Beddington-DeAngelis Functional Response Research Articles

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge 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.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

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

The additional food and protection zone provided to predator and prey respectively are introduced into a reaction-diffusion predation model with nonlinear functional response in this paper. We mainly concentrate on the combined effects of additional food and protection zone on the existence and uniqueness of solution for the model. It is shown that, compared with the models with classical Beddington-DeAngelis functional response, the integration of additional food and protection zone into Beddington-DeAngelis functional response causes the model to produce new critical values for the existence of positive solutions. Moreover, we derive that the model permits a unique positive solution when the strength of mutual interference among predators or the quantity of additional food is large, where, the effect of quantity of additional food on the positive solution is a new finding. The results shed light on that the combination of additional food and protection zone is an effective and environmentally-friendly strategy in preserving biodiversity.

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

The dynamical behaviors of an epidemic model based on the susceptible–infected–susceptible (SIS) model are investigated. The Beddington–DeAngelis functional response is used for the infection rate to present the dependence of the transmission of the infection on the ratio of both susceptible and infected populations. A Caputo fractional derivative is applied to show the existence of memory in nature affects population dynamics. The disease-free and endemic equilibrium points are obtained as the equilibrium points that describe the condition when the population is free from the disease or the disease exists in the population throughout time. The existence, uniqueness, non-negativity, and boundedness are proven which state the biological validity of the mathematical model. The local and global stability of each equilibrium point is studied including the basic reproduction number and its influence on the dynamical behaviors. Some numerical simulations are portrayed to explore more about the dynamics of the model which are relevant to the analytical findings. The partial rank correlation coefficient is presented to investigate the dominant parameter with respect to the basic reproduction number and the density of susceptible and infected populations. The parameter continuations are demonstrated to show the impact of infection rate and the inhibitory effect which lead to the occurrence of forward bifurcations. The memory effect is also demonstrated numerically to show the changes in convergence rate due to the changes in memory strength.

Chaos and extinction risks of sexually reproductive generalist top predator in a seasonally forced food chain system with Allee effect.

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

Animals have evolved fearful behavior throughout a lengthy evolutionary history, and such action should be the result of interactions between prey and predator. At present, the fear factor considered by many scholars in predator–prey models is reasonable to a certain extent, for example in some unicellular and bacterial species, but for most species, the fear factor should be density-dependent of both species due to sexual reproduction. Therefore, based on the existing works, this paper puts forward some more reasonable conditions that should be satisfied by fear factor, and makes theoretical analysis and numerical simulations for specific type of expression. The study shows that fear plays an important role in enriching model dynamics, and that the model dynamics are sensitive to moderate levels of fear. Lower levels of fear maintain the same dynamics as the model without fear, higher levels of fear can exclude periodic solutions (or limit cycles) to promote model stability, while moderate levels of fear may lead to the emergence of Hopf bifurcations. By selecting a different set of parameters, the Hopf bifurcation in our model can be both supercritical and subcritical, contrasting to the typical predator–prey model in which it frequently appears supercritical.

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

By applying the integrated pest management strategy to address pest control problems, Filippov pest-natural enemy systems have been extensively studied. However, little attention has been paid to pest changing rate and pest resistance in these studies. Therefore, this paper proposes a complex and more realistic Filippov pest-natural enemy system with curved discontinuity boundary by considering Beddington–DeAngelis functional response and pest resistance, as well as incorporating pest changing rate in the threshold control index. Based on the relevant theory of the Filippov system, we analyze a sextic equation and obtain that the proposed Filippov system can have at most six sliding segments. By using qualitative techniques, we discuss the sliding segment and pseudo-equilibrium bifurcations. Numerically, we investigate the local and global sliding bifurcations. It can be detected that one new local sliding bifurcation occurs, which could be called the pseudo-Hopf bifurcation but differs from the traditional pseudo-Hopf bifurcation as it involves two sliding segments with opposite stabilities and generates a sliding limit cycle. We also discover two interesting global sliding bifurcations with novel bifurcation processes, which may be termed triple limit cycle bifurcations as they both relate to the appearance of three nested limit cycles.

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

The aim of this article is investigating the spatio-temporal patterns of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses. First, by utilizing user-friendly version of Crandall–Rabinowitz bifurcation theory as an analytical method, the spatio-temporal patterns of positive steady state are obtained. Then, by regarding the cross-diffusion coefficient d1 as a bifurcation parameter, we will derive a bifurcation theorem for the corresponding nonlinear cross-diffusion model. Moreover, by applying spectrum theory, perturbation of simple eigenvalues and linearized stability, it is discovered that the bifurcation solutions possess local stability near the bifurcating point in proper conditions. These theoretical results mean that the cross-diffusion mechanism can create a coexistence environment for the preys and predator under some circumstances. Finally, a numerical example is proposed to verify our results.

Global Dynamics of Two-Species Amensalism Model with Beddington–DeAngelis Functional Response and Fear Effect

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.

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

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

Hopf bifurcation in an age-structured predator–prey system with Beddington–DeAngelis functional response and constant harvesting

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.

Continuous dependence of stationary distributions on parameters for stochastic predator–prey models

AbstractThis 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.

Population dynamics in a reaction–diffusion-advection predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a two-species predator–prey model in advective heterogeneous environments with the Beddington–DeAngelis interaction term, where the Danckwerts boundary conditions are imposed. Applying the comparison principle for predator–prey system and persistence theory, we draw a clear picture on the long-time dynamics, which also indicates the existence and non-existence of the coexistence steady state solutions. Specially, the uniqueness of the coexistence steady state solutions is considered under certain conditions.

Effect of non-linear harvesting and delay on a predator-prey model with Beddington-Deangelis functional response

Effect of non-linear harvesting and delay on a predator-prey model with Beddington-Deangelis functional response

Bifurcation and spatial patterns driven by predator-taxis in a predator-prey system with Beddington-DeAngelis functional response

Bifurcation and spatial patterns driven by predator-taxis in a predator-prey system with Beddington-DeAngelis functional response

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Periodic dynamics of predator-prey system with Beddington–DeAngelis functional response and discontinuous harvesting

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

Complex dynamics of a four-species food-web model: An analysis through Beddington-DeAngelis functional response in the presence of additional food

Abstract The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

Stability Analysis of E-epidemic SIT Model with Beddington-DeAngelis Functional Response for Wireless Sensor Network

Stability Analysis of E-epidemic SIT Model with Beddington-DeAngelis Functional Response for Wireless Sensor Network

Pattern dynamics in a reaction–diffusion predator–prey model with Allee effect based on network and non-network environments

In this paper, we establish a predator–prey model with Beddington–Deangelis functional response. For this model, firstly, the existence of the positive equilibrium point and the conditions for the Turing instability are studied. Then the amplitude equation is derived through weakly nonlinear analysis, and the relationship between the selection of pattern and the coefficients of the amplitude equation is obtained. At the same time, through a large number of numerical simulations, we verify the accuracy of the theoretical analysis. Actually, we mainly choose to change the values of parameters r and d2 to study the sensitivity of the pattern to them. When the pattern tends to be stable, there could be the pattern of spots, coexistence of spots and stripes or stripes. Even if they are spot pattern, the spot density will also be different due to the selection of parameters. Finally, we simulate and compare that the network structure(mainly BA and WS) has a certain influence on the time required for pattern stabilization and the distribution of node density. The final results show that the growth rate of prey, diffusion coefficients and network structure all play an important role in the formation of Turing pattern.