and maintain this new equilibrium by a constant control, a technique of theory of optimal control can be applied to construct a feedback control system.

Strict ESS for n-species systems

A system of n asexual populations is considered where both intra- and interspecific frequency-dependent game conflicts with lack of information take place. The concept of a strict n-species ESS is introduced which implies local asymptotic stability of the replicator dynamics of pure phenotypes. The dynamical concept of strict stability is also introduced which turns out to be equivalent to the strict n-species ESS concept. The above notions are also related to similar concepts considered in the literature for the same biological situation.

Game-theoretic methods for functional response and optimal foraging behavior

We develop a decision tree based game-theoretical approach for constructing functional responses in multi-prey/multi-patch environments and for finding the corresponding optimal foraging strategies. Decision trees provide a way to describe details of predator foraging behavior, based on the predators sequence of choices at different decision points, that facilitates writing down the corresponding functional response. It is shown that the optimal foraging behavior that maximizes predator energy intake per unit time is a Nash equilibrium of the underlying optimal foraging game. We apply these game-theoretical methods to three scenarios: the classical diet choice model with two types of prey and sequential prey encounters, the diet choice model with simultaneous prey encounters, and a model in which the predator requires a positive recognition time to identify the type of prey encountered. For both diet choice models, it is shown that every Nash equilibrium yields optimal foraging behavior. Although suboptimal Nash equilibrium outcomes may exist when prey recognition time is included, only optimal foraging behavior is stable under evolutionary learning processes.

Application of change-point problem to the detection of plant patches.

In ecology, if the considered area or space is large, the spatial distribution of individuals of a given plant species is never homogeneous; plants form different patches. The homogeneity change in space or in time (in particular, the related change-point problem) is an important research subject in mathematical statistics. In the paper, for a given data system along a straight line, two areas are considered, where the data of each area come from different discrete distributions, with unknown parameters. In the paper a method is presented for the estimation of the distribution change-point between both areas and an estimate is given for the distributions separated by the obtained change-point. The solution of this problem will be based on the maximum likelihood method. Furthermore, based on an adaptation of the well-known bootstrap resampling, a method for the estimation of the so-called change-interval is also given. The latter approach is very general, since it not only applies in the case of the maximum-likelihood estimation of the change-point, but it can be also used starting from any other change-point estimation known in the ecological literature. The proposed model is validated against typical ecological situations, providing at the same time a verification of the applied algorithms.

Stock estimation, environmental monitoring and equilibrium control of a fish population with reserve area

For sustainable exploitation of renewable resources, the separation of a reserve area is a natural idea. In particular, in fishery management of such systems, dynamic modelling, monitoring and control has gained major attention in recent years. In this paper, based on the known dynamic model of a fish population with reserve area, the methodology of mathematical systems theory and optimal control is applied. In most cases, the control variable is fishing effort in the unreserved area. Working with illustrative data, first a deterministic stock estimation is proposed using an observer design method. A similar approach is also applied to the estimation of the effect of an unknown environmental change. Then it is shown how the system can be steered to equilibrium in given time, using fishing effort as an open-loop control. Furthermore, a corresponding optimal control problem is also solved, maximizing the harvested biomass while controlling the system into equilibrium. Finally, a closed-loop control model is applied to asymptotically control the system into a desired equilibrium, intervening this time in the reserve area.

Optimal Forager against Ideal Free Distributed Prey

The introduced dispersal-foraging game is a combination of prey habitat selection between two patch types and optimal-foraging approaches. Prey’s patch preference and forager behavior determine the prey’s survival rate. The forager’s energy gain depends on local prey density in both types of exhaustible patches and on leaving time. We introduce two game-solution concepts. The static solution combines the ideal free distribution of the prey with optimal-foraging theory. The dynamical solution is given by a game dynamics describing the behavioral changes of prey and forager. We show (1) that each stable equilibrium dynamical solution is always a static solution, but not conversely; (2) that at an equilibrium dynamical solution, the forager can stabilize prey mixed patch use strategy in cases where ideal free distribution theory predicts that prey will use only one patch type; and (3) that when the equilibrium dynamical solution is unstable at fixed prey density, stable behavior cycles occur where neither forager nor prey keep a fixed behavior.

Monitoring and control in a spatially structured population model

In the paper methods of Mathematical Systems Theory are applied to the dynamical analysis of a harvested population with a reserve area. Although the methodology also applies to rather general spatially structured populations, for a concrete interpretation, we consider a fish population living in a free fishing area and in a reserved area, with migration between them. Using a fishing effort model based on logistic growth in both areas, from the catch, by the construction of an auxiliary system called observer, we dynamically estimate the total fish stock. A similar method also applies to the case of a changing environment, when there is a time-dependent abiotic environmental effect described by an additional exosystem. Furthermore, we also consider the problem of steering the population into a desired new equilibrium. To this end an optimal control problem is set up, which is numerically solved using an optimal control toolbox developed for MatLab.