Population dynamics are often subject to random independent changes in the environment. For the two strategy stochastic replicator dynamic, we assume that stochastic changes in the environment replace the payoffs and variance. This is modeled by a continuous time Markov chain in a finite atom space. We establish conditions for this dynamic to have an analogous characterization of the long-run behavior to that of the deterministic dynamic. To create intuition, we first consider the case when the Markov chain has two states. A very natural extension to the general finite state space of the Markov chain will be given.

A further generalization of the stochastic replicator dynamic derived by Fudenberg and Harris [12] is considered. In particular, a Poissonian integral is introduced to the fitness to simulate the affects of anomalous events. For the two strategy population, an estimation of the long run behavior of the dynamic is derived. For the population with many strategies, conditions for stability to pure strict Nash equilibria, extinction of dominated pure strategies, and recurrence in a neighborhood of an internal evolutionary stable strategy are derived. This extends the results given by Imhof [17].Asymptotic stochastic stability; evolutionarily stable strategy; invariant measure; Lyapunov function; Nash equilibrium; recurrence; jump stochastic differential equation 92D15; 60H10; 60J40; 92D25 IntroductionConsider a two-player symmetric game, where a ij is the payoff to a player using strategy S i against an opponent employing strategy S j , and define A = (a ij ), as the payoff matrix. We define ∆ n = y ∈ R n : y i > 0 for all i and y i = 1 as the n-dimensional simplex and ∆ n as it's closure. Within a population we assume that every individual is programmed to play a pure strategy S i . Let r i (t) be the size of the subpopulation that plays strategy S i at time t, which we denote as the i th subpopulation. Furthermore, define r(t) := (r 1 (t), . . . , r n (t)) T , R(t) := i r i (t) and s(t) := (s 1 (t), . . . , s n (t)) T where s i (t) := r i (t)/R(t) the i th subpopulation frequency of the population . When an agent in the i th subpopulation is randomly matched with another player from the entire population, As(t) i is the average payoff for this individual, and we take this to be the fitness of the player. We assume growth is proportional to fitness:ṙ i (t) = r i (t) As(t) i , and henceṡThis is the replicator dynamic.Foster and Young [30] appear to be the first to use a stochastic differential equation to describe replicator dynamic, which they do by injecting a Brownian term directly into the replicator equation. Considering a biological perspective, Fudenberg and Harris [12] derived a continuous time stochastic replicator dynamic by first assumingfor σ i ∈ R + and the W i (t) are pairwise independent standard Wiener processes. Itô's lemma then yields ds i (t) = j =i s i (t)s j (t)As(t) i − As(t) j dt + σ 2 j s j (t) − σ 2 i s i (t) dt + σ i dW i (t) − σ j dW j (t) .(1)

I propose a stochastic SIS and SIRS system to include a Poisson measure term to model anomalies in the dynamics. In particular, the positive integrand in the Poisson term is intended to model quarantine. Conditions are given for the stability of the disease free equilibrium for both systems.

The replicator–mutator dynamic was originally derived to model the evolution of language, and since the model was derived in such a general manner, it has been applied to the dynamics of social behavior and decision making in multi-agent networks. For the two type population, a bifurcation point of the mutation rate was derived, displaying different long-run behaviors above and below this point. The long-run behavior would naturally be subjected to noise from the environment, however, to date, there does not exist a model that dynamically accounts for the effects of the environment. To account for the environmental impacts on the evolution of the populace, mutation rates above and below this bifurcation point are switched according to a continuous-time Markov chain. The long-run behaviors of this model are derived, showing a counterintuitive result that the majority of initial conditions will favor the dominated type.

Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environmental switching parameter in several common evolutionary dynamics models including the replicator dynamic for linear symmetric and asymmetric landscapes, the Moran process, and incentive dynamics. We give a simple relationship between the probability of environmental switching, the relative fitness gain, and the effect on long term behavior in terms of fixation probabilities and long term outcomes for deterministic dynamics. We also discuss cases where the dynamic changes, for instance a population evolving under a replicator dynamic switching to a best-reply dynamic and vice-versa, giving Lyapunov stability results.