Complex patterns in a predator–prey model with self and cross-diffusion

In this paper, we present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self as well as cross-diffusion in a Beddington–DeAngelis-type predator–prey model. The instability of the uniform equilibrium of the model is discussed, and the sufficient conditions for the instability with zero-flux boundary conditions are obtained. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to stripes-spots, but also to hot/cold spots, stripes and wave pattern replication. This may enrich the pattern formation in cross-diffusive predator–prey model.     

Global existence of classical solutions to a cross-diffusion predator–prey system with a free boundary

This paper is concerned with a cross-diffusion predator–prey system with a free boundary over a one-dimensional habitat. The free boundary shows the spreading front of the prey and predator which implies that the velocity of the expanding front is proportional to the gradients of the prey and predator. By the contraction mapping principle, \(L^{p}\) estimates and Schauder estimates of parabolic equations, the local and global existence and uniqueness of classical solutions are established for this system.     

Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system

This paper deals with a generalized predator–prey system with cross-diffusion and homogeneous Neumann boundary condition, where the cross-diffusion is included in such a way that the prey runs away from the predator. We first give a priori estimate for positive steady states to the system. Then we obtain the non-existence result of non-constant positive steady states. Finally, we investigate the stability of constant equilibrium point and the existence of non-constant positive steady states. It is shown that the system admits a non-constant positive steady state provided that either of the self-diffusions is large or the cross-diffusion is small.     

Steady state in a cross-diffusion predator–prey model with the Beddington–DeAngelis functional response

The purpose of this paper is to study the existence of steady state in a linear cross-diffusion predator–prey model with Beddington–DeAngelis functional response. The proofs mainly rely on Fixed point index theory and analytical techniques.     

The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey

This paper considers the effect of immigration and refuge on the dynamics of a three species system in which one predator feeds on one of two competing species. Immigration is assumed only for the species which is not attacked by the predator.The main results address the stability of the system. Namely, it is shown that increasing the number of refuges stabilizes the system, whereas the opposite holds true by increasing the immigration rate. Also, one result about the persistence of the system and one concerning the global stability of the coexistence equilibrium are presented. Some numerical simulations illustrate the obtained results.     

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.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

Harvesting of a prey–predator fishery in the presence of toxicity

In this model we discuss the bioeconomic harvesting of a prey–predator fishery in which both the species are infected by some toxicants released by some other species. Here both the species are harvested where we use the usual catch-per-unit-effort hypothesis. The dynamical behaviour of the exploited system is examined. The possibility of existence of a bionomic equilibrium is considered. The optimal harvesting policy is studied by using Pontryagin’s maximal principle. Some numerical examples and the corresponding solution curves are studied to illustrate the results of the model. Finally, the existence of limit cycle is discussed.     

Novel dynamics of a predator–prey system with harvesting of the predator guided by its population

A predator–prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. The proposed model is a nonsmooth dynamic system with switches between the original predator-prey model (free subsystem) and a model with nonlinear harvesting (harvesting subsystem). We initially examine the dynamics of both the free and the harvesting subsystems, and then we investigate the dynamics of the switching system using theories of nonsmooth systems. Theoretical results showed that the harvesting subsystem undergoes multiple bifurcations, including saddle-node, supercritical Hopf, Bogdanov–Takens and homoclinic bifurcations. The switching system not only retains all of the complex dynamics of the harvesting system but also exhibits much richer dynamics such as a sliding equilibrium, sliding cycle, boundary node (saddle point) bifurcation, boundary saddle-node bifurcation and buckling bifurcation. Both theoretical and numerical results showed that, by implementing predator population guided harvesting, the predator and prey population could coexist in more scenarios than those in which the predator may go extinct for the continuous harvesting regime. They could either stabilize at an equilibrium or oscillate periodically depending on the value of the economic threshold and the initial value of the system.     

Solution of prey–predator problem by numeric–analytic technique

In this paper, an analytical expression for the solution of the prey–predator problem by an adaptation of the classical Adomian decomposition method (ADM). The ADM is treated as an algorithm for approximating the solution of the problem in a sequence of time intervals, i.e. the classical ADM is converted into a hybrid numeric–analytic method called the multistage ADM (MADM). Numerical comparisons with the classical ADM, and the classical fourth-order Rungge–Kutta (RK4) methods are presented.     

Evolution in predator–prey systems

We study the adaptive dynamics of predator–prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see that (i) the number of coexisting predators remains tight and the differences in traits from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we calculate the asymptotic rate at which the traits of the least and most “fit” predators in the population increase. This last result is obtained by comparison with a branching random walk killed to the left of a linear boundary and a finite branching–selection particle system.     

A sufficient and necessary condition for the existence of positive solutions for a prey–predator system with Ivlev-type functional response

We deal with a predator–prey interaction model with Ivlev-type functional response which is subject to the homogeneous Dirichlet boundary condition. On the basis of the fixed point index theory, we give a sufficient and necessary condition for the existence of positive solutions of the model.     

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.     

Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.     

Multiple periodic solutions of a delayed predator–prey system with stage structure for the predator

By using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a delayed predator–prey system with stage structure for the predator is established.     

Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.