Effect of Climate Change and Fear in aFractional Prey-Predator Model withCrowley and Martin Functional Response

Global climate change is intensifying, with rising temperatures signifi-cantly impacting ecological relationships. Understanding the connectionbetween temperature and predation is imperative to address the threatof extinction resulting from excessive temperature increases. Inthispaper, we introduce a fractional prey-predator model incorporatingthe Caputo fractional operator. Our model considers prey-predatorinteractions based on Crowley and Martin’s functional response,account-ing for the fear effect induced by predation. We analyze the non-negativity, existence, uniqueness, and boundedness of solutions withinour model, considering both classical and Caputo derivative scenar-ios. Additionally, we investigate the local stability of each equilibriumunder both integer and fractional order conditions, emphasizing theglobal stability of the coexistence positive equilibrium point in bothcontexts. Our examination treats predation as a time-dependent func-tion, with the temperature function reflecting climate change andelucidating how rising temperatures contribute to predation. Throughnumerical simulations, we explore the impacts of fear on prey behav-ior and population dynamics, and illustrate how climate change,especially rising temperatures, intricately affects species relationships. Mathematics Subject Classification:26A33 , 34C60 , 34D20 , 92D25 , 37N25