Middle Predator Research Articles

Dynamic relationship between the mutual interference and gestation delays of a hybrid tritrophic food chain model

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell– Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response. doi:10.1017/S144618111700044X

DYNAMIC RELATIONSHIP BETWEEN THE MUTUAL INTERFERENCE AND GESTATION DELAYS OF A HYBRID TRITROPHIC FOOD CHAIN MODEL

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

The Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populations

We define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.

Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Fear

In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to identify the chaotic behavior of the system.

Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study

Defense mechanisms are very important to all animal life. Predators in every biome must eat to survive. With predators being top on the food chain and always on the lookout for a meal, prey must constantly avoid being eaten. In this paper, we have proposed and analyzed a tri-trophic predator–prey model of one prey and two predator exhibiting group defense mechanism. We have assumed Monod-Haldane functional response for interaction between species due to group defense ability of prey and middle predator. We have performed Kolmogorov and Hopf bifurcation analysis for the model system. Linear and global stability of the model system have been analyzed. Lyapunov exponents are computed numerically and 2D scan for different parameters of the model have performed to characterize the complex behavior of the model system. The numerical simulations shows the chaotic and periodic oscillations of the model system for certain range of parameter. We have drawn bifurcation diagrams for different parameter values which shows the complex dynamical behavior of model system. This work is an attempt to study the effect of group defense mechanism of prey in predator population and effect of immigration within top predator population is investigated. It is also observed that in the presence of group defense, the model system stabilizes after adding a small amount of constant immigration within top predator population.

Impulsive Effect on Tri-Trophic Food Chain Model with Mixed Functional Responses under Seasonal Perturbations

In this paper, dynamical effect of seasonal variations and impulsive perturbations on a hybrid tri-trophic food chain model has been investigated. In this model mixed functional responses (Holling type-II and modified Leslie–Gower type) are considered. Using Floquet theory and small amplitude perturbation techniques, it is found that a prey and middle predator free periodic solution is globally asymptotically stable if the impulsive period lies in between two critical values. Further, using comparison results of impulsive differential inequalities, it is derived that the model system is permanent under certain conditions. Moreover, the model system is simulated numerically to find the influence of the seasonal forcing and impulsive perturbation. Complex dynamical behavior has been observed for biologically feasible parametric values.

Predator interference effects on biological control: The “paradox” of the generalist predator revisited

An interesting conundrum in biological control questions the efficiency of generalist predators as biological control agents. Theory suggests, generalist predators are poor agents for biological control, primarily due to mutual interference. However field evidence shows they are actually quite effective in regulating pest densities. In this work we provide a plausible answer to this paradox. We analyze a three species model, where a generalist top predator is introduced into an ecosystem as a biological control, to check the population of a middle predator, that in turn is depredating on a prey species. We show that the inclusion of predator interference alone, can cause the solution of the top predator equation to blow-up in finite time, while there is global existence in the no interference case. This result shows that interference could actually cause a population explosion of the top predator, enabling it to control the target species, thus corroborating recent field evidence. Our results might also partially explain the population explosion of certain species, introduced originally for biological control purposes, such as the cane toad (Bufo marinus) in Australia, which now functions as a generalist top predator. We also show both Turing instability and spatio-temporal chaos in the model. Lastly we investigate time delay effects.

Long time dynamics of a three-species food chain model with Allee effect in the top predator

The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the model, that is upper semicontinuous in the Allee threshold parameter m. Next, we numerically investigate the decay rate to a target attractor, that is when m=0, in terms of m. We find decay estimates that are O(mγ), where γ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m. Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.

Interspecies competition between prey and two different predators with Holling IV functional response in diffusive system

This paper deals with a prey–middle predator–top predator ecosystem model with Holling type IV predator response in the unreserved zone. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. The global stability analysis is carried out. It is observed that if the intrinsic growth rate of prey population crosses a certain critical value, the system enters into Hopf bifurcation. The existence of bionomic equilibrium of the system has been discussed. Further, we study a path of optimal harvesting policy by introducing the Pontryagins maximum principle. Moreover we have found out a condition for diffusive instability of a locally stable equilibrium. Finally, some numerical simulations are performed to justify analytical findings.

Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response

In this paper, we study a strongly coupled reaction-diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. An intra-species competition b2 among the second predator is introduced to the food chain model. This parameter produces some very interesting result in linear stability and Turing instability. We first show that the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system when the intra-species competition exists among the second predator. The positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction-diffusion system, hence the instability is driven solely from the effect of cross diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions. Numerically, we conduct a one parameter analysis which illustrate how the interactions change the existence of stable equilibrium, limit cycle, and chaos. Some interesting dynamical phenomena occur when we perform analysis of interactions in terms of self-production of prey and intra-species competition of the middle predator. By numerical simulations, it illustrates the existence of nonuniform steady solutions and new patterns such as spot patterns, strip patterns and fluctuations due to the diffusion and cross diffusion in two-dimension.

Existence and global stability of positive periodic solution of tri-trophic food chain with middle predator migratory in nature

Predator avoidance is the most important factor inducing vertical migration among zooplankton. Light intensity at the surface water also plays an important role to shape the predation pressure on zooplankton as most of the fishes show visually oriented predation. In the present paper, we assume that zooplankton can switch between vulnerable and invulnerable state through vertical migration. This switching is regulated by light intensity, predator abundance and food abundance in surface water. We propose a tri-trophic food chain model incorporating migratory behavior of middle predator. Further, we study the non-autonomous version of the proposed model by incorporating the seasonal variation of the rate parameters of the model. The sufficient conditions for the existence of positive periodic solutions are obtained. We also obtain the conditions for the global attractivity of positive periodic solutions. The numerical experiments are carried out to substantiate analytical findings.

Turing patterns and long-time behavior in a three-species food-chain model

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator–prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L2(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.

Revisited Hastings and Powell model with omnivory and predator switching

The effect of omnivory in predator–prey system is debatable regarding its stabilizing or destabilizing characteristics. Earlier theoretical studies predict that omnivory is stabilizing or destabilizing depending on the condition of the system. The effect of omnivory in the food chain system is not yet properly understood. In the present paper, we study the effect of omnivory in a tri-trophic food chain system on the famous Hastings and Powell model. Omnivory enhances the chance of predator switching between prey and middle predator. The novelty of this paper is to study the effect of predator switching of the top predator which is omnivorous in nature. Our results suggest that in the absence of switching, an increase of omnivory stabilizes the system from chaotic dynamics, however, if we further increase the strength of omnivory, the system becomes unstable and middle predator goes to extinction. It is also observed that the predator switching enhance the stability and persistence of all populations.

ROLE OF PROTECTION IN A TRI-TROPHIC FOOD CHAIN DYNAMICS

In this paper, the role of protection in stabilizing the tri-trophic food chain dynamics has been explored. The density-dependent protection is provided to bottom prey or middle predator or both. It favors the oscillations damping and has the potential to control the chaotic fluctuations of population density. The bifurcation diagrams have been drawn with respect to protection parameter. They exhibit coexistence of all three species in the form of periodic solutions. The coexistence in the form of stable equilibrium is possible for higher values of protection parameters. Further increase in protection parameters may lead to extinction of one or two species. A two-parameter bifurcation diagram has also been drawn. The Poincaré Maps further confirm the role of protection in controlling the chaos.

Persistence and stability for the three species ratio-dependent food–chain model

In this paper, we have proposed and analyzed a tritrophic food chain model composed of a prey, a middle predator and a top predator. Ratio-dependent functional response is considered to model the interactions among the species of the system. Mathematical analysis of model equation with regard to the nature of equilibria, boundedness and persistence of the solution are carried out. To verify the analytical findings numerical simulation is performed. Furthermore, global stability of the system is shown graphically. It has been observed from numerical simulation that prey population decreases in the absence of top predators. However, presence of top predators in the system causes an increase in prey population with a depression in the population of middle predators.

Challenges of living in the harsh environments: A mathematical modeling study

Abstract We propose and analyze a mathematical model, which mimics community dynamics of plants and animals in harsh environments. The mathematical model exploits type IV functional responses whose idiosyncrasies have been recognized only in recent years. The interaction of the middle predator with the top predator is cast into Leslie–Gower scheme. Linear and non-linear stability analyses are performed to get an idea of the stability behavior of the model food chain. It turns out that carrying capacity of the prey and the immunity parameter of the middle predator are two crucial parameters governing the model. Availability of alternative food options to the generalist predator also plays a key role in deciding the model dynamics. Simulation runs performed on this model provide insight into population dynamics of monkeys of macaque family found in northern Japan. These monkeys are social animals which reproduce sexually. The characteristic feature of the model dynamics is that the generalist predator (macaque monkeys) is able to avoid impending extinction frequently and recovers at a rate which falsify threats from exogenous external forces; extreme weather conditions, etc.

Migratory effect of middle predator in a tri-trophic food chain model

A tri-trophic food chain model in a two-patch environment is considered. Although tri-trophic food chain model is well studied, the study considering migration of middle predator is lacking. To the best of our knowledge, the present investigation is the first study in this direction. Both prey and predator density-dependent migrations are considered to observe the effects on stability and persistence of the system. We observe that migration of middle predator has the ability to control chaos in tri-trophic food chain model. Our results indicate that the chance of predator extinction enhances for prey density-dependent middle predator migration. Copyright © 2010 John Wiley & Sons, Ltd.

A ratio-dependent food chain model and its applications to biological control

While biological controls have been successfully and frequently implemented by nature and human, plausible mathematical models are yet to be found to explain the often observed deterministic extinctions of both pest and control agent in such processes. In this paper we study a three trophic level food chain model with ratio-dependent Michaelis–Menten type functional responses. We shall show that this model is rich in boundary dynamics and is capable of generating such extinction dynamics. Two trophic level Michaelis–Menten type ratio-dependent predator–prey system was globally and systematically analyzed in details recently. A distinct and realistic feature of ratio-dependence is its capability of producing the extinction of prey species, and hence the collapse of the system. Another distinctive feature of this model is that its dynamical outcomes may depend on initial populations levels. Theses features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator as a pest, and top predator as a biological control agent) where the simultaneous extinctions of pest and control agent is the hallmark of their successes and are usually dependent on the amount of control agent. Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study the questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.