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In this chapter, we investigate a two-player zero-sum game with separated impulsive dynamics. We study both qualitative and quantitative games. For the qualitative games, we provide a geometrical characterization of the victory domains. For the quantitative games, we characterize the value functions using the Isaacs partial differential inequalities. As a by-product, we obtain a new result of existence of a value for impulsive differential games. The main tool of our approach is the notion of impulse discriminating domain , which is introduced and discussed extensively here.

In the game Φ_ N with simple motions, the pursuer P and the coalition E _N = E _1, E _2, …, E _N of evaders move in a plane with constant speeds 1, β_1,β_2, …,β_ N . The average distance from a point to a set of points is defined as a weighted sum of the corresponding Euclidean distances with given positive constant weights. P strives to minimize the distance to E _N and terminates the game when the distance shortening is not guaranteed. First, we describe several conditions that are met by the states on the terminal manifold M Ф_ N of Ф N depending on the index of evaders caught there. Then, we study Ф_2 in detail. This game is a game of alternative pursuit since there are three different terminal sub-manifolds: P catches E _1 ( E _2) on M ^1 _Ф2 ( M ^2 ^Ф2) and all players are apart on M ^θ _Ф2. We set up and study associated games Ф^1 _2 (Ф^2 _2) and Ф^θ _2 with the payoffs equal to the average distance to E ^2 at instants when the state reaches M ^1 _Ф2 ( M ^2 _Ф2) and M ^θ _Ф2 correspondingly. It is shown that Ф_2 is strategically equivalent to the associated game with the minimal value.

An important step in coevolution occurs when a new mutant clone arises in a resident population of interacting individuals. Then, according to the ecological density dynamics resulting from the ecological interaction of individuals, mutants will go extinct or replace some resident clone or work their way into the resident system. One of the main points of this picture is that the outcome of the selection process is determined by ecological dynamics. For simplicity, we start out one from resident species described by a logistic model, in which the interaction parameters depend on the phenotypes of the interacting individuals. Using dynamic stability analysis we will answer the following purely ecological questions: After the appearance of a mutant clone, (1) what kind of mutant cannot invade the resident population, (2) and what kind of mutant can invade the resident population? (3) what kind of mutant is able to substitute the resident clone, (4) and when does a stable coexistence arise? We assume that the system of mutants and residents can be modelled by a Lotka-Volterra system.We will suppose that the phenotype space is a subset of R^n and the interaction function describing the dependence of the parameters of the Lotka-Volterra dynamics on the phenotypes of the interacting individuals is smooth and mutation is small. We shall answer the preceding questions in terms of possible mutation directions in the phenotype space, based on the analysis of ecological stability. Our approach establishes a connection between adaptive dynamics and dynamical evolutionary stability.

Previous work on the theory of continuous evolutionary games and adaptive dynamics has shown that a species can evolve to an evolutionarily stable minimum on a frequency-dependent adaptive landscape (e.g., Brown and Pavlovic [ 7 ], andAbrams etal . [ 3 ], [ 4 ]). While such stable minima are convergent stable, they can be invaded by rare alternative strategies. The significance of such stable minima for biology is that they produce “disruptive selection” which can potentially lead to speciation (Metz et al. [ 20 ], Geritz et al. [ 16 ], Cohen et al . [ 11 ], and Mitchell [ 21 ]. Previous analyses of Lotka-Volterra competition communities indicate that stable minima and speciation events are more likely to occur when the underlying resource distribution is broader than the resource utilization functions of the competing species. Here, I present an analysis based on a resource-consumer model which allows individuals to adaptively vary resource use as a function of competitor density and strategy. I show that habitat specialization, stable minima, community invasibility, and sympatric speciation are more likely when individuals are more efficient at converting resources into viable offspring. Conversely, factors that inhibit conversion efficiency inhibit speciation and promote competitive exclusion. This model suggests possible links between species diversity and factors influencing the resource conversion efficiency, such as climate, habitat fragmentation, and environmental toxins.

Modified versions of a wellknown model for coexistence are used to examine the conditions that determine the relative abundance of species that are in an evolutionarily stable state. Relative abundance is a term used to refer to the ranking of the number of individuals present within trophically similar species in an ecosystem. We use the G -function approach to understand why relative abundance relationships take the form so often found in field data. We assume that the ecosystem is at or near an evolutionary equilibrium and seek evolutionarily stable strategies to identify a coalition of individual species. In order to have a coalition greater than one, the G -function must produce frequency dependence, implying that the fitness of any given individual depends on the strategies used by all individuals in the population. This is an essential element of the evolutionary game. Otherwise, evolution would drive the population to a single strategy (i.e., a coalition of one) that is an optimal or group fitness strategy. We start with a classical version of the Lotka-Volterra competition equation that is not frequency dependent and make it frequency dependent in three different ways, thus allowing for the modeling of relative abundance. The first two methods involve a single resource niche and rely on modifications of the competitive effects to provide for a coalition of two or more. These models yield relative abundance distribution curves that are generally convex and are not typical of most field data. The third method creates several resource niches, and the simulated results generally create concave curves that are much closer to the field data obtained for natural systems.

Our objective is to determine the evolutionarily stable strategy [ 14 ] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [ 16 ]. Compared to [ 18 ], the innovation lies in the fact that random arrival times are allowed. In this first part, we investigate scramble competition: the game still yields simple Charnov-like strategies [ 4 ]. Thus we attempt to compute the optimal longterm mean rate γ* [ 11 ] at which resources should be gathered to achieve the maximum expected fitness: the assumed symmetry among foragers allows us to express γ* as a solution of an implicit equation, independent of the probability distribution of arrival times. A digression on a simple model of group foraging shows that γ*_N can be simply computed via the classical graph associated to the marginal value theorem— N is the size of the group. An analytical solution allows us to characterize the decline in efficiency due to group foraging, as opposed to foraging alone: this loss can be relatively low, even in a “bad world,” provided that the handling time is relatively long. Back to the original problem, we then assume that the arrivals on the patch follow a Poisson process. Thus we find an explicit expression of γ* that makes it possible to perform a numerical computation: Charnov’s predictions still hold under scramble competition. Finally, we show that the distribution of foragers among patches is not homogeneous but biased in favor of bad patches. This result is in agreement with common observation and theoretical knowledge [ 1 ] about the concept of ideal free distribution [ 12 , 22 ].

our objective is to determine the evolutionarily stable strategy [ 13 ] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [ 15 ]. Compared to [ 17 ], the innovation lies in the fact that random arrival times are allowed. In this second part, we add interference to the model: it implies that a “passive” Charnov-like strategy can no longer be optimal. A dynamic programming approach leads to a sequence of wars of attrition [ 13 ] with random end times. This game is solved in Appendix A. Under some conditions that prevail in our model, the solution is independent of the probability law of the horizon. As a consequence, the solution of the asynchronous foraging problem investigated here, expressed as a closed loop strategy on the number of foragers, is identical to that of the synchronous problem [ 17 ]. Finally, we discuss the biological implications such as a possible connection with the genetic variability in the susceptibility to interference observed in [ 22 ].

We explore the persistence of corn oil sensitivity in a population of the flour beetle Tribolium castaneum using evolutionary game methods that model population dynamics and changes in the mean strategy of a population over time. The strategy in an evolutionary game is a trait that affects the fitness of the organisms. Corn oil sensitivity represents such a strategy in the flour beetle. We adapt an existing model of the ecological dynamics of T. castaneum into an evolutionary game framework to explore the persistence of corn oil sensitivity in the population. The equilibrium allele frequencies resulting from the evolutionary game are evolutionarily stable strategies and compare favorably with those obtained from the experimental data.

Diet provides an important source of niche partitioning that promotes species coexistence and biodiversity. Often, one species selects for a scarcer but more nutritious food (Thomson gazelle) while another opportunistically consumes low-and high-quality foods indiscriminately (African buffalo). In addition to choosing a diet (selective versus opportunistic), organisms have co-adapted digestion physiologies that vary in size and the throughput rate at which food passes through the gut. We combine these elements into a game of resource competition. We consider a vector-valued strategy with elements of gut size and throughput rate. To the forager, food items now have three properties relating to the value of a particular strategy: profitability (energy gained per unit handling time), richness (energy gained per unit bulk), and ease of digestion (energy gain per unit of passage time). When foraging on foods that differ in profitability, richness, and ease of digestion, adjustment or modulation of gut size and throughput rate leads to digestive-system specialization. Modulation of digestive physiology to a particular food type causes different food types to become antagonistic resources. Adjustment of gut volume and processing thus selects for different degrees of diet specialization or opportunism, and thus may promote niche diversification. This in turn sets the stage for disruptive or divergent selection and may promote sympatric speciation.

Scarcity of water has become a major issue facing many nations around the world. To improve the efficiency of water usage there has been considerable interest in recent years in trading water. A major issue in trading water rights is the problem of how an allocation system can be designed in perpetuity that also has desirable properties at each point of time. This is an issue of the time consistency of the contract to trade water. In this chapter we develop a model of dynamic recontracting of water rights and study time consistency properties of the resultant contracts using the ideas of Filar and Petrosjan [7].