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.     

Effect of delay on a predator–prey model with parasitic infection

In the paper an eco-epidemic system with delay and parasitic infection in the prey is investigated. The conditions for asymptotic stability of steady states are derived and the length of the delay preserving the stability is also estimated. Further, the criterion for existence of Hopf-type small amplitude periodic oscillations of the predator and prey biomass is derived. Numerical results indicate that the delay does not affect the stability of the system in the process but makes all populations oscillate more intensively. In addition, the results show that the recovery makes the levels of the infected prey and the predator become lower but makes the sound prey higher in limit time.     

Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.     

Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting

In this article, we investigate a prey– predator model with Allee effect and state-dependent impulsive harvesting. We obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (1.2) by means of the geometry theory of semicontinuous dynamic system and the method of successor function. We also obtain that system (1.2) exhibits the phenomenon of heteroclinic bifurcation about parameter $\alpha $ . The methods used in this article are novel and prove the existence of order-1 periodic solution and heteroclinic bifurcation.     

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.     

Biological control of a predator–prey system through provision of a super predator

In this paper, we make a systematic analysis of the dynamics of a predator–prey system with type-II functional response, in which the predator growth rate is affected by the presence of a super predator. The main aim of this research is to study the consequences of the presence of a super predator on the system dynamics. The existence and stability of the different possible equilibrium points are studied, and we conclude that the maximum consumption rate of a super predator plays a key role in determining the eventual state of the ecosystem. A detailed bifurcation analysis is carried out through numerical simulations, and we observe that theoretically it is possible to control the dynamics of the system by manipulating the consumption rate of the super predator.     

Optimal predator control policy and weak Allee effect in a delayed prey–predator system

Nonlinear Dynamics - In this article, a system of delay differential equations to represent the predator–prey dynamics with weak Allee effect in the growth of predator population is...     

Influence of prey refuge on predator–prey dynamics

A diffusive predator–prey system with Michaelis–Menten type functional response subject to prey refuge is considered. Bifurcation analysis of Hopf and Turing are carried out in detail. In particular, Turing domain is given in the two parameters space. The obtained results show that the refuges used by prey have great influence on the pattern formation of the populations. More specifically, as prey refuge being increased, spotted pattern and coexistence of spotted and stripe-like pattern emerge. It is also proved that the pattern is not dependent on the initial conditions, which means the pattern is controlled by the intrinsic mechanism.     

Impact of noise on pattern formation in a predator–prey model

In this paper, we investigated the influence of color noise on pattern formation in a predator–prey model. When the model has no noise, it exhibits wave dynamics. A series of numerical simulations showed that break-up of waves will emerge when noise is added. Furthermore, stationary pattern can be induced by noise. The obtained results may point out that noise can have great effect on spatial complexity of ecosystems.     

The genic mutation on dynamics of a predator–prey system with impulsive effect

In this work, we consider a genic mutational predator?Cprey system with birth pulse and impulsive cutting on prey population at different moments. All the solutions of the investigated system are proved to be uniformly ultimately bounded. The conditions of the globally asymptotically stable predator-extinction boundary periodic solution of the investigated system are obtained. The permanent conditions of the investigated system are also obtained. Finally, numerical simulations are inserted to illustrate the results. Our results present that the genic mutational rate plays an important role on the permanence of the investigated system. Our results also provide reliable tactic basis for the practical biological economics management.     

Effort dynamics of a delay-induced prey–predator system with reserve

This paper describes a prey?Cpredator fishery system with prey dispersal in a two-patch environment, one of which is a free fishing zone and the other a protected zone. The proposed system reflects the dynamic interaction between the net economic revenue and the fishing effort used to harvest the population in presence of a suitable tax. Local as well as global stability of the system is analyzed. The optimal taxation policy is formulated and solved with the help of Pontryagin??s maximal principle. The objective of the paper is to achieve the sustainability of the fishery, keeping the ecological balance, and maximize the monetary social benefit. The dynamical behavior of the delay system is further analyzed through incorporating discrete type gestational delay of predators, and the existence of Hopf bifurcation phenomenon is checked at the interior equilibrium point. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Theoretical results are verified with the help of numerical examples and graphical illustrations.     

Dynamics in two nonsmooth predator–prey models with threshold harvesting

Considering a good pest control program should reduce the pest to levels acceptable to the public, we investigate the threshold harvesting policy on pests in two predator–prey models. Both models are nonsmooth and the aim of this paper is to provide how threshold harvesting affects the dynamics of the two systems. When the harvesting threshold is larger than some positive level, the harvesting does not affect the ecosystem; when the harvesting threshold is less than the level, the model has complex dynamics with multiple coexistence equilibria, limit cycle, bistability, homoclinic orbit, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation, and discontinuous Hopf bifurcation. Firstly, we provide the complete stability analysis and bifurcation analysis for the two models. Furthermore, some numerical simulations are given to illustrate our results. Finally, it is found that harvesting lowers the level of both species for natural enemy–pest system while raises the densities of both species for the pest–crop system. It is seen that the threshold harvesting policy of the enemy system is more effective than the crop system.     

Turing patterns in a predator–prey model on complex networks

Nonlinear Dynamics - Predator–prey model with modified Leslie–Gower and Holling type III schemes governed by reaction–diffusion equations can exhibit diversified pattern...     

Turing patterns induced by self-diffusion in a predator–prey model with schooling behavior in predator and prey

Nonlinear Dynamics - This article considers a reaction–diffusion predator–prey model with schooling behavior both in predator and prey species and subject to the homogeneous Neumann...     

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...     

Spatiotemporal dynamics of a predator–prey model

Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. In this paper, we consider a Holling?CTanner model with spatial diffusion. Choosing appropriate parameter values in parameter spaces, we obtain rich patterns, including spotted, black-eye, and labyrinthine patterns. The numerical results show that predator?Cprey system can exhibit complicated behavior.