Effect of Climate Change and Fear in a Fractional Prey-Predator Model with Crowley and Martin Functional Response

Global climate change is amplifying with each passing day, notably as rising temperatures increasingly influence ecological relationships. Therefore, a thorough understanding of the connection between temperature and predation is imperative to address the threat of extinction resulting from excessive temperature increases. In this paper, we introduce a fractional prey-predator model incorporating the Caputo fractional operator. Our model considers prey-predator interactions based on Crowley and Martin's functional response, taking into account the fear effect induced by predation. We analyze the non-negativity, existence, uniqueness, and boundedness of solutions within our model, considering both classical and Caputo derivative scenarios. Furthermore, we investigate the local stability of each equilibrium under both integer and fractional order conditions. In particular, we emphasize the global stability of the coexistence positive equilibrium point in both contexts. Our examination considers predation as a time-dependent function, with the temperature function reflecting climate change and elucidating how elevating temperatures contribute to predation. Through numerical simulations, we validate our theoretical findings, explore the impacts of fear on prey behavior and population dynamics, and illustrate how climate change, especially rising temperatures, intricately affects species relationships. Mathematics Subject Classification: 26A33 , 34C60 , 34D20 , 92D25 , 37N25.