Additive Allee Effect Research Articles

Bifurcation in a single-species logistic model with addition Allee effect and fear effect-type feedback control

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population.

Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population

In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.

Stability and bifurcation in a two-patch commensal symbiosis model with nonlinear dispersal and additive Allee effect

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant [Formula: see text] increases or [Formula: see text] decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches.

Stability and Bifurcation of a Gordon–Schaefer Model with Additive Allee Effect

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor’s theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases.

Modeling and Analysis of Fractional Order Logistic Equation Incorporating Additive Allee Effect

This study investigates the logistic model of a single species incorporating the additive Allee effect using Caputo fractional order differential equations. The Allee effect describes a positive correlation between individual fitness and population density at low densities. Populations subjected to the strong Allee effect can move towards extinction when their population is below a critical level. This study calculates the threshold level of the population suffering from the strong Allee effect. Various published studies are showing that fractional order models are more appropriate for explaining real-world phenomena than ordinary integer-order systems; therefore, this study involves the use of the Caputo fractional order derivative. Single-species models have been extensively used in mathematical biology, such as insect control, optimal biological resource planning, epidemic avoidance and control, and cell growth regulation. This study can help save vulnerable species from extinction and eliminate unwanted species by subjecting them to a strong Allee effect using artificial strategies.

Complex dynamics induced by additive Allee effect in a Leslie-Gower predator-prey model

Complex dynamics induced by additive Allee effect in a Leslie-Gower predator-prey model

Analysis of a prey-predator system incorporating the additive Allee effect and intraspecific cooperation

<abstract><p>To understand the influence of the Allee effect and intraspecific cooperation on the dynamics of a predator-prey system, we constructed a model using ordinary differential equations. Our research shows that the system exhibits more complex dynamics, including possible bistability between alternative semi-trivial states and an Allee effect for prey. The Allee effect can destabilize the system. The equilibrium points of the system could change from stable to unstable. Otherwise, even if the system were stable, it would take much longer time to reach a stable state. We also find that the presence of the Allee effect of prey increases the positive equilibrium density of the predator but has no effect on the positive equilibrium density of the prey. It should be noted that the influence of nonlinear predator mortality also causes the system to take a longer time to reach a steady state.</p></abstract>

DETERMINISTIC AND STOCHASTIC ANALYSIS OF A SIZE-DEPENDENT PHYTOPLANKTON–ZOOPLANKTON MODEL WITH ADDITIVE ALLEE EFFECT

In this paper, a deterministic size-dependent phytoplankton–zooplankton (PZ) model with additive Allee effect as well as its stochastic differential equation version is formulated to explore the growth dynamic of phytoplankton. A stability and Hopf bifurcation analysis is performed towards deterministic PZ model, and stochastic dynamic analysis is carried out on the stochastic PZ model, including the stochastic extinction, persistence in mean and the unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulation analysis. Numerical simulations reveal that the synergistic effect of plankton body size and Allee effect can have a complex impact on the growth of phytoplankton in the deterministic and stochastic environments. One of the most interesting findings suggests that the reduced phytoplankton cell size can trigger bistable phenomenon, while the increased phytoplankton cell size or the weakened Allee effect has the ability to destroy such bistable phenomenon.

Dynamical Analysis of a Predator–Prey Model with Additive Allee Effect and Migration

In this paper, a predator–prey model in which the prey has the additive Allee effect and the predator has artificially controlled migration is proposed. When the system introduces additive Allee effect and artificially controlled migration, more complicated dynamical behavior is obtained. The system can undergo saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Two limit cycles are found and discussed. The influence of the additive Allee effect and artificially controlled migration on the dynamics of the system is also presented. In detail, when the Allee effect is large, the prey will become extinct. When the artificially controlled migration rate is larger, the intensity of the prey (pest) will be smaller and the intensity of the predator will be larger. This indicates that artificially controlled migration can be effectively used to control the pest.

BIFURCATION ANALYSIS AND CHAOS CONTROL OF DISCRETE PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT

This paper presents a study of a discrete prey–predator model with additive Allee effect. The model is discretized using the forward Euler method, and the system is analyzed using bifurcation theory and chaos control. The equilibrium point of the discrete system is obtained through equilibrium analysis, and the stability of the equilibrium point is determined by analyzing the parameter conditions. The study also establishes the existence and direction of Neimark–Sacker bifurcation at the positive equilibrium point. The paper proposes two control strategies to manage chaotic behavior and Neimark–Sacker bifurcation: exponential control and hybrid feedback control. These control methods are demonstrated to be effective in controlling the chaotic behavior and bifurcation of the system through numerical simulation. Overall, the results of the study provide important insights into the dynamics of the discrete prey–predator model with additive Allee effect, as well as effective methods for controlling chaos and bifurcation in the system.

Singular Perturbation Analysis for a Holling–Tanner Model with Additive Allee Effect

In this article, a reaction–diffusion predator–prey equation with additive Allee effect acting on prey is investigated by twice singular perturbation analysis. Our focus is to study the traveling fronts of predator invasion under the assumption that the diffusion ability of the predator is much greater than that of the prey. Our results exhibit two kinds of traveling fronts for a Holling–Tanner system in two different limit cases. And these traveling fronts correspond to the heteroclinic connections between a saddle and either a stable node or a stable focus. In addition, we show that heteroclinic connections are formed in different ways for different limit cases. One is formed on the slow manifold, which has one time scale. While the other is formed by the intersection of the slow manifold and the fast manifold, which has two time scales. Furthermore, the existence of traveling front solutions in different limit cases is demonstrated through theoretical analysis and numerical simulation. The main tools are geometric singular perturbation theory and Bendixson’s criteria.

Dynamical Analysis of Two-Preys and One Predator Interaction Model with An Allee Effect on Predator

Mathematical modeling in biology is quite interesting in the field of real-world problems. This research paper focused on the interaction between two prey and one predator species. Here, our interaction is based upon the competition between two prey and one predator including an additive Allee effect in the predator population along with a Holling type II functional response. Further, this intuition allowed us to prove the positive invariance and boundedness of the model. This analysis shows that there are six equilibrium points including the coexistence of all three populations. Stability analyses are also derived and proved both locally and globally. Also in this paper, we discussed the optimal control approach to reduce the population affected by an Allee effect by the predator population. Numerical simulations are carried out to support our theoretical results.

Additive Allee effect on prey in the dynamics of a Gause predator–prey model with constant or proportional refuge on prey at low or high densities

Assuming that the intrinsic growth of prey is affected by an additive Allee effect, and from which a proportion to the population or critical size of prey is hidden from the predator if the quantity of prey is above or below the critical size, respectively, this paper proposes a predator–prey Gause model composed of two vector fields separated by the critical prey population size. Since the proposed model is not discontinuous, it could have a single interior equilibrium, belonging to one of the vector fields of the model, and a stable limit cycle formed between the two vector fields, and a possible extinction of the prey considering a strong Allee effect.

A predator–prey model with additive Allee effect and intraspecific competition on predator involving Atangana–Baleanu–Caputo derivative

A mathematical model of an interaction between two populations namely prey and predator is studied based on a Gause-type predator–prey model involving the additive Allee effect and intraspecific competition on the predator. A famous fractional operator called Atangana–Baleanu–Caputo fractional derivative (ABC) is employed to integrate the impact of the memory effect on the dynamical behavior of the model. The existence, uniqueness, non-negativity, and boundedness of the solution are given to confirm the biological feasibility and validity of the model. Three types of equilibrium points are identified on the origin, axial, and interior including their existence conditions. The Lyapunov direct method for the ABC model is used to investigate the asymptotic stability condition for each equilibrium point. The numerical simulations are provided to demonstrate the impact of several biological parameters on the dynamics of the solutions. The emergence of transcritical, saddle–node, and backward bifurcations driven by the Allee constant are provided which resulted in the appearance of bistability condition. A Hopf bifurcation as well as the evolution of the limit-cycle also occurs as the impact of the memory effect. Each analytical and numerical result is biologically interpreted to show the way that the density of both populations always balances in their ecosystem.

Spatio-temporal pattern selection in a prey–predator model with hunting cooperation and Allee effect in prey

In this work, we have studied the temporal as well as spatio-temporal dynamics of a prey–predator model with additive Allee effect in prey growth and hunting cooperation among the specialist predators. Hunting cooperation impacts both the stability and the existence scenario of the coexistence equilibrium of the temporal model. The temporal system is capable of exhibiting a wide range of local bifurcations such as transcritical, saddle–node, Hopf, Bogdanov–Takens bifurcations and a global Homoclinic bifurcation. For the diffusive system, the well-posedness is first proved. Then Turing instability is investigated to understand the relationship between the diffusion and the hunting cooperation behind the stationary pattern formation. By employing weakly nonlinear analysis, the amplitude equation of stationary solution is derived, and using the stability of amplitude equation, the pattern selection of several stationary patterns such as spot, stripe, and their combinations have been discussed under various parameter regimes. Extensive numerical simulations are carried out to verify the analytical results. Main contribution of this work is the identification of all possible stationary patterns depending upon the signs of the coefficients involved in the amplitude equation. In addition, some time-dependent nonhomogeneous irregular solutions are observed for parameter values in the Hopf domain or Turing–Hopf domain. With the help of time-series analysis of the obtained numerical results, we have established that the time and space-varying solution is spatio-temporal chaos.

Stability, bifurcation, and chaos control of predator-prey system with additive Allee effect

The current investigation focuses on the dynamics of a discrete-time predator-prey system with additive Allee effect. Discretization is accomplished by the use of a piecewise constant argument approach of differential equations. Firstly, we studied the existence and topological classification of equilibrium points. We then investigated existence and direction of period-doubling and Neimark-Sacker bifurcations in the system. Moreover, to control the chaos caused by bifurcation, we employ a hybrid control technique. Finally, all theoretical results are justified numerically.

Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect

<abstract><p>This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.</p></abstract>

Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect

Abstract We propose and analyze a Lotka-Volterra commensal model with an additive Allee effect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle-node bifurcations and transcritical bifurcations are derived by using Sotomayor’s theorem. Third, we give sufficient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee effect case, the additive Allee effect has a negative effect on the final density of both species, with increasing Allee effect, the densities of both species are decreasing. For the strong Allee effect case, the additive Allee effect is one of the most important factors that leads to the extinction of the second species. The additive Allee effect leads to the complex dynamic behaviors of the system.

Qualitative Analysis of a Spatiotemporal Prey‐Predator Model with Additive Allee Effect and Fear Effect

A diffusive predator‐prey system with both the additive Allee effect and the fear effect in the prey subject to Neumann boundary conditions is considered in this paper. Firstly, non‐negative and non‐trivial solution a priori estimations are shown. Furthermore, for specific parameter ranges, the absence of non‐constant positive solutions is demonstrated. Secondly, we use the linearized theory to investigate the stability of non‐negative constant solutions. The spatially homogeneous and non‐homogeneous periodic solutions, as well as non‐constant steady state solutions, are next investigated by using Allee effect parameters as the bifurcation parameter. Finally, numerical simulation is used to illustrate some theoretical results.

Turing instability of a diffusive predator-prey model along with an Allee effect on a predator

Modeling biological systems is a significant task in nature. This work concentrates on the relation between two prey species and one predator species model. Our interaction is based on a competition between two preys and one predator, with an additive Allee effect in the predator population. Many investigations have been done in stability properties of the population dynamics. However in this study, we demonstrated population instability as a result of the incorporating diffusive terms in the PDE system and we construct the amplitude equations of stationary patterns using the nonlinear multiple scale analysis approach and Taylor series expansion. Numerical results are also presented to validate our propositions.