Stochastic replicator dynamics subject to Markovian switching

Abstract

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.

On imitation dynamics in population games with Markov switching

“…In this paper, we provide sufficient conditions for the model parameters to ensure the emergence of the four different convergence outcomes described in the previous section for the deterministic replicator dynamics. To the authors' knowledge, a similar analysis has been performed only for the particular case of the replicator equation with Markov jumps in [19]. Thus, this work generalizes the existing results in the literature, providing theoretical tools to properly analyze Equation ( 15), shedding light on the role of the imitation rate function f and of the Markov transition rate matrix Q in determining the behavior of the system.…”

“…The first result is summarized in the following theorem, which generalizes the results in [19] by discussing stability or instability of the two equilibria x 1 = 0 and x 1 = 1.…”

“…As showed in [19], four different outcomes are possible: 1) Strategy 1 dominates: a > c and b > d; then the system has two equilibria: x 1 = 1 is asymptotically stable and x 1 = 0 is unstable. 2) Strategy 2 dominates: a < c and b < d; then the system has two equilibria: x 1 = 1 is unstable and x 1 = 0 is asymptotically stable.…”

“…Switched systems, in which the payoff structure of the game (termed mode) may instantly change to another mode, according to a continuous-time Markov jump process, have emerged as a valuable modeling framework to capture these phenomena [18]. However, to the best of our knowledge, theoretical results on imitation dynamics with Markov switching are few, and limited to the analysis of specific cases, such as the replicator equation [19].…”

The stochastic interactions between predator and prey under Markovian switching: competitive interaction between multiple prey

“…May [20] also noted that due to environmental fluctuation, birth rates, carrying capacity, competition coefficients and other parameters involved in a population model system exhibit random fluctuation to a greater or lesser extent. Therefore, many authors introduced stochastic perturbations into the deterministic models [16,19,29,33]. Takeuchi et al [28] considered the following predator-prey model with telegraph noise based on model (1.1):…”