Species Food Chain Model Research Articles

Harvesting Optimal Policy in Three Species Food Chain Model As an Optimal Control Problem with Path Constraints

In this paper, we address the issue of harvesting prey and intermediate predators in a tritrophic food chain. Our approach is based on a model that characterizes the interactions among the three species. We assume that the intermediate predator has alternative food sources (it is a generalist), while the top predator relies solely on the intermediate predator (it is a specialist). This model has been previously explored in the literature, but representing the harvesting effort as a scalar control variable. In this study, we treat it as a vector variable, offering a more comprehensive representation, particularly relevant for terrestrial species hunting. Our primary objective is to determine optimal harvesting policies that ensure the persistence of all three species. To achieve this, we formulate the problem as an optimal control problem with a finite horizon and path constraints. We present a numerical example solved using ICLOCS2.

Effect of prey refuge in a three species food chain model with Beddington-DeAngelis functional response

Effect of prey refuge in a three species food chain model with Beddington-DeAngelis functional response

A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation

In this article, we study the dynamics of a three species food chain model for a predator–prey system with hunting cooperation in predators; due to fear of predation risk, prey population shows anti-predator behavior. Fear of middle-predator affects the growth rate of prey and the fear of top-predator reduces the predation of middle-predator on prey species. The prey and middle-predator are assumed to take refuge against predation. Numerical results confirm the stabilizing role of refuge whereas hunting cooperation destroys stability and induces limit cycle oscillations. We find that the fear of middle-predator on prey has tendency to stabilize the system while the fear of top-predator on middle-predator creates instability in the system. Further, time delays are considered in the processes of taking refuges and hunting cooperations. Delay in hunting cooperations creates multiple stability switches in the system. Furthermore, seasonal forcing of parameters related to refuges and hunting cooperations depicts a more realistic scenario. We investigate the behaviors of seasonally forced system in the absence as well as presence of time delays. The seasonally forced system exhibits different types of periodic solutions. Complex bursting patterns are observed due to periodicity in refuge of middle-predator (cooperation of top-predator).

Studies of different types of bifurcations analyses of an imprecise two species food chain model with fear effect and non-linear harvesting

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can’t be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences.

Stability analysis of a three species food chain model with linear functional response via imprecise and parametric approach

Currently, proper modeling of a biological system in an uncertain situation is a challenging task for researchers. Facing this challenge, in this article, a new concept of interval representation named as linear parametric representation is introduced by which a biological system in an uncertain situation can be formulated mathematically in a precise way. The aim of this work is to formulate a three species imprecise food chain model with one prey and two competing predators in an interval environment. Here, all the biological parameters except the conversion efficiency of the species are considered interval-valued. And interactions among the species are taken as Holling Type I functional response. Using the linear parametric representation of interval, a system in the interval form has been converted in the parametric form. Positivity and boundedness of the solutions of the proposed imprecise system are verified. Then, conditions of local stability, global stability and transcritical bifurcation of equilibrium points of the proposed system are established. Thereafter, all the theoretical results of this work are validated by a numerical example with interval-valued parametric data and results are presented pictorially. Finally, some biological implications of the obtained results are discussed and a fruitful conclusion has been made.

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

In this paper, we have discussed harvesting of prey and intermediate predator species. Both are subjected to Holling type I–V functional response. Conditions for local and global stability of the nonnegative equilibria are verified. The permanent coexistence criterion of the model system and existence of optimal equilibrium solution of the control problem are demonstrated. Maximum sustainable yield and maximal net present revenue are determined. To confirm analytical results, numerical solution has been carried out using the Matlab™ ODE solver ODE45 and the simulations show the model system reveals complex behavior (such as oscillations), which reflects the real situation. Recommendations for Resource Managers From our investigation of this study, we recommend to the management the following points. 1. Coexistence of the three species with harvesting, or persistence of the model system is possible provided that good management(treatment) of some factors (such as harvesting rate, growth rate of species, etc.) are performed. 2. The dynamics reveals complex behavior (such as oscillations), which reflects the real situation and it is sensitive to the above factors, especially the growth rate of the intermediate predator. 3. The policy makers should recommend the optimal effort to be applied and the optimal stock to harvest. This indicates that maximum profit will attain while securing sustainability of the three species in the ecosystem.

Study of Stability and Bifurcation of Three Species Food Chain Model with Non-monotone Functional Response

In this research article, we consider a tri-trophic food chain model with one prey and two predators such as- prey, intermediate predator and top predator. In this model, the prey and intermediate predator follows non-monotonic functional response; top predator consumes prey and intermediate predator following Holling type I functional response. The positivity, boundedness of solutions of the proposed model and stability conditions of different equilibrium points are discussed here. Then using Center Manifold theorem, the nature of non-hyperbolic type equilibrium points are discussed. After that, different local bifurcations such as- Saddle-node, Transcritical and Hopf bifurcations are studied theoretically as well as numerically by considering half-saturation constant and death rate of intermediate predator as the bifurcation parameters. Finally, the dynamics of the proposed model has been illustrated with the help of some numerical simulations.

Dynamic analysis of a stochastic four species food-chain model with harvesting and distributed delay

A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest in each species and the effect of time delay for interaction between species are considered, which makes the model more applicable in real situations. Specifically, we address the stochastic global dynamics behavior, including the existence of global positive solutions, stochastic ultimate boundedness, extinction with probability one, persistence in mean and global stability. The asymptotic stability in the probability distribution is obtained, and the criterion for the existence and non-existence of the optimal harvesting strategy is also derived. Furthermore, this paper can provide reference for the research of general n-species stochastic food-chain models.

Additive Allee effect of top predator in a mathematical model of three species food chain

In the present paper, the Hastings and Powell model of three species food chain model is modified incorporating Allee effect. Additive type Allee effect has been used in order to modify the three-species food chain model. We investigate the impact of Allee in the food chain model. Dynamical behaviours such as existence of steady states and their local stability analysis, bifurcation analysis of the proposed three species model have been discussed. Numerical examples are provided to justify the analytical results. Regions of stability of different equilibrium points have been plotted which are in coherence with the results derived analytically. From this study, we find that chaos in the three species food chain can be controlled when prey is subject to the strength of the Allee effect. Bifurcation diagrams show that the model turns from stable to periodic doubling and periodic doubling to chaos due to the strength of Allee effect. We observe that this modified HP model is lesser chaotic in case of increase in severity of Allee effect.

Chaos in Beddington–DeAngelis food chain model with fear

In the current paper, the effect of fear in three species Beddington–DeAngelis food chain model is investigated. A three species food chain model incorporating Beddington-DeAngelis functional response is proposed, where the growth rate in the first and second level decreases due to existence of predator in the upper level. The existence, uniqueness and boundedness of the solution of the model are studied. All the possible equilibrium points are determined. The local as well as global stability of the system are investigated. The persistence conditions of the system are established. The local bifurcation analysis of the system is carried out. Finally, numerical simulations are used to investigate the existence of chaos and understand the effect of varying the system parameters. It is observed that the existence of fear up to a critical value has a stabilizing effect on the system; otherwise it works as an extinction factor in the system.

Controlling chaos in three species food chain model with fear effect

In this article, we study the impact of fear on the dynamics of a three species food chain model. We propose a model with the assumption that the growth rate of intermediate predator reduces at the cost of fear due to top predator, and the growth rate of prey is suppressed due to the fear of the intermediate predator. We carry out the existence of equilibria, local stability analysis and bifurcation analysis. Our numerical simulation reveals that for a low cost of the fear, system remains chaotic while increase in fear factor leads to stability. Even the large cost of fear causes the population to become extinct. We conclude that the fear effect can stabilize the chaotic dynamics of the system.

Harvesting in tri-trophic food chain stabilises the chaotic dynamics-conclusion drawn from Hastings and Powell model

The paper explores a tri-trophic food chain model with harvesting in the species. The curiosity of this paper is to observe chaotic dynamics and its control. We perform the local stability analysis of the equilibrium points. The Hopf bifurcation analysis and global stability around the interior equilibrium point are also performed. Our numerical simulations reveal that the three species food chain model induces chaos from period-doubling, limit cycle and stable focus for increasing values of half saturation constant. We conclude that chaotic dynamics can be controlled by the harvesting parameter. We apply basic tools of non-linear dynamics such as Poincare section and Lyapunov exponent to identify chaotic behaviour of the system.

Harvesting in tri-trophic food chain stabilises the chaotic dynamics-conclusion drawn from Hastings and Powell model

The paper explores a tri-trophic food chain model with harvesting in the species. The curiosity of this paper is to observe chaotic dynamics and its control. We perform the local stability analysis of the equilibrium points. The Hopf bifurcation analysis and global stability around the interior equilibrium point are also performed. Our numerical simulations reveal that the three species food chain model induces chaos from period-doubling, limit cycle and stable focus for increasing values of half saturation constant. We conclude that chaotic dynamics can be controlled by the harvesting parameter. We apply basic tools of non-linear dynamics such as Poincare section and Lyapunov exponent to identify chaotic behaviour of the system.

Effects of Additional Food on the Dynamics of a Three Species Food Chain Model Incorporating Refuge and Harvesting

AbstractIn this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin’s maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate (m1), harvesting rate (E), the intrinsic growth rate of prey (r), carrying capacity of prey (k) and conservation rate of prey (c1) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system.

Global dynamics in a stochastic three species food-chain model with harvesting and distributed delays

This paper proposes a stochastic three species food-chain model with harvesting and distributed delays. Some criteria for the global dynamics of all positive solutions, including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, are established by using the stochastic integral inequalities, Lyapunov function method, and the inequality estimation technique. Furthermore, the effects of harvesting are discussed, the optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY for short) are obtained. Finally, numerical examples are carried out to illustrate our main results.

Refugia and Allee Effect in Prey Species Stabilize Chaos in a Tri-Trophic Food Chain Model

In this paper a mathematical model is proposed and analyzed to study the dynamics of tri-trophic food chain model with refugia and allee effect in prey species. Criteria for local stability, instability and global stability of the non-negative equlibria are obtained. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed. Further we pay attention to the chaotic dynamics which is produced by half-saturation constant. Our numerical simulations reveal that the three species food chain model without refugia and allee effect induced chaos from stable focus for increasing the value of half-saturation constant. We conclude that chaotic dynamics can be controlled by the Allee and refugia parameters.

Detection of turing patterns in a three species food chain model via amplitude equation

Study of pattern formation in three species spatio-temporal food chain models with the help of amplitude equations within Turing domain is a much less explored area compared to that of two species prey-predator models. These models play a vital role in understanding the resource-consumer interactions at the successive trophic levels of respective ecosystems. In this work, we consider the spatio-temporal extension of a three species food chain model with Holling type-II functional response to describe the grazing pattern of both the intermediate and top predator populations. Conditions for Turing instability are derived. The thresholds for emergence and stability of various Turing patterns near the Turing bifurcation boundary are obtained with the help of amplitude equations using weakly nonlinear analysis. The analytical results are further verified with the help of extensive numerical simulations for the complete nonlinear model and both the results are compared for different sets of parameter values.

Stability in the mean of a stochastic three species food chain model with general L[formula omitted]vy jumps

A stochastic three species food chain model with general Le´vy jumps is investigated in the paper. The criterion on the global stability in the mean with probability one for each species is established. The results show that the dynamics of the model can be significantly changed by Le´vy jumps, and strongly indicate that the general jump N(dt, du) is more complicated than compensated jump N˜(dt,du).

Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response

Predator interference, or a decline in the per predator consumption rate as predator density increases, is generally considered a stabilizing mechanism in two-species predator-prey models. There is significant debate, as to whether prey handling predators, might interfere in the hunting process of prey searching predators, or whether these are mutually exclusive events. In the current manuscript, a three species food chain model, with strong top predator interference is considered. We prove that in terms of explosive instability/finite time blow up, sufficient interference by prey handling predators always tends to destabilize the system. The dynamics of a time delayed version, as well as the spatially explicit model are also explored. We use our results to comment on a certain paradox in ecological theory, as well as provide further insight into the nature of predator interference, and exploding populations of invasive species.

On “small” data blow-up in a three species food chain model

In Parshad et al. (2015) we showed that both the ODE and PDE versions of a classical three species food chain model, possess solutions that blow-up in finite time, for sufficiently large initial data. The object of the current work is to show that solutions to the PDE model can blow-up in finite time for small initial data as well. To the best of our knowledge, this is the first example of “diffusion induced” blow-up phenomenon for small initial data, in a three species system, where the corresponding ODE also has a possibility to blow-up for large initial data.