Modeling cooperative evolution in prey species using the snowdrift game with evolutionary impact on prey–predator dynamics

Cooperative evolution refers to the collaborative process by which two or more individuals or species interact and adapt together some specific behavior, leading to mutual benefits for their survival and reproduction. In this article, we employ the snowdrift game to model cooperative evolution in prey species and examine its evolutionary impact on the dynamics of prey–predator interactions. The model system consists of a replicator equation that captures the evolution of cooperative behavior in prey species, along with two growth equations of prey and predator incorporating the factors that influence cooperative evolution on prey reproduction. Our analysis reveals that in long time run, the prey population becomes polymorphic, allowing cooperative and defective individuals to coexist and persist over time. However, a very low cooperative proportion poses a risk of extinction for both prey and predator. Again if majority of prey species exhibit cooperative behavior, the predator species still faces possible extinction due to the strong group defense mechanism employed by prey. At some intermediate level of cooperative proportions, the proposed model system exhibits more rich and complex dynamics such as oscillatory patterns, bi-stable phenomena, homoclinic orbit, and saddle–node bifurcations of limit cycles. Further, we explore the transient dynamics of the proposed system through which one can anticipate the potential shifts in population size and can identify critical thresholds in promoting sustainable coexistence.