Stage-structured Predator Prey System Research Articles

Role of multiple time delays on a stage-structured predator–prey system in a toxic environment

In this paper, a stage-structured predator–prey model is considered in a toxic environment, emphasizing on the affect of the toxic substances on growth of the species in presence of time delays. The positivity and boundedness of solutions have been discussed. The analytical and numerical simulations of the proposed model show that the coexistence equilibrium point is stable for low and moderate values of the delay parameter associated with the predator’s maturity. Even though there is an interval of the maturation time in which the system exhibits oscillatory behaviour, a longer maturity duration may lead to the extinction of the predator species. The impact of toxicity on the proposed system is also investigated. The stability of the system repeatedly switches as the time delay, related to the changes of toxicity incurred in the adult predator, varies, and finally becomes unstable for a significant value of it. Besides that, the delay related to the prey’s toxicity has also led some exciting results.

Stability Analysis of a Prey-Predator Model with Prey Refuge and Fear of Adult Predator

This paper is concerned with a Holling-II stage-structured predator-prey system in which predators are divided into an immature and mature predators. The aim is to explore the impact of the prey's fear caused by the dread of mature predators in a prey-predator model including intraspecific competitions and prey shelters. The theoretical study includes the local and global stability analysis for the three equilibrium points of the system and shows the prey's fear may lead to improving the stability at the positive equilibrium point. A numerical analysis is given to ensure the accuracy of the theoretical outcomes and to testify the conditions of stability of the system near the non-trivial equilibrium points.

Stability Analysis of a Stage-Structure Predator–Prey Model with Holling III Functional Response and Cannibalism

This paper considers the time taken for young predators to become adult predators as the delay and constructs a stage-structured predator–prey system with Holling III response and time delay. Using the persistence theory for infinite-dimensional systems and the Hurwitz criterion, the permanent persistence condition of this system and the local stability condition of the system’s coexistence equilibrium are given. Further, it is proven that the system undergoes a Hopf bifurcation at the coexistence equilibrium. By using Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium and the coexistence equilibrium are globally asymptotically stable, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Some numerical simulations are carried out to illustrate the main results.

The (De)Stabilizing effect of juvenile prey cannibalism in a stage-structured model.

Cannibalism, or intraspecific predation, is the act of an organism consuming another organism of the same species. In predator-prey relationships, there is experimental evidence to support the existence of cannibalism among juvenile prey. In this work, we propose a stage-structured predator-prey system where cannibalism occurs only in the juvenile prey population. We show that cannibalism has both a stabilizing and destabilizing effect depending on the choice of parameters. We perform stability analysis of the system and also show that the system experiences a supercritical Hopf, saddle-node, Bogdanov-Takens and cusp bifurcation. We perform numerical experiments to further support our theoretical findings. We discuss the ecological implications of our results.

Chaos Detection and Optimal Control in a Cannibalistic Prey–Predator System with Harvesting

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

Dynamics of a discrete-time stage-structured predator–prey system with Holling type II response function

A discrete-time model of predator–prey dynamics with Holling type II response function is proposed. Each species considered has its age structure which is typical for natural communities. The bifurcations, dynamic modes and a possibility of its shifting are studied for the proposed model. The stability loss of a non-trivial fixed point is realised according to Neimark–Sacker and Feigenbaum scenarios. The phase space structure of multistability areas in which a variation of current numbers of prey or predator can shift the dynamic modes is analysed using basins of attraction. Particular attention is paid to the investigation of the feasible values of the parameters in which the model has a biological meaning. To understand the mechanisms of interspecific interaction influence on each species dynamics in a community, we compare cases with and without interspecific interaction. The presence of interspecific interaction in the community increases the variety of emerging dynamic modes of the predator population size.

Qualitative analysis and control for predator-prey delays system

Abstract In this paper, the dynamics of a stage-structured predator-prey system with two time delays and Monod–Haldane response function are considered. By taking the two time delays as the bifurcation parameter, the local stability of the interior equilibrium is established and the conditions for existence of the Hopf bifurcation are obtained. Furthermore, based on the normal form method and center manifold theorem, the direction of the Hopf bifurcation and the stability of the bifurcation period solutions are investigated. In addition, by using parameter disturbance and state feedback control to act on the system, we succeeded in controlling the Hopf bifurcation of the original system. Numerical examples are given to carry out to support the obtained theoretical findings.

Deadly competition and life-saving predation: the potential for alternative stable states in a stage-structured predator-prey system.

Predators often undergo complete ontogenetic diet shifts, engaging in resource competition with species that become their prey during later developmental stages. Theory posits that this mix of stage-specific competition and predation, termed life-history intraguild predation (LHIGP), can lead to alternative stable states. In one state, prey exclude predators through competition (i.e. juvenile competitive bottleneck), while in the alternative, adult predators control prey density to limit competition and foster coexistence. Nevertheless, the interactions leading to these states have not been demonstrated in an empirical LHIGP system. To address this gap, we manipulated densities of cannibalistic adult cyclopoid copepods (Mesocyclops edax) and their cladoceran prey (Daphnia pulex) in a response-surface design and measured the maturation and survival of juvenile copepods (nauplii). We found that Daphnia reduced and even precluded both nauplii maturation and survival through depletion of a shared food resource. As predicted, adult copepods enhanced nauplii maturation and survival through Daphnia consumption, yet this positive effect was dependent on the relative abundance of Daphnia as well as the absolute density of adult copepods. Adult copepods reduced nauplii survival through cannibalism at low Daphnia densities and at the highest copepod density. This work demonstrates that predation can relax a strong juvenile competitive bottleneck in freshwater zooplankton, though cannibalism can reduce predator recruitment. Thus, our results highlight a key role for cannibalism in LHIGP dynamics and provide evidence for the interactions that drive alternative stable states in such systems.

Bifurcation analysis of a delayed predator-prey system with stage structure and Holling-II functional response

This paper is concerned with a stage-structured predator-prey system with Holling-II functional response and two delays. Choosing a possible combination of the two delays as the bifurcation parameter, the existence of the Hopf bifurcation of the system is discussed. Furthermore, the properties of the Hopf bifurcation such as the direction and the stability are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are presented to justify the theoretical results.

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACTA stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.

Dynamic analysis of a stage-structured predator–prey system with disturbed time delay and birth pulse

In this paper, a stage-structured predator–prey system with birth pulse and disturbed time delay is investigated. The conditions of the prey-extinction periodic solution of the system which are globally attractive have been obtained. Furthermore, the sufficient conditions for the permanence of the system are established. Finally, numerical analysis is given to confirm the theoretical results.

Bifurcation analysis in a stage-structured predator–prey model with maturation delay

In this paper, we investigate the impact of maturation delay on the positive equilibrium solutions in a stage-structured predator–prey system. By analyzing the characteristic equation we derive the conditions for the emergence of Hopf bifurcation. By applying the normal form and the center manifold argument, the direction as well as the stability of periodic solutions bifurcating from Hopf bifurcation is explored. Results show that maturation delay can change the nature of the positive equilibrium solutions, and the loss of equilibrium stability occurs as a consequence of Hopf bifurcation. When Hopf bifurcation takes place, periodic solution arises and is further demonstrated to be asymptotically stable. In addition, the periodic solutions appear only for intermediate maturation delay, that is, there exists a delay window, outside of which the positive equilibrium is locally stable. Furthermore, numerical analysis shows that Hopf bifurcation is favored by a superior competition for adult predators to juveniles, a smaller mortality on juvenile and/or adult predators, and a higher resource carrying capacity. Interestingly, increasing food carrying capacity can lead to the emergence of irregular chaotic dynamics and regular limit cycles.

Stability and bifurcation in a stage-structured predator–prey system with Holling-II functional response and multiple delays

In this paper, we analyse a delayed Holling-II predator–prey system with stage-structure for the prey. At first, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analysing the distribution of the roots of the associated characteristic equation. Then, the explicit formula that determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions from the Hopf bifurcation are established by using the normal form method and centre manifold argument. Finally, some numerical simulations are carried out to support the main theoretical results.

Data Processing for Dynamic Consequences of Prey Refuge in a Predator-Prey System with Stage Structure and Time Delay

—In this paper, the effect of prey refuge on the dynamic consequences of the stage-structured predator-prey system with time delay are studied. The results indicate that the prey refuge play an important role in population dynamics, the extinction and coexistence of predator and prey population. The results show that the equilibrium density of immature and mature prey populations increase with increasing in prey refuge and the prey refuge has a clearly stabilizing effect on the predator-prey system with stage structure and time delay under a restricted set of conditions. The Data process is also analysized and obtained.

Permanence of a stage-structured predator–prey system with impulsive stocking prey and harvesting predator

In this paper we propose a stage-structured predator prey system with impulsive stocking immature prey and harvesting predator at different moment. We analyze the global attractivity of the mature prey-extinction periodic solution, and obtain sufficient conditions for the permanence of the system. Numerical simulations are also inserted to verify the feasibility of the theoretical results. Moreover, the obtained results show that impulsive stocking immature prey or harvesting predator may play a key role on the permanence of the system and provide tactical basis for the biological resources management.

Stability in a predator–prey model with Crowley–Martin function and stage structure for prey

This paper is concerned with a stage-structured predator–prey system with Crowley–Martin functional response. By analyzing the characteristic equation, the local stability of the equilibria is investigated. The persistence of system and global asymptotic stability of the positive equilibrium are carried out. Finally, several numerical simulations to illustrate the theoretical analysis are also given.

Stability and Bifurcation Analysis in a Stage-Structured Predator-Prey Model with Delay

Hopf bifurcation occurs in most of dynamics systems when the influence from the past state varies. In modeling population dynamics, it is more reasonable taking into account the time delays. In this paper, a stage-structured predator-prey system with delay is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

Global stability of a stage-structured predator–prey system

An autonomous stage-structured predator–prey system (stage structure for both predator and prey) with discrete delay is studied in this paper. By using an iterative method, the global stability of the interior equilibrium point of the system is investigated. Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones.

Dynamical analysis of a stage structured predator–prey system with impulsive diffusion and generic functional response

Based on the predator–prey system with stage structure and generalized functional response, an impulsive differential system to model the process of periodical diffusion at different fixed time is proposed and investigated. By using comparison theorem of impulsive differential equation and some analysis techniques, sufficient conditions ensuring the existence of a predator-extinction positive periodic solution and the permanence of the system are established. Numerical simulation shows that impulses and functional response affect the dynamics of the system, which may be very complex.

Dynamics of a density-dependent stage-structured predator–prey system with Beddington–DeAngelis functional response

In this paper, we study the dynamics of a stage-structured predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. First, we derive a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of hyperbolic curves. Then, we provide a sufficient condition to assure the local asymptotic stability of this positive equilibrium by constructing a Lyapunov function. Afterward, by iteratively making use of the comparison theorem, we propose a sufficient condition to assure its global attractiveness. Finally, by investigating types of the ω-limit set instead of making use of the persistence theory, we prove that the predator coexists with the prey permanently if and only if positive equilibria exist.