A Fractional-Order Food Chain Model with Omnivore and Anti-Predator

A fractional-order food chain model is proposed in this article. The model is built by prey, intermediate predator, and omnivore. It is assumed that intermediate predator only eat prey and omnivore can consume prey and intermediate predator. But, prey has the ability called as anti-predator behavior to escape from both predators. For the first discussion, it is found that all solutions are existential, uniqueness, boundedness, and non-negative. Further, we analyze the existence condition and local stability of all points, that is point for the extinction of all populations, both predators, intermediate predator, omnivore, and point for the existence of all populations. We also investigate the global stability of all points, except point for the extinction of all populations and both predators. Finally, we preform several numerical solutions by using the nonstandard Grunwald-Letnikov approximation to demonstrate the our analytical results

The dynamics of a delayed generalized fractional-order biological networks with predation behavior and material cycle

In this paper, a delayed generalized fractional-order biological networks with predation behavior and material cycle is comprehensively discussed. Some criteria of stability and bifurcation for the present system is presented. Moreover some results of two delays are obtained. Finally, some numerical simulations are presented to support the analytical results

Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative

This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle

(R2020) Dynamical Study and Optimal Harvesting of a Two-species Amensalism Model Incorporating Nonlinear Harvesting

This study proposes a two-species amensalism model with a cover to protect the first species from the second species, with the assumption that the growth of the second species is governed by nonlinear harvesting. Analytical and numerical analyses have both been done on this suggested ecological model. Boundedness and positivity of the solutions of the model are examined. The existence of feasible equilibrium points and their local stability have been discussed. In addition, the parametric conditions under which the proposed system is globally stable have been determined. It has also been shown, using the Sotomayor theorem, that under certain parametric conditions, the suggested model exhibits a saddle-node bifurcation. The parametric conditions for the existence of the bionomic equilibrium point have been obtained. The optimal harvesting strategy has been investigated utilizing the Pontryagins Maximum Principle. The potential phase portrait diagrams have been provided to corroborate the acquired findings

Influence of impreciseness in designing tritrophic level complex food chain modeling in interval environment

Abstract In this paper, we construct a tritrophic level food chain model considering the model parameters as fuzzy interval numbers. We check the positivity and boundedness of solutions of the model system and find out all the equilibrium points of the model system along with its existence criteria. We perform stability analysis at all equilibrium points of the model system and discuss in the imprecise environment. We also perform meticulous numerical simulations to study the dynamical behavior of the model system in detail. Finally, we incorporate different harvesting scenarios in the model system and deploy maximum sustainable yield (MSY) policies to determine optimum level of harvesting in the imprecise environment without putting any unnecessary extra risk on the species toward its possible extinction

Pursuit-Evasion Dynamics for Bazykin-Type Predator-Prey Model with Indirect Predator Taxis

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Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey

In this paper, a dynamical analysis of a fractional-order predator-prey model with infectious diseases in prey is performed. First, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. We also show that the model has at most five equilibrium points, namely the origin, the infected prey and predator extinction point, the infected prey extinction point, the predator extinction point, and the co-existence point. For the first four equilibrium points, we show that the local stability properties of the fractional-order system are the same as the first-order system, but for the co-existence point, we have different local stability properties.We also present the global stability of each equilibrium points except for the origin point. We observe an interesting phenomenon, namely the occurrence of Hopf bifurcation around the co-existence equilibrium point driven by the order of fractional derivative. Moreover, we show some numerical simulations based on a predictor-corrector scheme to illustrate the result of our dynamical analysis

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

[EN]The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper