This research delves into the generalized Beddington host–parasitoid model, which includes an arbitrary parasitism escape function. Our analysis reveals three types of equilibria: extinction, boundary, and interior. Upon examining the parameters, we discover that the first two equilibria can be globally asymptotically stable. The boundary equilibrium undergoes period-doubling bifurcation with a stable two-cycle and a transcritical bifurcation, creating a threshold for parasitoids to invade. Furthermore, we determine the interior equilibrium’s local stability and analytically demonstrate the period-doubling and Neimark–Sacker bifurcations. We also prove the permanence of the system within a specific parameter space. The numerical simulations we conduct reveal a diverse range of dynamics for the system. Our research extends the results in [Kapçak et al., 2013] and applies to a broad class of the generalized Beddington host–parasitoid model.

This paper examines the relationship between herbivores and plants with a strong Allee effect. When the plant reaches a particular size, the herbivore attacks it. We use the logistic equation to model plant growth and analyze its behavior without herbivores before investigating their interactions. Our study investigates the equilibrium points and their stability, discovering that different fixed points can become unstable due to various bifurcations such as transcritical, saddle-node, period-doubling, and Neimark–Sacker bifurcations. We have identified the Allee threshold, which, if exceeded, can cause both populations to become extinct below that level. However, we have discovered a coexistence equilibrium that is locally asymptotically stable for a range of parameter values above that threshold. Our additional numerical simulations suggest that this area of stability can be expanded. Our results indicate that this system is highly responsive to its parameters. We compare our findings to those of a system without strong Allee effects and conduct numerical simulations to verify our results. By including the Allee effect in the plant population, we enrich the local and global dynamics of the system.

This paper studies the dynamics of a class of host-parasitoid models with host refuge and the strong Allee effect upon the host population. Without the parasitoid population, the Beverton–Holt equation governs the host population. The general probability function describes the portion of the hosts that are safe from parasitism. The existence and local behavior of solutions around the equilibrium points are discussed. We conclude that the extinction equilibrium will always have its basin of attraction which implies that the addition of the host refuge will not save populations from extinction. By taking the host intrinsic growth rate as the bifurcation parameter, the existence of the Neimark–Sacker bifurcation can be shown. Finally, we present numerical simulations to support our theoretical findings.

. We investigate global dynamics of the following systems of difference equations 𝑥 𝑛+1 = 𝑥 𝑛 /(𝐴 1 + 𝐵 1 𝑥 𝑛 + 𝐶 1 𝑦 𝑛 ) , 𝑦 𝑛+1 = 𝑦 2𝑛 /(𝐴 2 + 𝐵 2 𝑥 𝑛 + 𝐶 2 𝑦 2𝑛 ) , 𝑛 = 0, 1, .. . , where the parameters 𝐴 1 , 𝐴 2 , 𝐵 1 , 𝐵 2 , 𝐶 1 , and 𝐶 2 are positive numbers and the initial conditions 𝑥 0 and 𝑦 0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation:

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number We establish the global asymptotic stability of the disease-free equilibrium (GAS) for We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for . For the transcritical bifurcation appears. For we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number .

<jats:p>We present the bifurcation results for the difference equation <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mrow> <mo>/</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>a</mi> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mi>f</mi> </mrow> </mfenced> </math> </jats:inline-formula> where <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>a</mi> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>f</mi> </math> </jats:inline-formula> are positive numbers and the initial conditions <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> </jats:inline-formula> are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.</jats:p>