Abstract We propose a fractional-order predator-prey model with maturation delay in prey species and Beddington-DeAngelis functional response. We have also incorporated a prey refuge and fear effect. There is a research gap as very few analyses have examined the impacts of fractional order, prey refuge, and fear on population fluctuation in the presence of time delays. In this analysis, the stability behaviour of the coexisting equilibrium is explored, taking delay as the bifurcating parameter for the system. Our primary focus is to check whether time delay, fractional order, prey refuge, and fear effect can stabilize a destabilized system or destabilize a stable system when the parameters are varied. We observe that (i) fractional order has a destabilizing effect, (ii) time delay has a destabilizing effect for fixed memory, (iii) prey refuge has a stabilizing effect; however, switching of instability may also occur, and (iv) depending on the other parameter set fear factor may have stabilizing effect, destabilizing effect, and switching of instability. All the theoretical analyses are confirmed by extensive numerical simulation.

This study investigates the dynamical behaviour of a prey-predator system with two competing predators, incorporating the Beddington–DeAngelis functional response and the effects of environmental toxicants. Analytical analysis ensures the boundedness of solutions, guaranteeing biologically feasible population dynamics. Equilibrium points are identified, and their stability is examined using local and global stability analyses. Numerical simulations validate the analytical findings, demonstrating that as the competition coefficient b1 increases, the system transitions from a stable equilibrium to periodic oscillations and eventually to chaotic behaviour. Furthermore, the impact of the toxicant uptake rate d1 is explored to assess its role in system stability. The results indicate that low levels of toxicant absorption promote oscillatory dynamics, while higher values of d1 suppress population growth and restore stability. This highlights the dual role of toxicants in ecological systems, where moderate exposure disrupts equilibrium, but excessive accumulation can lead to stabilization. Bifurcation diagrams and time-series simulations further reinforce these transitions, revealing critical thresholds where stability is lost or regained. The study provides valuable insights into the complex interplay between toxicant dynamics, predator-prey interactions, and bifurcation phenomena. The findings emphasize the ecological implications of toxicant exposure and interspecies competition, offering potential applications in environmental management and conservation strategies.

In view of the importance of predator-dependent functional response and fear of prey induced by powerful predators, we construct a delayed prey–predator model with fear and Beddington–DeAngelis functional response. The existence, uniqueness, and global asymptotic stability of equilibrium points are investigated and some criteria are established. Next, Hopf bifurcation analysis is executed, and the critical values of such bifurcation parameters as fear and delay for the determinate system are obtained. Then we extend it to a random environment and study the boundedness of expectation of solutions and the global asymptotic stability. Finally, the main findings are validated by numerical examples. It is worth noting that the specific influences of fear by predator, time delay, and white noise are explored numerically. Simulation figures intuitively exhibit that fear, delay, and white noise bring serious influences on the stability of the system. Fear from predator leads to a lower equilibrium state of prey and predator, and it can change the system stability from unstable to stable after exceeding a certain critical value. The time delay has a significant impact on the system stability by producing Hopf bifurcations accompanied by limit cycles, and even lead to multiple stabilities. Larger white noise can change the system stability from stable to unstable.

Complex Dynamics of a Quad-Trophic Food Chain Model with Beddington–DeAngelis Functional Response, Fear Effect and Prey Refuge

Understanding the dynamics of the propagation of computer viruses is crucial for the development of effective cybersecurity strategies. This paper proposes a novel compartmental model based on the traditional SIR framework, which incorporats the Beddington-DeAngelis functional response to capture the inhibitory effects of the modern operating systems' built-in security mechanisms. We rigorously establish the well-posedness of the model by proving the non-negativity and boundedness of solutions. The existence and stability of equilibrium points are thoroughly analyzed using the Hurwitz criterion, Lyapunov functions, and Dulac criterion. Numerical simulations validate our theoretical findings and demonstrate the significant role of the system self-protection parameters in controlling virus spread. Unlike previous models, our approach provides explicit connections between the model parameters and real-world cybersecurity metrics, thus offering practical insights for network defense strategies. The model's ability to maintain a persistent infected state aligns with the observed behaviors of modern malware, thus providing a more realistic representation of the dynamics of computer viruses. The study employs advanced visualization techniques, including three-dimensional surface plots to elucidate the complex interactions between protection parameters, thus providing actionable insights for the design and implementation of cybersecurity policies.

The growth of prey and predators in the real world is highly dependent on the environmental factors that are of high stochasticity, and a recent field experiment showed that the fear of predators can greatly reduce the reproduction of prey. In this paper, we developed a stochastic predator-prey system with general fear effect function and Beddington-DeAngelis functional response. We provided threshold criteria for the existence of a unique invariant measure with ergodic property, and testified that the transition probability function of the solution approaches to the invariant measure at an exponential rate. These results improve and generalize some recent outcomes greatly. Analytical results and numerical simulations indicate that stochasticity has an important impact on the stability of the system.

Stability and Hopf bifurcation for a multi-delay PSIS eco-epidemic model with saturation incidence and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

Structure of positive solutions for a reaction-diffusion model with additional food and protection zone

Untangling the memory and inhibitory effects on SIS-epidemic model with Beddington–DeAngelis infection rate

Dynamics of a stochastic and periodic virus model with Beddington-DeAngelis functional response

This paper investigates the dynamics of a tritrophic food chain model incorporating an Allee effect, sexually reproductive generalist top predators, and Holling type IV and Beddington-DeAngelis functional responses for interactions across different trophic levels. Analytically, we explore the feasible equilibria, their local stability, and various bifurcations, including Hopf, saddle-node, transcritical, and Bogdanov-Takens bifurcations. Numerical findings suggest that higher Allee intensity in prey growth leads to the inability of species coexistence, resulting in a decline in species density. Likewise, a lower reproduction rate and a higher strength of intraspecific competition among top predators also prevent the coexistence of species. Conversely, a rapid increase in the reproduction rate and a decrease in the strength of intraspecific competition among top predators enhance the densities of prey and top predators while decreasing intermediate predator density. We also reveal the presence of bistability and tristability phenomena within the system. Furthermore, we extend our autonomous model to its nonautonomous counterpart by introducing seasonally perturbed parameters. Numerical analysis of the nonautonomous model reveals that higher seasonal strength in the reproduction rate and intraspecific competition of top predators induce chaotic behavior, which is also confirmed by the maximum Lyapunov exponent. Additionally, we observe that seasonality may lead to the extinction of species from the ecosystem. Factors such as the Allee effect and growth rate of prey can cause periodicity in population densities. Understanding these trends is critical for controlling changes in population density within the ecosystem. Ecologists, environmentalists, and policymakers stand to benefit significantly from the invaluable insights garnered from this study. Specifically, our findings offer pivotal guidance for shaping future strategies aimed at safeguarding biodiversity and maintaining ecological stability amidst changing environmental conditions. By contributing to the existing body of knowledge, our study advances the field of ecological science, enhancing the comprehension of predator-prey dynamics across diverse ecological conditions.

Impact of both-density-dependent fear effect in a Leslie–Gower predator–prey model with Beddington–DeAngelis functional response

Sliding dynamics of a Filippov ecological system with nonlinear threshold control and pest resistance

Spatio-temporal patterns and global bifurcation of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results.

This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, reproductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In addition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/33/abstr.html

In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β(a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (a)$$\\end{document} are assumed to be piecewise functions related to their maturation period τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.

Abstract This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.