Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects

This paper is concerned with a delayed predator–prey diffusive system with Neumann boundary conditions. The bifurcation analysis of the model shows that Hopf bifurcation can occur by regarding the delay as the bifurcation parameter. In addition, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are also discussed by employing the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, the effect of the diffusion on bifurcated periodic solution is considered.     

Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect

In this paper, we consider a ratio-dependent predator–prey system with diffusion. And we mainly discuss the following problems: (1) stability and Hopf bifurcation analysis of the positive equilibrium for the reduced ODE system; (2) Diffusion-driven instability of the equilibrium solution; (3) Hopf bifurcations for the corresponding diffusion system with homogeneous Neumann boundary conditions. In order to verify our theoretical results, some numerical simulations are also included, respectively.     

An optimal control problem for a prey–predator system with a general functional response

An optimal control problem is studied for a prey–predator system with a general functional response. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The number of switching points of the optimal control is discussed in terms of the sign of a specific constant.     

Weak-renormalized solutions for predator–prey system

In this article, we consider the predator–prey system with Dirichlet boundary conditions which is used in the modelling of ecology. Under the assumptions of no growth conditions and integrable data, we prove the existence of weak-renormalized solutions to the predator–prey system.     

Turing pattern formation in a predator–prey system with cross diffusion

The paper explores the impacts of cross-diffusion on the formation of spatial patterns in a ratio-dependent predator–prey system with zero-flux boundary conditions. Our results show that under certain conditions, cross-diffusion can trigger the emergence of spatial patterns which is however impossible under the same conditions when cross-diffusion is absent. We give a rigorous proof that the model has at least one spatially heterogenous steady state by means of the Leray–Schauder degree theory. In addition, numerical simulations are performed to visualize the complex spatial patterns.     

Traveling wavefronts for a two-species predator–prey system with diffusion terms and stage structure

In this paper, we considered an important model describing a two-species predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the system and obtained the locally stable conditions. And by using the approach introduced by Canosa [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17 (1973) 307–313] and the method of upper and lower solutions, we studied the existence of traveling wavefronts, connecting the zero solution with the positive equilibrium of the system. Our results show that the traveling wavefronts exist and appear to be monotone. Finally, we given a conclusion to summarize the overall achievements of the work presented in the paper.     

Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.     

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.     

A singular reaction–diffusion system modelling prey–predator interactions: Invasion and co-extinction waves

We consider a singular reaction–diffusion system arising in modelling prey–predator interactions in a fragile environment. Since the underlying ODEs system exhibits a complex dynamics including possible finite time quenching, one first provides a suitable notion of global travelling wave weak solution. Then our study focusses on the existence of travelling waves solutions for predator invasion in such environments. We devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co-extinction tail for both species. Under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere, such a property arising for every admissible wave speeds.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food

Spatiotemporal dynamics of a predator–prey system in presence of spatial diffusion is investigated in presence of additional food exists for predators. Conditions for stability of Hopf as well as Turing patterns in a spatial domain are determined by making use of the linear stability analysis. Impact of additional food is clear from these conditions. Numerical simulation results are presented in order to validate the analytical findings. Finally numerical simulations are carried out around the steady state under zero flux boundary conditions. With the help of numerical simulations, the different types of spatial patterns (including stationary spatial pattern, oscillatory pattern, and spatiotemporal chaos) are identified in this diffusive predator–prey system in presence of additional food, depending on the quantity, quality of the additional food and the spatial domain and other parameters of the model. The key observation is that spatiotemporal chaos can be controlled supplying suitable additional food to predator. These investigations may be useful to understand complex spatiotemporal dynamics of population dynamical models in presence of additional food.     

Two-patches prey impulsive diffusion periodic predator–prey model

In this paper, we study a periodic predator–prey system with prey impulsive diffusion in two patches. On the basis of comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the predator–prey system where predator have not other food source are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results. A conclusion is given in the end.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

Permanence and periodic solutions for a diffusive ratio-dependent predator–prey system

This paper considers a diffusive predator–prey model, in which there is a ratio-dependent functional response with Holling III type. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. The existence of a unique globally stable periodic solution is also presented.     

The effect of predator competition on positive solutions for a predator–prey model with diffusion

A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large m$m$. Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when k$k$ goes to ∞$\infty$, we find that there is at most a positive solution for any c∈R$c\in R$ when k$k$ is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension.     

Qualitative behaviour of a ratio-dependent predator–prey system

This paper deals with the qualitative properties of an autonomous system of differential equations, modeling ratio-dependent predator–prey interactions.This model differs from traditional ratio dependent models essentially in the predator mortality term, the death rate of the predator is not constant but instead increases when there is overcrowding.We incorporate delay(s) into the system. The most important observation is that as the delay(s) is (are) increased the originally asymptotic stable interior equilibrium loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place: small amplitude periodic solutions arise.