Individuals Strategies and Predator–Prey Game Models in Deterministic and Random Settings

ABSTRACT The authors propose a new predator–prey game model by integrating two optional strategies into prey species: cooperative and isolation strategies. An investigation of the evolutionary impact on predator–prey system dynamics is given. The model utilizes a replicator equation to track changes in the frequency of cooperative strategy among preys, along with two population growth equations detailing variations in prey and predator biomass. Interestingly, both cooperative and isolated prey strategies can coexist within the prey species. As the predation rate increases, preys are more inclined to adopt cooperative strategies, thereby boosting predator biomass. Abundant resources then encourage preys to opt for cooperative strategies to secure resources and defend against predators. Under specific conditions, the system exhibits intricate dynamics involving bistability, transcritical, and Hopf bifurcations. A novel fractional‐order predator–prey game model is then derived using a continuous‐time random walk based on a potential physical stochastic process, which is different than the traditional fractional‐order models. The influence of fractal dimension on system dynamics is analyzed, showing that it can either diminish or eliminate cooperative strategies within prey species and potentially lead to predator extinction. Interestingly, dimension affects the convergence speed of the system without altering its final steady state: As the fractal dimension and time change, the frequency of cooperative strategies adopted by preys, the biomasses of prey, and predator species ultimately stabilize toward the same steady state, but their rates of approaching to the steady state are different. In this case, only preys survive and cooperative and isolated strategies coexist in prey species. These models can have multiple applications in different settings, aiding with management of animal populations, pest control, tumor immune interactions, and infectious diseases.