Stage-structured Predator-prey Model Research Articles

Stability, bifurcation, and chaos of a stage-structured predator-prey model under fear-induced and delay

The motivation of this paper is inspired by the research of Zanette et al. (2011) [4], which reveals that the perception of predation risk can reduce the number of juvenile prey. To investigate how fear affects the behavior of prey population, we formulate a predator-prey model with a stage structure for prey in which adult prey is undergoing the interference of fear. First, the dynamic properties of the model are analyzed, including the existence and stability of equilibria, bifurcations dynamics, such as transcritical, saddle-node, and Hopf bifurcations. Our results reveal that fear can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. Numerical simulations and theoretical analysis indicated that when the level of fear reaches 25.4, the number of juvenile prey affected by fear decreases by 40% compared to those not affected by fear. Terrifyingly, the predators are on the brink of extinction in high levels of fear. Moreover, we examine the impact of time delay on fear in the previous model as prey needs some time to assess the predation risk. Some dynamic characteristics of this delayed model are also analyzed. Results indicate that the fear effect is conducive to generating a stable state, while delay is detrimental to generating a stable state, leading to limit cycles or even chaos.

Boundedness and Stabilization in a Stage-Structured Predator–Prey Model with Two Taxis Mechanisms

Boundedness and Stabilization in a Stage-Structured Predator–Prey Model with Two Taxis Mechanisms

Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge

The prey–predator interaction between organisms living in an ecosystem is greatly affected by the cooperation among the species as well as immature prey refuge in presence of time delay. This study has developed and analysed a stage-structured prey–predator model that includes refuge behaviour for prey and two types of cooperative behaviours, namely cooperative behaviour among prey as well as cooperative behaviour among predators, in addition to gestational delay. The presence of Saddle–node, Transcritical, and Hopf-bifurcations is explained by analytically and numerically. We examine the model parameter’s sensitivity to determine the species’ favourable and unfavourable impacts and to identify the most significant parameters. The dynamics of delay and non-delay systems are compared, and we recognise that a consistent increase in time delay causes chaotic oscillations with period-doubling fluctuations which can be regulated by the refuge for immature prey. However, we demonstrate that the immature prey refuge destabilises the system in absence of time delay while it acts as a control parameter in presence of delay. Finally, we wrap up the paper by highlighting the key biological implications of the numerical and analytical findings.

Role reversal in a stage-structured prey-predator model with fear, delay, and carry-over effects.

The present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov-Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmony.

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.

A stage-structured predator-prey model with Crowley-Martin functional response

<p style='text-indent:20px;'>A stage-structured predator-prey model with Crowley-Martin functional response is formulated and analyzed to investigate the impaction of predator maturity delay and predator interference on the dynamics of the system. It is shown that the net reproduction number of the system determines its threshold dynamics: the system is permanent if and only if the net reproduction number is larger than one unit, otherwise the predators go extinct; Moreover, given that the system is permanent, the unique coexistence equilibrium is shown to be globally asymptotically stable provided that the predator's interference is large enough; Numerical simulations are also performed to explore the effects of predator maturity delay and predator's interference on the stability of interior equilibrium. It is indicated that an increase of maturity delay of the predator will enlarge its extinction risk and may also enrich dynamics of the system in the sense of bringing stability switches of the coexistence equilibrium; and that an increase of predator's interference may lead to stability switch of coexistence equilibrium from unstable to stable, implying that a large predator's interference can be stabilizing.</p>

A stage structured predator-prey model with periodic orbits

<p style='text-indent:20px;'>This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference.</p>

Predator–prey ecological system with group defense and anti-predator traits of the preys: Synergies between two important ecological actions

During smart anti-predator adaptations, preys use their size, fleeing capability and herd in their evolutionary tug-of-war with the predators. Predator–prey ecology is teeming with such examples where predators can be defeated or even killed by the group of victims, that is, preys, banding together. In this paper, we focus on a stage-structured predator–prey model with herd behavior and anti-predator phenomenon that results in the lesser ecological opportunities for predation in the presence of the smart anti-predatory constraints. It is our purpose in this paper to illustrate the combined effects of the herd behavior and anti-predator strategy on the predator–prey ecology. By constructing the model equations, we present an analytical study of the system and the occurrence of Hopf and transcritical bifurcation for the anti-predator rate as a bifurcation parameter. Conditions for co-existence under global stable interior equilibrium as well as permanence region for the survival of all species is adequately discussed. Also, the extinction of predators is explored via numerical simulations, that is, the prey population survives successfully in the habitat due to their defense mechanism and anti-predator behavior. In order to better understand the role of anti-predator traits in predator–prey system with square root functional response, we compare the theoretical and numerical results of the proposed model with a similar predator–prey system having Holling type-I functional response. Our findings show that the predator–prey dynamics under prey’s herd behavior and anti-predator tactics can have large influence on shifting the paradigm of predator–prey relationship by turning the predator–prey dynamics into the prey dominant dynamics, which is indeed, due to the smart synergy between both ecological actions. When underdog, sticking together, fights back, that would be a headache to deal with, no matter how menacing the predator is.

Stability and Hopf Bifurcation Analysis of a Stage-Structured Predator–Prey Model with Delay

In this work, a Lotka–Volterra type predator–prey system with time delay and stage structure for the predators is proposed and analyzed. By using the permanence theory for infinite dimensional system, we get that the system is permanent if some conditions are satisfied. The local and global stability of the positive equilibrium is presented. The existence of Hopf bifurcation around the positive equilibrium is observed. Further, by using the normal form theory and center manifold approach, we derive the explicit formulas determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation. Numerical simulations are carried out by Matlab software to explain the theoretical results. We find that combined time delay and stage structure can affect the dynamical behavior of the system.

Study of Dynamical Behavior of a Delayed Stage-Structured Predator-Prey Model with Disease in Prey

The present paper deals with the dynamics of a stage-structured predator-prey model, with a ratio-dependent functional response including gestational delay in the predator. The prey is carrying an infection which affects the predator adversely. The boundedness of solutions and the stability of equilibrium points have been investigated. There is a Hopf-bifurcation arising out of the variation in the time-delay parameter. Numerical simulations of phase-plane diagrams, and bifurcation diagrams illustrate the dependence of the system on the delay -time. The effect of the disease transmission from prey to predator has also been illustrated through simulations.

Stability analysis and Hopf bifurcation of a brown planthopper–rice model under the effect of monsoon

In this paper, we studied the brown planthopper (BPH) infestation on rice under the effects of monsoon using a two-time scales stage-structured predator–prey model. We divided the BPH population into two distinct stages, mature and juvenile, owing to the significant biological differences between the two. We considered a heterogeneous environment composed of two adjacent rice fields. Using the two-time scales, we reduced the model into an aggregated model with smaller dimensions and topological equivalent. The positivity and boundedness of the model’s solution were established. We conducted the dynamical analysis of the BPH–rice ecosystem through the local and global stability of the model’s equilibria. Moreover, the effects of rice fields area and monsoon were analyzed using Hopf bifurcation theory. We performed numerical simulations to illustrate the theoretical results. Finally, we provided some emergence discussions from the ecological perspective to give useful information for preventing and managing BPH on rice.

Dynamics of stage-structured prey–predator model with prey refuge and harvesting

ABSTRACT A stage-structured prey–predator model with prey refuge is assumed, where predators have two stages as immature and mature. We consider here the harvesting on prey only and analyze the model using Beddington-DeAngelis functional response. Local stability as well as global stability at an interior equilibrium point are investigated in the proposed model. We choose different types of bifurcation like Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation in the formulated model. Moreover, in this paper, we check the optimality of the harvesting function with the help of optimal control. We provide the numerical simulation of the proposed model for the confirmation of analytical results.

The dynamics and analysis of an SIS disease of a stage-structured prey-predator model with a prey refuge

In this paper, the dynamical behaviour of an {epidemiological system} has been investigated. A stage-structured prey-predator model includes harvest and refuge for only prey, the disease of type (SIS) is just in the immature of the prey and the disease is spread by contact and by external source has been studied. transmission of infectious diseases in the prey populations has been described by the linear type. While Lotka-Volterra functional response is used to describe the predation process of the whole prey population. This model has been represented by a set of nonlinear differential equations. solution's existence, uniqueness and boundedness have been studied. The local and global stability conditions of all the equilibrium points have been confirmed. As a final point, numerical simulation has been used to study the global dynamics of the model.

EXTINCTION AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE AND STAGE STRUCTURE OF PREY

<abstract> The interaction between predator and prey is an important part of ecological diversity. This paper presents a stage-structured predator-prey model to study how stochastic environments affect population dynamics. Holling Ⅱ functional response is also incorporated in the proposed theoretical framework. Specifically, by using the theory of stochastic stability, we provide conditions for the stochastic system to suffer extinction or to have a unique ergodic stationary distribution. Besides, numerical simulations are also employed to verify the validity of the theoretical results. </abstract>

Periodic solution of a stage-structured predator-prey model with Crowley-Martin type functional response

&lt;abstract&gt;&lt;p&gt;In this paper, the existence of positive periodic solution of stage-structured predator-prey model with Crowley-Martin type functional response is investigated. The prey population fall into two categories: mature and immature prey. The predator population is dependent only on mature prey and is influenced by Crowley-Martin type functional response. Based on the Mawhin's coincidence degree theory and nontrivial estimation techniques for a priori bounds of unknown solutions to the operator equation $ Fz = \mu Nz $, we prove the existence of positive periodic solution. Finally, the effectiveness of our result is verified by an example and numerical simulation.&lt;/p&gt;&lt;/abstract&gt;

Predator–Prey System: Bachelor Herding of the Prey Imposes Ecological Constraints on Predation

Bachelor herd behavior is very common among juvenile animals who have not become sexually matured but have left their parent groups. The complex grouping or schooling behavior provides vulnerable juveniles refuge from predation and opportunities for foraging, especially when their parents are not within the area to protect them. In spite of this, juvenile/immature prey may easily become victims because of their greenness while on the other hand, adult prey may be invulnerable to attack due to their tricky manoeuvring abilities to escape from the predators. In this study, we propose a stage-structured predator–prey model, in which predators attack only the bachelor herds of juvenile prey while adult prey save themselves due to small predator–prey size ratio and their fleeing capability, enabling them to avoid confrontation with the predators. Local and global stability analysis on the equilibrium points of the model are performed. Sufficient conditions for uniform permanence and the impermanence are derived. The model exhibits both transcritical as well as Hopf bifurcations and the corresponding numerical simulations are carried out to support the analytical results. Bachelor herding of juvenile prey as well as inaccessibility of adult prey restricts the uncontrolled predation so that prey abundance and predation remain balanced. This investigation on bachelor group defence brings out some unpredictable results, especially close to the zero steady state. Altogether, bachelor herding of the juvenile prey, which causes unconventional behavior near the origin, plays a significant role in establishing uniform permanence conditions, also increases richness of the dynamics in numerical simulations using the bifurcation theory and thereby, shapes ecosystem properties tremendously and may have a large influence on the ecosystem functioning.

Dynamics of Stage-Structured Predator–Prey Model with Beddington–DeAngelis Functional Response and Harvesting

In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for τ=0 and τ≠0 (time delay τ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter τ. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when τ=0; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when τ≠0. Finally, examples with numerical simulations are given to illustrate the obtained results.

A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments

In this paper, we formulate a stage-structured predator-prey model with Holling-I and Crowley-Martin functional responses in deterministic and stochastic environments, where Holling-I and Crowley-Martin functional responses conform respectively to predator feeds on juvenile and adult prey. In the deterministic case, by discussing the existence and stability of equilibria as well as equilibrium point bifurcations, we observe that the model can possess more than one positive equilibrium and exhibit rich dynamics such as bistability and complex bifurcations, meaning that its dynamics is easily affected by the environmental perturbations. In the stochastic case, by constructing appropriate Lyapunov functions we establish respectively the sufficient conditions for the ergodic stationary distribution and extinction of the model. Moreover, for the bistability scenario between a positive equilibrium and an interior limit cycle in the absence of noise, we can numerically observe the phenomenon of noise-induced state frequent switching between two stochastic attractors in the bistable zone. Biologically, our results can partially explain the phenomenon that in real world, for the inevitably small environmental noise intensities, the biological populations may remain at least two patterns of survival to switch.

Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense

In the predator-prey system, predators can affect the prey population (1) by direct killing and (2) by inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. However, the anti-predator strategy is not the same for all individual preys of different life stages. Also, anti-predator behavior has both cost and benefit, but most of the mathematical models observed the dynamics by incorporating its cost only. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that adult preys are only adapting group defense as an anti-predator strategy when they are sensitive to predation. Group defense plays a positive role for adult prey by reducing their predation, but, on the negative side, it simultaneously decreases their reproductive potential. A parameter, anti-predator sensitivity is introduced to interlink both the benefit and cost of group defense. Our result shows that when adult preys are not showing anti-predator behavior, with an increase of maturation rate, the system exhibits a population cycle of abruptly increasing amplitude, which may drive all species of the system to extinction. Anti-predator sensitivity may exclude oscillation through homoclinic bifurcation and avert the prey population for any possible random extinction. Anti-predator sensitivity also decreases the predator population density and produces bistable dynamics. Higher values of anti-predator sensitivity may lead to the extinction of the predator population and benefit adult preys to persist with large population density. Below a threshold value of anti-predator sensitivity, it may possible to retain the predator population in the system by increasing the fear level of the predator. We also observe our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors, various types of bistabilities in different bi-parameter planes. Finally, we discuss the potential impact of our findings.

Numerical analysis of predator-prey model in presence of toxicant by a novel approach

It is assumed in this paper that the environmental toxicant affects both prey and predator population. We consider the Holling Type-I predator-prey model and stage-structured predator-prey model in presence of environmental toxicant in the form of Volterra integro-differential equations. We solve the system of equations by a novel approach Trapezoidal Base Homotopy Perturbation Method. Using the dynamical behaviour of the systems as a validating tool, we validate the above mentioned method as an efficient method to solve Volterra integro-differential predator-prey model.