Azmy Ackleh

Department of Mathematics, University of Louisiana at Lafayette

We investigate the dynamics of discrete-time predator–prey models under several biologically important mechanisms. We begin by replacing the under-compensatory Beverton–Holt prey growth function in an earlier model with an over-compensatory Ricker-type growth function. We analyze persistence and stability of the resulting system and establish conditions for global stability of the interior equilibrium. We then incorporate seasonal breeding by assuming that prey reproduction occurs at alternate time steps. We establish persistence conditions and characterize the resulting interior 2-cycle. Comparison with the corresponding nonseasonal model shows that seasonality can substantially reduce the parameter region associated with stable dynamics. Next, we examine the impact of an Allee effect in prey growth. The Allee effect introduces bistability and, under predator invasion, can produce coexistence dynamics that depend strongly on initial population levels. Predator introduction can also induce a Neimark–Sacker bifurcation, leading to stable quasi-periodic oscillations. Finally, we consider the evolution of predator resistance to toxicant exposure under lethal, sublethal, and mixed toxicant effects. Assuming a trade-off between resistance and prey capture ability, we investigate how evolutionary adaptation alters persistence and stability. Resistance evolution can enable predator persistence under conditions that would otherwise lead to extinction; however, under lethal toxicant effects, evolution can also produce multiple stable boundary equilibria and, paradoxically, lead to predator extinction in situations where the predator would persist without resistance evolution. Together, these results illustrate how over-compensatory growth, seasonality, Allee effects, and evolutionary adaptation can fundamentally alter the long-term dynamics of discrete predator–prey systems.