Delayed Predator Prey Model Research Articles

Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Existing literature established stability in a delayed logistic Lotka–Volterra predator- prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size h=1m,m∈Z+. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay τ−sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.

Stability and Hopf Bifurcation of a Delayed Predator–Prey Model with a Stage Structure for Generalist Predators and a Holling Type-II Functional Response

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots λ=±iω(ω>0), we just need to consider iω(ω>0) due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings.

Dynamics of a stochastic delay predator-prey model with fear effect and diffusion for prey

We establish a stochastic delay predator-prey model with fear effect and prey diffusion, and investigate the dynamic behavior of the model. Initially, we use Itô's formula to prove the existence and uniqueness of a global positive solution and the stochastic ultimate boundedness of the system. Subsequently, we provide sufficient conditions for the extinction and persistence of prey and predator, and discuss three possible scenarios. Beside, we construct a suitable Lyapunov function and obtain the conditions for the existence of an ergodic stationary distribution. Finally, we present some numerical simulations to verify the analytical results.

Bifurcation Analysis of a Delayed Predator–Prey Model With Square Root Response Functions

Bifurcation Analysis of a Delayed Predator–Prey Model With Square Root Response Functions

Controlling chaotic dynamics of a delayed Hassell–Varley type predator–prey model with non-linear harvesting efforts in prey by using imprecise biological parameters

In this article, we have studied the dynamical behaviors of a delayed predator–prey model with non-linear harvesting effort using imprecise parameters. A method has been proposed for dealing with these imprecise parameters using the parametric form of the interval numbers. The proposed predator–prey model represents the Hassell–Varley predation and Michaelis–Menten prey harvest. We have studied the feasibility of various equilibrium points and their stability. The study reveals that the delay model exhibits stable limit cycle fluctuations due to variation in the delay parameter. Numerical simulations are performed to illustrate all analytical results.

Hopf and Turing–Hopf bifurcation analysis of a delayed predator–prey model with schooling behavior

Hopf and Turing–Hopf bifurcation analysis of a delayed predator–prey model with schooling behavior

A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion

This study investigates the emergence of Turing patterns in a predator–prey system with time delay at variable carrying capacity and prey harvesting in the presence of cross-diffusion. The impact of time delay and periodic diffusion on the stability of the equilibrium point and corresponding Turing pattern is primarily explored. Theoretical implications of Turing instability conditions caused by time delay and periodic diffusion are examined, followed by numerical simulations for ecologically meaningful control parameter values. The effects of carrying capacity specification parameters, time delay, and prey-harvesting effort on the emerging pattern are reported. Cross-diffusion results in a wide range of spatiotemporal pattern creation, including spot and stripe patterns, as well as both coexist under the zero flux border condition in unharvested or harvested dynamics. Furthermore, an intertwined pattern emerges under certain parameters, including time delay. The final comparison highlights the significant roles played by periodic diffusion, time delay, and harvesting effort in the evolution of the Turing pattern.

Explicit impacts of harvesting on a fractional-order delayed predator–prey model

Explicit impacts of harvesting on a fractional-order delayed predator–prey model

Stability and Hopf bifurcation analysis of a delayed predator–prey model with age-structure and Holling III functional response

Stability and Hopf bifurcation analysis of a delayed predator–prey model with age-structure and Holling III functional response

Three-Species Predator–Prey Stochastic Delayed Model Driven by Lévy Jumps and with Cooperation among Prey Species

Our study focuses on analyzing the behavior of a stochastic predator–prey model with a time delay and logistic growth of prey, influenced by Lévy noise. Initially, we establish the existence, uniqueness, and boundedness of a positive solution that spans globally. Subsequently, we explore the conditions under which extinction occurs, and identify adequate criteria for persistence. Finally, we validate our theoretical findings through numerical simulations, which also helps illustrate the dynamics of the stochastic delayed predator–prey model based on different criteria.

Bifurcation in a delayed predator–prey model with Holling type IV functional response incorporating hunting cooperation and fear effect

Bifurcation in a delayed predator–prey model with Holling type IV functional response incorporating hunting cooperation and fear effect

Dynamics of a Delayed Predator–Prey Model with Prey Refuge, Allee Effect and Fear Effect

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.

Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response

In the predator–prey system, predators can affect the prey population by direct killing and predation fear. In the present study, we consider a delayed predator–prey model with fear and Beddington–DeAngelis functional response. The model incorporates not only the fear of predator on prey with an intraspecific competition relationship, but also fear delay and pregnancy delay. Apart from the local stability analysis of the equilibrium points of the model, we find that time delay can change the stability of the system and cause Hopf bifurcation. Taking time delay as the bifurcation parameter, the critical values of delays in several cases are derived. In addition, we extend it to the random environment and study the stochastic ultimate boundedness of the stochastic process. Finally, our theoretical results are validated by numerical simulation.

The impact of fear effect on the dynamics of a delayed predator–prey model with stage structure

In this paper, a stage structure predator–prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed. The prey populations are divided into two parts: juvenile prey and adult prey. From extensive experimental data, it has been found that prey fear of predators can alter the physiological behavior of individual prey, and the fear effect reduces their reproductive rate and increases their mortality. In addition, we also consider the presence of constant ratio refuge in adult prey populations. Moreover, we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors. We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria. Next by choosing the intra-specific competition coefficient of adult prey as bifurcation parameter, we demonstrate that Hopf bifurcation may occur near the positive equilibrium point. Then by taking the gestation delay as bifurcation parameter, the sufficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given. And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory. What’s more, numerical experiments are performed to test the theoretical results obtained in this paper.

HOPF BIFURCATION AND GLOBAL STABILITY OF A PREDATOR–PREY MODEL WITH ALTERNATIVE RESOURCE AND CANNIBALISM

In this study, we present a delayed predator–prey model with cannibalism including a prey, a juvenile predator and an adult predator. The adult predator hunts not only the prey but also the juvenile predator. The alternative resource is considered to reduce cannibalism in predator species. The main results are demonstrated by local stability and Hopf bifurcation. Sufficient criteria for local stability of the interior equilibrium point and the exhibition of a Hopf bifurcation are derived. Global stability is analyzed via designing a suitable Lyapunov function. Additionally, stability of the Hopf bifurcation is explored. Finally, numerical simulations are executed to validate the obtained results.

Qualitative Analysis in a Beddington–DeAngelis Type Predator–Prey Model with Two Time Delays

In this paper, we investigate a delayed predator–prey model with a prey refuge where the predator population eats the prey according to the Beddington–DeAngelis type functional response. Firstly, we consider the existence of equilibrium points. By analyzing the corresponding characteristic equations, the local stability of the trivial equilibrium, the predator–extinction balance, and the coexistence equilibrium of the system are discussed, and the existence of Hopf bifurcations concerning both delays at the coexistence equilibrium are established. Then, in accordance with the standard form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and stability of bifurcating period solutions are derived. Finally, representative numerical simulations are performed to validate the theoretical analysis.

Hopf bifurcation of a delayed predator–prey model with Allee effect and anti-predator behavior

This paper proposes a diffusive predator–prey model with Allee effect, time delay and anti-predator behavior. First, the existence and stability of all equilibria are analyzed and the conditions for the appearance of the Hopf bifurcation are studied. Using the normal form and center manifold theory, the formulas which can determine the direction, period and stability of Hopf bifurcation are obtained. Numerical simulations show that the Allee effect can determine the survival abundance of the prey and predator populations, and anti-predator behavior can greatly improve the stability of the coexisting equilibrium.

Impact of time delay and cooperation strategy on the stability of a predator–prey model with Holling type III functional response

In this paper, we propose and analyze a delayed predator–prey model with Holling type III functional response taking into account cooperation behavior in predators. The time delay is introduced in the attack rate to represent the time necessary to trigger the attack. Each analytical result is followed by an ecological interpretation. We investigate the effect of hunting cooperation on both the number and the level of positive steady states. We observe that the level of the positive equilibrium decreases when increasing the hunting cooperation parameter. Then, we study the impact of the delay as well as the cooperation in hunting on the dynamics of the system. We prove that the presence of delay in the attack rate induces stability switches around the coexisting equilibrium when predators cooperate. In addition, we consider the discrete delay as a bifurcation parameter and prove that the model undergoes a Hopf-bifurcation at the coexisting equilibrium when the delay crosses some critical values. Numerical simulations are presented to confirm our analytical findings.

Spatially nonhomogeneous periodic patterns in a delayed predator–prey model with predator-taxis diffusion

In this paper, we study the effect of time delay on the dynamics of a diffusive predator–prey model with predator-taxis under Neumann boundary condition. The joint effect of predator-taxis and delay can lead to spatially nonhomogeneous periodic patterns via spatially nonhomogeneous Hopf bifurcations. It is also shown that there exist double Hopf bifurcations due to the interaction either between homogeneous and nonhomogeneous or between nonhomogeneous Hopf bifurcations with different modes, which cannot occur for the system with only either delay or predator-taxis diffusion.

Dynamical Analysis of a Delayed Diffusive Predator–Prey Model with Additional Food Provided and Anti-Predator Behavior

We studied a delayed predator–prey model with diffusion and anti-predator behavior. Assume that additional food is provided for predator population. Then the stability of the positive equilibrium is considered. The existence of Hopf bifurcation is also discussed based on the Hopf bifurcation theory. The property of Hopf bifurcation is derived through the theory of center manifold and normal form method. Finally, we analyze the effect of time delay on the model through numerical simulations.