Reaction-diffusion Predator-prey System Research Articles

Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system.

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (subT mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of subT-mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

ABSTRACTIn this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator–prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator–prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.

Dynamics analysis of a delayed reaction-diffusion predator-prey system with non-continuous threshold harvesting

This paper deals with a delayed reaction-diffusion predator–prey model with non-continuous threshold harvesting. Sufficient conditions for the local stability of the regular equilibrium, the existence of Hopf bifurcation and Turing bifurcation are obtained by analyzing the associated characteristic equation. By utilizing upper-lower solution method and Lyapunov functions the globally asymptotically stability of a unique regular equilibrium and asymptotically stability of a unique pseudoequilibrium are studied respectively. Further, the boundary node bifurcations are studied. Finally, numerical simulation results are presented to validate the theoretical analysis.

Qualitative Analysis of a Predator-prey Model with Weak Allee Effect

In the past decades, the research of reaction-diffusion equations has been one of the dominant themes due to its universal existence and importance. The applications of reaction-diffusion equations play an active role in physics, chemistry, biological populations and medicine. In various ecological systems, the predator- prey model is an important branch. In this paper, a predator-prey reaction-diffusion system with weak Allee effect and homogeneous Neumann boundary condition is considered. First, it is proved that the unique positive constant steady state is stable, and the methods to study local stability are based on local linearization techniques. Second, a prior-estimate of positive steady state is given by Harnack inequality and Maximum principle. Also, we found that system has no non-constant positive solution when the diffusion coefficients meeting certain conditions. Finally, we establish the existence of positive solutions to the system mainly using the fixed point index theory in cone. Through the research, we find out the conditions for the existence of positive steady state, which provide theoretical basis for the improvement of yield predators.

Stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting

The goal of this paper is to study the stability and traveling waves of stage-structured predator-prey reaction-diffusion systems of Beddington-DeAngelis functional response with both nonlocal delays and harvesting. By analyzing the corresponding characteristic equations, the local stability of various equilibria is discussed. We reduce the existence of traveling waves to the existence of a pair of upper-lower solutions by using the cross iteration method and the Schauder’s fixed point theorem. The existence of traveling waves connecting the zero equilibrium and the positive equilibrium is then established by constructing a pair of upper-lower solutions.

Stability and traveling fronts for a food chain reaction-diffusion systems with nonlocal delays

This paper is purported to investigate a food chain reaction-diffusion predator-prey system with nonlocal delays in a bounded domain with no flux boundary condition. We investigate the global stability and find the sufficient conditions of global stability of the unique positive equilibrium for this system. The derived results show that delays often restrain stability. Using the method of linearizing this system, we see that the zero equilibrium is unstable. Moreover, by constructing upper-lower solutions, we find that there exist traveling wavefronts which connect the zero equilibrium and positive equilibrium when the wave speed is large enough and the prey intrinsic growth rate and the death rate of the predator are relatively big.

Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model

Time delay is an essential ingredient of spatio-temporal predator–prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator–prey reaction–diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction–diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge–Kutta scheme.

Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis

This paper proves the global existence and boundedness of solutions to a general reaction–diffusion predator–prey system with prey-taxis defined on a smooth bounded domain with no-flux boundary condition. The result holds for domains in arbitrary spatial dimension and small prey-taxis sensitivity coefficient. This paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions. Applications to models from ecology and chemotaxis are discussed.

Stability and bifurcation of a reaction–diffusion predator–prey model with non-local delay and Michaelis–Menten-type prey-harvesting

We investigate a reaction–diffusion predator–prey system with homogeneous Neumann boundary conditions and non-local delay due to predator gestation. By analysing the corresponding characteristic equations, we establish the local stability of the positive steady state. We also discuss the existence of Hopf bifurcations at the positive steady state. We derive sufficient conditions for the global stability of the positive steady state of the proposed problem using the Lyapunov functional. Numerical simulations illustrate the results and reveal that as the discrete delay τ increases, the species may tend to extinction. Changes in the harvesting effort E and non-local delay β can transform an unstable system into a stable one.

Spatial, temporal and spatiotemporal patterns of diffusive predator–prey models with mutual interference

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction–diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction–diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Bifurcation analysis of a diffusive predator–prey model with ratio-dependent Holling type III functional response

The spatial, temporal, and spatiotemporal dynamics of a reaction–diffusion predator–prey system with ratio-dependent Holling type III functional response, under homogeneous Neumann boundary conditions, are studied in this paper. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented. For the reaction–diffusion model, firstly the parameter regions for the stability or instability of the unique constant steady state are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next, it is proved that the model exhibits Hopf bifurcation, which produces temporal inhomogeneous patterns. Finally, the existence and nonexistence of nonconstant steady- state solutions are established by bifurcation method and energy method, respectively. Numerical simulations are presented to verify and illustrate the theoretical results.

Stability and Hopf bifurcation in a reaction–diffusion predator–prey system with interval biological parameters and stage structure

This paper deals with a delayed reaction–diffusion predator–prey system combined with stage structure for prey and interval biological parameters. By taking the sum of delays as the bifurcation parameter, the local stability of the equilibrium points is investigated and the condition of Hopf bifurcation is obtained. In succession, using the normal form theory and the center manifold reduction for partial functional differential equations, we derive the explicit formulas determining stability, direction and other properties of bifurcating periodic solutions. Some numerical simulations are also included to illustrate our theoretical results.

Dynamics in a diffusive predator–prey system with strong Allee effect and Ivlev-type functional response

The dynamics of a kind of reaction–diffusion predator–prey system with strong Allee effect in the prey population is considered. We prove the existence and uniqueness of the solution and give a priori bound. Hopf bifurcation and steady state bifurcation are studied. Results show that the Allee effect has significant impact on the dynamics.

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

Traveling Wave Solution in a Diffusive Predator-Prey System with Holling Type-IV Functional Response

We establish the existence of traveling wave solution for a reaction-diffusion predator-prey system with Holling type-IV functional response. For simplicity, only one space dimension will be involved, the traveling solution equivalent to the heteroclinic orbits inR3. The methods used to prove the result are the shooting argument and the invariant manifold theory.

Dynamics of n+1-dimensional impulsive reaction-diffusion predator-prey system with Holling type III functional response

An n+1-dimensional impulsive reaction-diffusion periodic predator-prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.

Strong Allee effect in a diffusive predator–prey system with a protection zone

A reaction–diffusion predator–prey system with strong Allee effect and a protection zone for the prey is considered. Dynamics and steady state solutions of the system are analyzed. In particular it is shown that the overexploitation phenomenon can be avoided if the Allee effect threshold is low and the protection zone is large.

The role of space in stage-structured cannibalism with harvesting of an adult predator

The dynamics of a predator–prey reaction–diffusion system with Holling type III functional response, where the predator has two stages, a juvenile stage and an adult stage, is proposed and analyzed to study the effect of predation with non-constant harvesting of an adult predator. Our analysis leads to different thresholds in terms of the model parameters acting as conditions under which the organisms associated with the system cannot thrive even in the absence of predation. Local stability of the system is obtained in the absence of one or more organisms and in the presence of all the organisms. Moreover, it is shown that the system undergoes Hopf bifurcation when the intrinsic growth rate of herbivorous prey crosses certain critical value. Computer simulations have been carried out to illustrate various analytical results.

TRAVELING WAVES IN A REACTION-DIFFUSION PREDATOR-PREY SYSTEM WITH NONLOCAL DELAYS

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with nonlocal delays. By introducing a partially exponential quasi-monotonicity condition and a new cross iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. By constructing a desirable pair of upper-lower solutions, we establish the existence of traveling wave solutions. Finally, some numerical examples are carried out to illustrate the theoretical results.

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.