Middle Predator Research Articles

Complex dynamics of a four-species food-web model: An analysis through Beddington-DeAngelis functional response in the presence of additional food

Abstract The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

An investigation of the parameter space in a tri-trophic food chain model with refuge

Aside from humans, the instinct to run and hide from dangerous situations is ingrained in individuals of many other species. Taking refuge is a very common precautionary strategy among prey species to avoid direct contact with their predators. In this article, we investigate a three-species food chain model where the logistically growing basal prey seeks refuge to avoid the middle predator. To assess the impact of varying levels of refuge, we explore the system dynamics in the biparametric space of refuge and another important system parameter related to the half-saturation constant of the basal prey. The isospike diagrams along with the largest Lyapunov exponent diagrams show a stabilizing role of refuge. Apart from this, we observe an abundance of organized periodic structures embedded inside the chaotic zone of the biparametric space. The most notable ones include the shrimp-shaped structures and an “elliptic disc” shaped structure. The article also explores the coexistence between multiple pairs of coexisting attractors and one instance of spike-bubbling phenomena. The basins of attraction for the coexisting attractors also have a self-similar pattern. We also modify the present model by replacing the density-dependent prey refuge with constant proportion of prey refuge, and then explore the dynamics of the modified model within the same biparametric space and find similar complexity in its dynamics. We compare the effects of both types of refuge on the density variations of all three species when they show stable coexistence and observe that density-dependent refuge is more beneficial to the system from the standpoint of species conservation.

Stability and bifurcation analysis of delayed tri‐trophic food chain model in poisoned environment with fear effect and additional food

One of the most important goals of theoretical ecologists is to find a strategy for controlling the chaos in ecological models to maintain healthy ecosystems. We investigate the influence of fear and the supply of additional food to predators in a delayed tri‐trophic food chain model, where interference among the species is determined by Holling type‐II functional response. The proposed model incorporates the fact that the growth rates of prey and middle predators are decreased due to the fear of middle and special predators, respectively. The additional food is supplied to both predators by either nature or external agencies. Furthermore, the poisoned prey in the predator's environment is taken into account. In the proposed model, we use two time delays: one for the growth term of the middle predator and another for the gestation delay of the special predator. We determine the conditions for the existence of ecologically feasible equilibrium points and their local and global stability. In addition, we establish the conditions for the existence of Hopf bifurcation around interior equilibrium to seek periodic behavior in nondelayed and delayed models. Numerical investigations are performed to justify the proposed theoretical findings through phase portraits, time series of solutions, and bifurcation diagrams. We observe that the chaotic dynamics of a tri‐trophic food chain model can be controlled by the proper choice of the fear effect and by supplying additional food parameters. As a result, our findings provide more ecological insights into the dynamics of the delayed tri‐trophic food chain model with the fear effect and additional food supply.

Fear and delay effects on a food chain system with two kinds of different functional responses

For food chain system with three populations, direct predation is the basic interaction between species. Different species often have different predation functional responses, so a food chain system with Holling-II response for middle predator and Beddinton–DeAngelis response for top predator is proposed. Apart from direct predation, predator population can significantly impact the survival of prey population by inducing the prey’s fear, but the impact often possesses a time delay. This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence. By use of Lyapunov functional method and bifurcation theory, the positiveness and boundedness of solutions, local and global behavior of species, the system stability around the equilibrium states and various kinds of bifurcation are investigated. Numerically, some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained. It is observed that fear and delay can not only stabilize, but also destabilize the system, which depends on the magnitude of the fear and delay. The system varies from unstable to stable due to the continuous increase of the prey’s fear by middle predator. Small fear induced by top predator or small delay of the prey’s fear can stabilize the system, while they are sufficiently large, the system stability is to be destroyed. Simultaneously, the conversion rate can also change the stability and even make the species to be extinct. Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited, which results in the convergence of the species from one stable equilibrium to another.

Dynamics in the biparametric spaces of a three-species food chain model with vigilance

In nature, vigilance is a common anti-predator strategy prey individuals employ to protect themselves from a possible attack. It gives them sufficient time to take cover or flee. Yet this seemingly profitable strategy, which lowers the number of successful predatory attacks, has a significant impact on the prey's growth rate. Researchers attribute this to reduced foraging time. In this article, we study a three-species food chain model in which both the basal prey and middle predator show active vigilance against their respective predators. Apart from the positivity, boundedness, and local stability analysis of the equilibrium points of the model, we find the conditions for global stability and the occurrence of Hopf bifurcation around the coexisting equilibrium point. Bifurcation diagrams and their corresponding Lyapunov exponent diagrams (largest and second-largest) illustrate diverse dynamical scenarios ranging from stable coexistence to periodic to chaotic oscillations. The isospike diagrams and their associated largest Lyapunov exponent diagrams in the biparametric space of basal prey's and middle predator's vigilance reveal a broader picture of the effects of different levels of vigilance on the dynamics of the system. A great deal of attention has been paid to unwrapping the complex structural beauties like shrimp-shaped periodic structures inside the chaotic sea and multistability between several sets of attractors in the overlapping zones of periodic windows. We observe that multiple coexisting attractors possess fractal basin boundaries. We also explore the dynamics of the system in two other biparametric spaces. The density variation maps at the end show the effect of vigilance on the equilibrium density of all three species. Our results suggest that vigilance promotes regularity and coexistence, but too high levels of vigilance can cause species extinction. For better visualization of the system dynamics, we provide a few high-resolution animations of the phase portraits and the basins of attraction in the supplementary material.

Complex Dynamics of a Three-Species Food Chain Model with Fear and Allee Effect

In ecology, predator–prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings–Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.

Impact of vigilance on the density variations in a food chain model

Anti-predator behavior in the form of vigilance greatly influences the dynamics of a predator-prey system. In the present work, we investigate the impact of prey vigilance in a three-species food chain model where both prey and middle predator use vigilance as a survival strategy in the presence of their respective predators. We present basic mathematical results such as local and global stability, bifurcation behavior of the system. We explore the variation of the densities of the populations in different bi-parameter spaces and observe that vigilance plays a crucial role in the survival and extinction of the populations.

Qualitative and Dynamical Analysis of a Bionomic Fishery Model with Prey Refuge.

Predation and escaping from predation through hiding are two fundamental phenomena in ecology. The most common approach to reducing the chance of predation is to use a refuge. Here, we consider a three species fishery model system with prey refuge induced by a Holling type-II functional response. These three species of fish populations are named prey, middle predator, and top predator. Harvesting is employed in most fishery models to achieve both ecological and commercial benefits. Research proves that non-linear harvesting (Michaelis-Menten type) returns more realistic outcomes. So, we have combined the Michaelis-Menten type of harvesting efforts for all populations. Uniform boundedness conditions for the solutions of the model are discussed. The existence conditions for possible equilibrium points with stability are presented. We explain the dynamical behavior at each equilibrium point through bifurcation analysis. The persistent criteria of the system are examined. Bionomic equilibrium and optimal harvesting control using Pontryagin's maximum principle are calculated. For validation of the model in the real world, we have implemented this in the freshwater ecosystem of Lake Victoria. Extraction of native fish species and ecological balances are the foremost solicitude of Lake Victoria. We may resolve this concern partially by implementing prey refuge, since it may sustain the ecology of Lake Victoria, and therefore also its economical importance. Lake Victoria is acclaimed worldwide for the trade of fishing. Also, it provides the largest employment in east-central Africa and is beneficial to fishing equipment manufacturers. So, the bionomic equilibrium and harvesting control have significant applications in the fisheries. All the analytical studies are verified by numerical simulations. We have plotted phase portraits, bifurcation diagrams, Lyapunov exponents to explore the dynamics of the proposed model.

Deterministic and Stochastic Fractional-Order Hastings-Powell Food Chain Model

In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting is proposed and analysed. The interaction between prey, middle predator and top predator population is investigated. In order to clarify the characteristics of the proposed model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed model are examined. Some sufficient conditions that ensure the local and global stability of equilibrium points are obtained. By using stability analysis of the fractional-order system, it is proved that if the basic reproduction number , the predator free equilibrium point is globally asymptotically stable. The occurrence of local bifurcation near the equilibrium points is investigated with the help of Sotomayor’s theorem. Some numerical examples are given to illustrate the theoretical findings. The impact of harvesting on prey and the middle predator is studied. We conclude that harvesting parameters can control the dynamics of the middle predator. A numerical approximation method is developed for the proposed stochastic fractional-order model.

A phytoplankton-zooplankton-fish model with chaos control: In the presence of fear effect and an additional food.

The interplay of phytoplankton, zooplankton, and fish is one of the most important aspects of the aquatic environment. In this paper, we propose to explore the dynamics of a phytoplankton-zooplankton-fish system, with fear-induced birth rate reduction in the middle predator by the top predator and an additional food source for the top predator fish. Phytoplankton-zooplankton and zooplankton-fish interactions are handled using Holling type IV and II responses, respectively. First, we prove the well-posedness of the system, followed by results related to the existence of possible equilibrium points. Conditions under which a different number of interior equilibria exist are also derived here. We also show this existence numerically by varying the intrinsic growth rate of phytoplankton species, which demonstrates the model's vibrant nature from a mathematical point of view. Furthermore, we performed the local and global stability analysis around the above equilibrium points, and the transversality conditions for the occurrence of Hopf bifurcations and transcritical bifurcations are established. We observe numerically that for low levels of fear, the system behaves chaotically, and as we increase the fear parameter, the solution approaches a stable equilibrium by the route of period-halving. The chaotic behavior of the system at low levels of fear can also be controlled by increasing the quality of additional food. To corroborate our findings, we constructed several phase portraits, time-series graphs, and one- and two-parametric bifurcation diagrams. The computation of the largest Lyapunov exponent and a sketch of Poincaré maps verify the chaotic character of the proposed system. On varying the parametric values, the system exhibits phenomena like multistability and the enrichment paradox, which are the basic qualities of non-linear models. Thus, the current study can also help ecologists to estimate the parameters to study and obtain such important findings related to non-linear PZF systems. Therefore, from a biological and mathematical perspective, the analysis and the corresponding results of this article appear to be rich and interesting.

Dynamical Behaviors of a Fractional-Order Three Dimensional Prey-Predator Model

In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.

Dynamics of a predator-prey model with Beddington-DeAngelis functional response, omnivory and predator switching

In this paper, a predator-prey model has been developed by considering Beddington-DeAngelis functional response among the interaction of three species prey, middle predator and top predator. It is assumed that middle predator consumes only prey, but top predator consumes prey as well as middle predator. The concept of omnivory has been introduced in this model. Also, it is assumed that due to omnivory property, the top predator has a chance to prefer the food (prey) to consume. Here two types of predator switching behavior such as preferential switching and density dependent switching have been introduced. Different possible equilibrium points are evaluated and the stability of the system has been investigated around these points. Theoretical analysis of for the existence condition of Hopf bifurcation of the system has been studied with respect to $w_4$. It is found that the increase of death rate of middle and top predator may be responsible for the extinction of both the species or that particular species. It is observed that omnivory may lead the system towards stability. It is also observed that the predator switching behaviour may strengthen the stability and persistence of all populations. Finally, some numerical simulation results have been presented to validate the analytical findings.

Coexistence in a four-species food web model with general functional responses

The dynamics of an intraguild predation system is analyzed. Two shared resources, a middle predator and a top predator are considered in the model. Parameter conditions for coexistence in the intraguild system are given. The coexistence of species, through the existence of an equilibrium point or a stable limit cycle or an invariant torus is shown. Conditions to have Andronov-Hopf and Hopf-Hopf bifurcation are given. Then, limit sets and complex dynamical behaviors, can exhibit in the system. For a wide family of functional responses, these results are valid. Numerical simulations with varied functional responses are given and different limit sets are shown.

Comparison between two tritrophic food chain models with multiple delays and anti-predation effect

Exploring the predator–prey linkage in food chain system is the most familiar research work in population biology. Recently, some research experiments show that predator–prey interaction not only governed by direct hunting but also influenced by some indirect effect such as fear effect (felt by prey) that may change the physiological behavior of prey. Based upon this fact, we consider a tritrophic food chain model incorporating with anti-predation response (fear effect) and multiple time delays for biomass conversion from prey to middle predator and middle to top predator. We analyze the resulting delay differential equations and explore how the anti-predation response level affects the population dynamics. We also investigate the effect of delay parameters, for which the model system switches its stability through Hopf-bifurcation. We compare all of our results between two different food chain models consisting of two different functional responses. Some numerical simulations are performed to validate the effectiveness of the derived theoretical results.

Complex dynamics of delay-induced plankton–fish interaction exhibiting defense

In this paper, we propose a delayed plankton–fish model with Monod–Haldane-type functional response. Many prey organisms have developed defense mechanisms against predation. Predators also exhibit ways of coping with variable environmental conditions that enter in adaptation in the event of a shortage of prey. We assume prey and middle predators exhibit defense ability towards their predators. Time delay is incorporated in middle predator as an adaptation to cope with variable environmental conditions. Analytically, we study the feasibility and boundedness of the model system by comparison principle followed by local and global stability conditions. We also investigate the stability and direction of bifurcating periodic solution by normal form theory and center manifold arguments. Numerically, we study the effect of defense and time delay on the stability behavior of the system. Our numerical results reveal that the increase in time delay system becomes unstable as well as stable under different parametric restrictions. We observe that due to defense ability in prey and middle predator system shows extinction in top predator.

Chaos in a cyclic three-species predator–prey system with a partial consumption of superpredator

This paper aims at the detailed numerical analysis of a cyclic three species predator–prey model where the prey consumes only a part of the super-predator population. Such a model exists only when the prey acts as an omnivore. Here, we have investigated the dynamical behaviour of the prey, middle predator and super-predator. All the possible equilibrium points of the model are computed and the existence and stability condition of the equilibrium states are determined. The phase portraits are generated for different sets of parameter values. The long term behaviour of the system is investigated by studying the bifurcation structure and nature of the attractors, thereby identifying the domain of chaos, as each of the control parameter is varied independently. Finally, we show that a transition from chaotic domain to escape or vice-versa of the predator in a small region of the parameter plane leads to a fractal structure.

Stability and bifurcation analysis of Filippov food chain system with food chain control strategy

In the present work, we introduce a control model to describe three species food chain interaction model composed of prey, middle predator, and top predator. The middle predator preys on prey and the top predator preys on middle predator. The control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep stability of the food chain, by prohibiting fishing when the population density drops below a prescribed threshold. The behavior of the system stability of the regular, virtual, pseudo-equilibrium and tangent points are discussed. The complicated non-smooth dynamic behaviors (sliding and crossing segment and their domains) are analyzed. The bifurcation set of pseudo-equilibrium and the sliding crossing bifurcation have been investigated. Our analytical findings are verified through numerical investigations.

Stability and dynamics of a fractional-order three-species predator-prey model.

In this paper, a fractional-order predator-prey mathematical model has been developed considering Holling type II functional response. Here, we have investigated the interaction dynamics of prey, middle predator and top predator. We assume that the middle predator consumes only the prey, and the top predator consumes only the middle predator. Here, the logistic growth of prey has been considered. Then, different possible equilibrium points of our proposed system are determined. The stability of our proposed system is investigated around the equilibrium points. Then, some numerical simulations results are presented for better understanding the dynamics of our proposed model. It is found that the fractional-order derivative can improve the stability of our proposed system.

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator

In this paper, we propose and analyze a three-species predator–prey system in the presence of additional food for top predator. It is assumed that the middle predator is acting as a prey as well as a predator and the top predator consumes both prey as well as middle predator. It is also considered that a constant amount of additional food for top predator exists in the ecosystem. The effects of harvesting of top predator are investigated. The existence and stability conditions of the equilibria have been discussed analytically. The Hopf bifurcation analysis of the system with respect to predation rate of prey to the top predator and the harvesting effort have been analyzed both analytically and numerically. Pontryagin’s maximum principle is used to determine the optimal harvesting of top predator population to maximize the discounted net revenue. From our analysis, it is seen that the additional food has a significant impact to prevent the extinction risk of top predator population and also to increase revenue collection. Finally, some numerical results have been given in support of our analytical findings.