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This paper deals with the host–parasitoid model, where the logistic equation governs the host population growth, and a proportion of the host population can find refuge. The equilibrium points' existence, number, and local character are discussed. Taking the parameter regulating the parasitoid's growth as a bifurcation parameter, we prove that Neimark–Sacker and period-doubling bifurcations occur. Despite the complex behavior, it can be proved that the system is permanent, ensuring the long-term survival of both populations. Furthermore, it was observed that the presence of the proportional refuge does not significantly influence the system's behavior compared to the system without a proportional refuge. [ABSTRACT FROM AUTHOR]

We examine a discrete-time model of plant-herbivore interaction, in which the population of plants follows the Ricker law and is affected by strong Allee effects. We discuss the equilibrium points, including their characteristics and number. Additionally, we analyze in detail the local behavior of the solutions around the equilibrium points. Under certain initial conditions, both populations may go extinct. We conduct research demonstrating the occurrence of transcritical and period-doubling bifurcations, resulting in a stable two-cycle at the boundary equilibrium. However, the interior equilibrium becomes unstable due to the emergence of the Neimark-Sacker bifurcation and period-doubling. We confirm the latter's existence through analytical methods and illustrate the period-doubling bifurcation through numerical simulations. [ABSTRACT FROM AUTHOR]

Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May's host–parasitoid model's solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population's intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings. [ABSTRACT FROM AUTHOR]

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

We investigate the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the difference equation ... where the parameters ;C are positive numbers and the initial conditions x1 and x0 are arbitrary nonnegative numbers such that x-1 + x0 > 0. We determine the basin of attraction of fixed point and period-two solutions. The associated map is not defined at the (0; 0): However, we show that there exist period two solutions on the axis that are locally asymptotically stable and two continuous invariant curves passing through the point (0; 0); which are boundaries of the basins of attractions of these period two solutions, such that every solution starting on these two curves or in the region between these two curves is attracted to the point (0; 0). [ABSTRACT FROM AUTHOR]