Stability and bifurcation in a generalized delay prey–predator model

The present paper considers a generalized prey–predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.     

Hopf bifurcation of a predator–prey system with predator harvesting and two delays

In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

Computational analysis on Hopf bifurcation and stability for a consumer–resource model with nonlinear functional response

We consider a consumer–resource model with nonlinear functional response and reaction–diffusion terms. By taking the growth rate of the resource as the parameter, we give a computational and theoretical analysis on Hopf bifurcation emitting from the positive equilibrium for the model and discuss the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions by space decomposition and vector operation techniques. It is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Some numerical examples are presented to support and illustrate our theoretical analysis.     

Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.     

A delayed ratio-dependent predator–prey model of interacting populations with Holling type III functional response

In this article, we study a ratio-dependent predator–prey model described by a Holling type III functional response with time delay incorporated into the resource limitation of the prey logistic equation. This investigation includes the influence of intra-species competition among the predator species. All the equilibria are characterized. Qualitative behavior of the complicated singular point (0,0) in the interior of the first quadrant is investigated by means of a blow-up transformation. Uniform persistence, stability, and Hopf bifurcation at the positive equilibrium point of the system are examined. Global asymptotic stability analyses of the positive equilibrium point by the Bendixon–Dulac criterion for non-delayed model and by constructing a suitable Lyapunov functional for the delayed model are carried out separately. We perform a numerical simulation to validate the applicability of the proposed mathematical model and our analytical findings.     

Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities

Poor dispersion characteristics of rockets, due to the orientation of the launcher for multiple launch rocket system (MLRS) departing from that intended, have always restricted the MLRS development for several decades. Orienting control is a key technique to improve the dispersion characteristics of rockets. The purpose of this paper is to propose an orienting control method for launcher of the MLRS in a salvo firing. Because the MLRS is a typical nonlinear system, the major difficulty in designing the orienting controller lies in the nonlinearity. To deal with the nonlinearity, the concept of computed torque control is introduced. The MLRS equation of motion is established using Lagrange method. The inner loop feedforward and the outer loop feedback are adopted to design the controllers for the azimuth and elevation axes of MLRS. By combining the inner and outer control loops together, the PID-computed torque controller is designed. The numerical simulation is implemented to show the control performance, and then, the effectiveness and applicability of the proposed controller are demonstrated by the firing experiment of a salvo of three rockets.     

A delay-induced predator–prey model with Holling type functional response and habitat complexity

Nonlinear Dynamics - A delay-induced predator–prey system with the effect of habitat complexity and Holling type functional response is proposed. The dynamical behaviors of the presented...     

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.     

Bifurcation and stability analysis in predator–prey model with a stage-structure for predator

A predator–prey system with Holling type II functional response and stage-structure for predator is presented. The stability and Hopf bifurcation of this model are studied by analyzing the associated characteristic transcendental equation. Further, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from positive equilibrium is derived by the normal form theory and center manifold argument. Some numerical simulations are also given to illustrate our results.     

Nonlinear dynamics of a delayed Leslie predator–prey model

The present paper is concerned with a delayed Leslie predator–prey model. The conditions of boundedness of the solutions of the system, existence, and stability of the equilibrium of the system are investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. The extensive simulations carried out show that the bifurcations arise around the positive equilibrium.     

Global stability of periodic solutions for a discrete predator–prey system with functional response

The purpose of this paper is to study the existence and global stability of a periodic solution for a discrete predator–prey system with the Beddington–DeAngelis functional response and predator cannibalism. By using the continuation theorem, the existence conditions of at least one periodic solution are obtained, and the sufficient conditions, which ensure the global stability of the positive periodic solution, are derived by constructing a special Lyapunov function.     

Bifurcations of a predator–prey system with cooperative hunting and Holling III functional response

Nonlinear Dynamics - In this paper, we consider the dynamics of a predator–prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known...