Optimal harvesting policy of a stochastic predator–prey model with time delay

This paper concerns the optimal harvesting of a stochastic delay predator–prey model. Sufficient and necessary conditions for the existence of an optimal control are established. The optimal harvesting effort and the maximum value of the cost function are obtained as well. Some numerical tests are given to illustrate the main results.     

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.     

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.     

Stability and bifurcation analysis for a fractional prey–predator scavenger model

In this study, we consider a fractional prey–predator scavenger model as well as harvesting by a predator and scavenger. We prove the positivity and boundedness of the solutions in this system. The model undergoes a Hopf bifurcation around one of the existing equilibria where the conditions are met for the occurrence of a Hopf bifurcation. The results show that chaos disappears in this biological model. We conclude that the fractional system is more stable compared with the classical case and the stability domain can be extended under fractional order. In addition, a suitable amount of prey harvesting and a fractional order derivative can control the chaotic dynamics and stabilize them. We also present an extended numerical simulation to validate the results.     

Optimal control of effort of a stage structured prey–predator fishery model with harvesting

This paper describes a prey–predator fishery model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. We have incorporated state feedback controller to stabilize the model system in the case of positive economic interest. Fishing effort used to harvest the adult prey and predator populations is used as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.     

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.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

Dynamics of a delayed predator–prey model with predator migration

Based on the availability of prey and a simple predator–prey model, we propose a delayed predator–prey model with predator migration to describe biological control. We first study the existence and stability of equilibria. It turns out that backward bifurcation occurs with the migration rate as bifurcation parameter. The stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the positive equilibrium may be delay-dependent. Moreover, delay can switch the stability of the positive equilibrium. When the positive equilibrium loses stability, Hopf bifurcation can occur. The direction and stability of Hopf bifurcation is derived by applying the center manifold method and the normal form theory. The main theoretical results are illustrated with numerical simulations.     

Optimal harvesting policy for a stochastic predator–prey model

Optimal harvesting of a stochastic predator–prey model is considered in this paper. Sufficient and necessary criteria for the existence of optimal harvesting strategy are obtained. At the same time, the optimal harvest effort and the maximum of sustainable yield are given.     

Dynamic complexities of predator–prey system in a pulsed chemostat

In this paper, we study a food chain model with Holling III and Monod type functional response under periodic pulsed conditions, which contains with predator, prey and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in prey and predator. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

The present paper aims to provide a detailed qualitative analysis of a non-smooth Gause predator–prey model. In this model, the saturating functional response function with a discontinuity at a critical prey density was employed to show the effects of a prey refuge on the population dynamic behavior. Analysis of this model revealed rich dynamics including locally (or globally) stable canard cycles, a locally (globally) stable pseudo-equilibrium, unbounded trajectories in which both populations go to infinity or the prey goes to infinity and the predator dies out eventually. The main purpose of the present work is to carry out a completely qualitative analysis for this model. In particular, two sets of sufficient conditions drive both populations to approach infinity and the sufficient and necessary conditions for all of the other main results are presented.     

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.     

Hopf bifurcation and center stability for a predator–prey biological economic model with prey harvesting

In this paper, we investigate Hopf bifurcation and center stability of a predator–prey biological economic model. By employing the local parameterization method, Hopf bifurcation theory and the formal series method, we obtain some testable results on these issues. The economic profit is chosen as a positive bifurcation parameter here. It shows that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Besides, we also find that the center of the biological economic model is always unstable. Finally, some numerical simulations are given to illustrate the effectiveness of our results.     

Qualitative analysis of Holling type II predator–prey systems with prey refuges and predator restricts

This paper is devoted to investigation of Holling type II predator–prey systems with prey refuges and predator restricts. Using a transformation technique, we change the system into a generalized Liénard system and give sufficient conditions to ensure the global stability of the positive equilibrium and existence and uniqueness of a stable limit cycle. We also find the property of alternation for phase structure of the system.     

Complex dynamics of a diffusive predator–prey model with strong Allee effect and threshold harvesting

In this paper, complex dynamics of a diffusive predator–prey model is investigated, where the prey is subject to strong Allee effect and threshold harvesting. The existence and stability of nonnegative constant steady state solutions are discussed. The existence and nonexistence of nonconstant positive steady state solutions are analyzed to identify the ranges of parameters of pattern formation. Spatially homogeneous and nonhomogeneous Hopf bifurcation and discontinuous Hopf bifurcation are proved. These results show that the introduction of strong Allee effect and threshold harvesting increases the system spatiotemporal complexity. Finally, numerical simulations are presented to validate the theoretical results.     

Codimension-two bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

In this paper, we investigate dynamical behaviours of a discrete predator–prey model with nonmonotonic functional response. Codimension-2 bifurcations associated with 1:2, 1:3 and 1:4 resonances are analyzed by using bifurcation theory. Codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams and phase portraits, which not only illustrate the validity of the theoretical results but also display some interesting complex dynamical behaviours, are obtained by numerical simulations.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.