Diffusive Predator Prey System Research Articles

Spatiotemporal dynamics induced by nonlocal competition in a diffusive predator-prey system with habitat complexity

Spatiotemporal dynamics induced by nonlocal competition in a diffusive predator-prey system with habitat complexity

Spatio-temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator-prey system and the diffusion-reaction equations

The coupled parabolic nonlinear evolution equations that describe the temporal-spatio dynamics of predator-prey systems and the nonlinear diffusion-reaction equations with diffusion effects have broad applications in biology, population dynamics, ecology, and many other fields where nonlinearity and diffusivity are key features of wave evolution. As a result, exact traveling wave solutions of such models are highly effective in numerical and analytical theories. In this article, we find the soliton solutions of the above stated biological models by contriving the enhanced modified simple equation method which is related to the exponential, hyperbolic, and trigonometric functions by balancing the highest order nonlinearity and diffusivity associated with biological groups. Graphical representations of some of the obtained solutions in three- and two-dimensional layouts are provided to estimate their behavior. The results revealed that the stated method is categorical, robust, and efficient in finding exact solutions to a variety of nonlinear evolution equations.

The Effect of Diffusion on the Dynamics of a Predator-Prey Chemostat Model

This paper deals with a diffusive predator-prey chemostat system which describes the growth of planktonic rotifers, Brachionus calyciflorus, feeding on unicellular green algae, Chlorella vulgaris. The dynamical behavior of this system is established in terms of the diffusion rate. The results show that there exist two critical diffusion rates which classify the dynamical behavior of this system into the following three scenarios: (i) for a large diffusion rate, all species will be washed out; (ii) for an intermediate diffusion rate, the predator goes extinct and the prey survives; (iii) for a small diffusion rate, all species coexist. Finally, our numerical results show that the solution of this system may undergo a steady-state bifurcation or Hopf bifurcation for a suitably small diffusion rate, which supplements our theoretical results.

Dynamical behavior in a reaction-diffusion system with prey-taxis

In this article, we study a diffusive predator-prey system with prey-taxis under homogeneous Neumann boundary conditions. We establish the existence and boundedness of nonnegative global solutions. Through comparison with the system without prey-taxis, we find that the positive constant equilibrium remains stable for positive prey-taxis, while negative prey-taxis makes it unstable.

SPATIOTEMPORAL DYNAMICAL ANALYSIS OF A PREDATOR–PREY SYSTEM WITH FEAR AND GROUP DEFENSE IN PREY

This paper considers a diffusive prey–predator system with fear and group defense in the prey population. Also, we consider that the mortality of predators is linear and quadratic. By using local stability analysis, we get the prerequisite of Turing instability. Using comprehensive numerical computations, we get non-Turing pattern formation in the system with linear death of predator. Turing patterns are obtained for the system with the quadratic death of the predator. The modeling technique of multiple scale analysis is used to determine amplitude equations near the Turing bifurcation origin for the model with the predator’s quadratic mortality rate. The amplitude equations stability leads to various Turing patterns such as spots, stripes, and mixed. The result focuses on changing the mortality rate linear to quadratic of a predator in the prey–predator system. The derived results support us in a more immeasurable understanding of prey–predator interaction dynamics in the actual world.

Hopf bifurcation induced by time delay and influence of Allee effect in a diffusive predator–prey system with herd behavior and prey chemotaxis

Hopf bifurcation induced by time delay and influence of Allee effect in a diffusive predator–prey system with herd behavior and prey chemotaxis

Bifurcationing Analysis of Predator-Prey Diffusive System Based on Bazykin Functional Response

A predator-prey diffusion system with a Bazykin functional response is studied. The existence of equilibrium points, the stability of normal number equilibrium points and the existence of Hopf bifurcationes are investigated for the proposed system, the existence of positive solutions in the system is discussed under Neumann boundary conditions, and the stability of constant equilibrium points is focused on under the condition of Hurwitz criterion. The results show that there exist positive equilibrium points in the system under Neumann boundary conditions, and the normal number equilibrium points are stable when specific conditions are satisfied, and the bifurcation points of Hopf bifurcationes and their orders are given.

Bifurcations in a Diffusive Predator-Prey System with Linear Harvesting

Bifurcations in a Diffusive Predator-Prey System with Linear Harvesting

Bifurcation, Pattern Recognition and Numerical Implementations for a Diffusive Predator-Prey System with Weak Allee Effect

Bifurcation, Pattern Recognition and Numerical Implementations for a Diffusive Predator-Prey System with Weak Allee Effect

On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation

<p style='text-indent:20px;'>The current paper provides a qualitative study of a diffusive predator-prey system with a hunting-cooperation functional response under homogeneous Neumann boundary conditions. The considered functional response explains a behavioral mechanism in which predators search for and hunt their prey cooperatively. We investigate the global attractor for nonnegative time-dependent solutions to the system and the nonpersistence leading to the extinction of the predator species in the system. Moreover, we study the local stability at all feasible nonnegative constant steady states and the occurrence of Hopf bifurcation (i.e., the existence of time-periodic orbits) in the system. Finally, we study the existence and nonexistence of nonconstant positive steady states in the system, and the limiting behavior of the positive steady states according to the diffusion rate. Through this study, we observe interesting results, such as bistability, extinction of the predator species induced by a large consumption rate, and stationary patterns in the system with hunting cooperation in predators.</p>

Stability and Spatiotemporal Dynamics of a Delayed Diffusive Predator-Prey System with Chemotaxis and Fear Effect

Stability and Spatiotemporal Dynamics of a Delayed Diffusive Predator-Prey System with Chemotaxis and Fear Effect

Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and generalist predator

<abstract><p>A delayed diffusive predator-prey system with nonlocal competition and generalist predators is considered. The local stability of the positive equilibrium and Hopf bifurcation at positive equilibrium is studied by using time delay as a parameter. In addition, the property of Hopf bifurcation is analyzed using the center manifold theorem and normal form method. It is determined that time delays can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillation of prey and predator population densities.</p></abstract>

Turing-Hopf Bifurcation of a Diffusive Predator-Prey System with Nonlocal Effect and Group Defense in Prey Against Generalist Predator

Turing-Hopf Bifurcation of a Diffusive Predator-Prey System with Nonlocal Effect and Group Defense in Prey Against Generalist Predator

Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and schooling behavior

<abstract><p>We consider a delayed diffusive predator-prey system with nonlocal competition in prey and schooling behavior in predator. We mainly study the local stability and Hopf bifurcation at the positive equilibrium by using time delay as the parameter. We also analyze the property of Hopf bifurcation by center manifold theorem and normal form method. Through the numerical simulation, we obtain that time delay can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillations of prey and predator's population densities. In addition, we observe that the increase of space area will not be conducive to the stability of the positive equilibrium $ (u_*, v_*) $, and may induce the inhomogeneous periodic oscillations of prey and predator's population densities under some values of the parameters.</p></abstract>

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

Stability and spatiotemporal dynamics of a diffusive predator–prey system with generalist predator and nonlocal intraspecific competition

In this paper, we investigate the influence of the nonlocal intraspecific competition on the stability and spatiotemporal dynamics for a diffusive predator–prey system with generalist predator. The stability region and the possible bifurcation curves are determined in the plane consisting of diffusion coefficients of the prey and the predator. Spatially inhomogeneous steady states with different spatial profiles and spatially inhomogeneous periodic solutions with mode-1 as the parameters are varied across the boundaries of the stability region.

Spatiotemporal patterns in a diffusive predator–prey system with Leslie–Gower term and social behavior for the prey

In this paper, we deal with a new approximation of a diffusive predator–prey model with Leslie–Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator–prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator–prey interaction in the presence of Leslie–Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self‐diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing‐driven instability, and Turing–Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.

Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal

We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a complete picture in the local diffusion case. We show that the survival of the species depends crucially on how the climate change speed compares with the minimal speed of some pulse type forced waves.

Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

AbstractWe investigate the effect of nonlocal intraspecific prey competition on the spatiotemporal dynamics of a Holling–Tanner predator–prey model with diffusion. We first establish the criteria for Hopf, Turing, double‐Hopf, and Turing–Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. For Turing–Hopf bifurcation, by analyzing the normal form truncated to the third order, we derive that, with strong nonlocal interaction, the system exhibits the tristable phenomena, that is, the coexistence of a stable spatially nonhomogeneous periodic orbit and two nonconstant stable steady states, as well as the existence of periodic orbits with two spatial wave frequencies induced by the nonlocal interaction. The main analytical difficulty arises from the nonlocal interaction that prevents the direct application of formulas for the coefficients of the normal form. Biologically, the emerging spatiotemporal patterns suggest that the global intraspecific competition can promote the coexistence of the prey and predator by allowing the prey maintain a critical total population size, which may provide an alternative approach in explaining the group formation of some prey species under the risk of predation. Some coexistence patterns are destabilized by introducing another prey species with local intraspecific competition, leading to the coexistence of two preys and one predator, namely, the prey with nonlocal interaction is concentrated at a single spatial location, and the other prey is distributed uniformly in the rest of the habitat. Accordingly, the predator is forced to change its behavior as well.