Trade-off and chaotic dynamics in a two-prey one-predator model with refuge, environmental noise and seasonal effects

In nature, poisonous prey species have developed toxins as an attack against predators. However, predator’s decisions to consume aposematic toxic prey are not solely determined by their toxicity. Predators often weigh the costs of ingesting toxins against the benefits of obtaining nutrients, with body size acting as a significant cue. In terms of avoiding predation, prey commonly respond by seeking refuge, a behavior that fluctuates considerably due to seasonality and environmental variability. In this study, we integrate these factors into the dynamics of two prey and one predator model. We explore the cost of fear to illustrate the trade-off dynamics of prey, and we adjust the predator’s death rate by integrating a function that serves predator loss resulting from encounters with toxic prey. Our model demonstrates bistability as a consequence of hysteresis effect and undergoes various bifurcations, including saddle–node, Hopf, Bautin, and cusp bifurcations. We observe that increasing prey refuge and reducing the level of fear contribute to system stabilization, yet chaotic behavior persists with respect to both parameters. While seasonality can introduce stability and periodicity, it may also give rise to complex and unpredictable dynamics. We comprehend that the system might undergo transitions through multiple periodic cycles as a result of changes in the degree of seasonality, yet it eventually settles into a periodic solution. Furthermore, our analysis using the stochastic model reveals that low noise levels maintain species densities close to deterministic equilibrium, whereas high noise levels result in broader dispersion or even extinction of species. The information presented in this article may help readers understand the biological knowledge gained by investigating interactions between predators and prey.