Qualitative analysis of a diffusive prey–predator model

In this work we examine a Lotka–Volterra model with diffusion describing the dynamics of multiple interacting prey and predator species. We show that the solution exists, and is unique, bounded, nonnegative, and globally defined. We also prove the non-existence of nonconstant steady state solutions if certain conditions are satisfied. For the particular case of two prey (e.g., engineered and native, respectively) and one common predator species, by performing a linear stability analysis about the initial native-dominant steady state, we determine under which conditions the engineered species invasion succeeds.     

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

In this paper, a diffusive predator–prey system with Holling III functional response and nonconstant death rate subject to Neumann boundary condition is considered. We study the stability of equilibria, and Turing instability of the positive equilibrium. We also perform a detailed Hopf bifurcation analysis to PDE system, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. In addition, some numerical simulations are carried out.     

Positive steady states of a diffusive predator–prey system with modified Holling–Tanner functional response

In this paper, we study a diffusive predator–prey system with modified Holling–Tanner functional response under homogeneous Neumann boundary condition. The qualitative properties, including the global attractor, persistence property, local and global asymptotic stability of the unique positive constant equilibrium are obtained. We also establish the existence and nonexistence of nonconstant positive steady states of this reaction–diffusion system, which indicates the effect of large diffusivity.     

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.     

Bifurcation analysis in a diffusive Segel–Jackson model

Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling predator–prey interaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns.     

Diffusive Holling–Tanner predator–prey models in periodic environments

In this paper, by using the Lyapunov method, we establish sufficient conditions for the global asymptotic stability of the positive periodic solution to diffusive Holling–Tanner predator–prey models with periodic coefficients and no-flux conditions.     

Analysis of a predator–prey model with modified Holling–Tanner functional response and time delay

In paper, a predator–prey model with modified Holling–Tanner functional response and time delay is discussed. It is proved that the system is permanent under some appropriate conditions. The local stability of the equilibria is investigated. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the positive equilibrium of the model.     

Bifurcation analysis in a discrete differential-algebraic predator–prey system

The discrete-time predator–prey biological economic system obtained by Euler method is investigated. Some conditions for the system to undergo flip bifurcation and Neimark–Sacker bifurcation are derived by using new normal form of differential-algebraic system, center mainfold theorem and bifurcation theory. Numerical simulations are given to show the effectiveness of our results and also to exhibit period-doubling bifurcation in orbits of period 2, 4, 8 and chaotic sets. The results obtained here reveal far richer dynamics in discrete differential-algebraic biological economic system. The contents are interesting in mathematics and biology.     

Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects

In this article, we study the Holling–Tanner predator–prey model with nonlinear diffusion terms under homogeneous Neumann boundary condition. The nonlinear diffusion terms here mean that the prey runs away from the predator, and the predator chases the prey. Nonexistence and existence of nonconstant positive steady states are obtained, which reveal that cross-diffusion can create spatial patterns even when the random diffusion fails to do so. Moreover, asymptotic behaviour of positive solutions as the cross-diffusion tends to ∞ is shown.     

Bifurcation analysis of an autonomous epidemic predator–prey model with delay

In this paper, a class of an autonomous epidemic predator–prey model with delay is considered. Its linear stability and Hopf bifurcation are investigated. Applying the normal form theory and center manifold theory, the explicit formulas for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.     

Dynamics for a diffusive prey–predator model with different free boundaries

To understand the spreading and interaction of prey and predator, in this paper we study the dynamics of the diffusive Lotka–Volterra type prey–predator model with different free boundaries. These two free boundaries, which may intersect each other as time evolves, are used to describe the spreading of prey and predator. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behaviors of global solution. Some sufficient conditions for spreading and vanishing are established. When spreading occurs, we provide the more accurate limits of $(u,v)$ as $t\to \infty$, and give some estimates of asymptotic spreading speeds of $u,v$ and asymptotic speeds of $g,h$. Some realistic and significant spreading phenomena are found.     

Existence of positive stationary solutions for a diffusive variable-territory prey–predator model

This paper concerns the existence of positive stationary solutions for a diffusive variable-territory prey–predator model, and completely settles an open problem of Wang and Pang (2009). The main result closes a gap in an earlier result (2011) by the authors.     

Asymptotic profiles of positive steady states for a diffusive predator–prey model with predator interference

This paper is concerned with positive steady states for a diffusive predator–prey model with predator interference in a spatially heterogeneous environment. We first establish the necessary and sufficient conditions for the existence of positive steady states. In order to get a better understanding of the structure of positive steady states, we further investigate the asymptotic profiles of positive steady states as some parameter tends to zero or infinity.     

Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition

This article discusses a predator–prey system with predator saturation and competition functional response. The local stability, existence of a Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained in the absence of diffusion. Further, we discuss the diffusion-driven instability, Hopf bifurcation for corresponding diffusion system with zero flux boundary condition and Turing instability region regarding the parameters are established. Finally, numerical simulations supporting the theoretical analysis are also included.