Stability Of Positive Equilibrium Point Research Articles

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

<abstract><p>A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.</p></abstract>

Almost periodic stability on a delay Nicholson’s blowflies equation

ABSTRACT By applying some novel inequality techniques, the properties of almost periodic function and the fluctuation lemma, this manuscript establishes the criteria for the existence and globally exponential stability of positive almost periodic solutions of a non-autonomous delayed Nicholson’s blowflies model under weaker conditions, which refine and complement some existing literature. In particular, when the influence of delay disappears and the addressed model is autonomous, the adopted assumption in the obtained result is a sharp condition ensuring the global exponential stability of positive equilibrium points. Moreover, some numerical examples are provided to illustrate the effectiveness and feasibility of the theoretical findings.

Dynamic modelling and optimal control of herd behaviour with time delay and media

ABSTRACT A dynamic model of herd behaviour with delay time and media is established and analysed to discover the latent mechanism that represents how capital flows in the stock market. We prove that solutions of the model are uniformly bounded, and the contagion threshold is obtained. The stability of positive equilibrium points of the model with zero and nonzero delay is discussed. An optimal control problem with media is formulated, and Pontryagin's maximum principle is applied to find an optimal strategy to control herding. Several numerical simulations show the effect of media and delay on herd behaviour. Finally, the practical meaning of the presented model is briefly discussed.

Dynamics and Hopf Bifurcation of a Chaotic Chemical Reaction Model

Taking into account an ideal mixture and a well-stirred reactor, some dynamical aspects for a 3-dimensional chaotic system are carried out. Positivity and boundedness of solutions are discussed. Equilibria are investigated and method of linearization is implemented for asymptotic behavior of system about these equilibria. Lyapunov function is constructed to prove the global stability of positive equilibrium point. Moreover, it is proved that system undergoes Hopf bifurcation about its interior (positive) equilibrium. An explicit criterion of Hopf bifurcation without finding the eigenvalues is used for the existence of Hopf bifurcation. Numerical simulation is presented for the illustration of theoretical discussion. Lyapunov dimension is approximated and maximum Lyapunov characteristic exponents are plotted to ensure the chaotic behavior of the model.

Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator

A predator–prey model with functional response Holling type II, Allee effect in the prey and a generalist predator is considered. It is shown that the model with strong Allee effect has at most two positive equilibrium points in the first quadrant, one is always a saddle point and the other exhibits multi-stability phenomenon since the equilibrium point can be stable or unstable. The model with weak Allee effect has at most three positive equilibrium points in the first quadrant, one is always a saddle point and the other two can be stable or unstable node. In addition, when the parameters vary in a small neighbourhood of system parameters the model undergoes different bifurcations, such as saddle–node, Hopf and Bogdanov–Takens bifurcations. Moreover, numerical simulation is used to illustrate the impact in the stability of positive equilibrium point(s) by adding an Allee effect and an alternative food sources for predators.

Memory and mutualism in species sustainability: A time-fractional Lotka-Volterra model with harvesting

We first present a predator-prey model for two species and then extend the model to three species where the two predator species engage in mutualistic predation. Constant effort harvesting and the impact of by-catch issue are also incorporated. Necessary sufficient conditions for the existence and stability of positive equilibrium points are examined. It is shown that harvesting is sustainable, and the memory concept of the fractional derivative damps out oscillations in the population numbers so that the system as a whole settles on an equilibrium quicker than it would with integer time derivatives. Finally, some possible physical explanations are given for the obtained results. It is shown that the stability requires the memory concept in the model.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

Mathematical Study About a Predator–Prey Model with Anti-predator Behavior

Species use their defense mechanisms to fight, kill or escape from predators. Each species shows its unique defense mechanism to avoid the predation. In this paper, we propose and analyze a three species predator–prey system with group defense mechanism. Conditions for boundedness and positivity of equilibrium points have been discussed. We establish conditions for stability of positive equilibrium points and periodic solutions via Hopf-bifurcation. The predator–prey system is numerically studied with the help of phase portrait, time evolution and bifurcation diagrams. Lyapunov exponent and sensitivity analysis ensure the chaotic behaviour of the model system. Period doubling and period halving bifurcations show dynamical complexities of the food chain system. This study suggests that the repercussion of better group defense by the intermediate predator is able to produce chaos in food chain models.

Role of Induced Volatile Emission Modelling Tritrophic Interaction

Plant population response to herbivore attack by releasing volatile organic compounds (VOCs) in a highly regulated fashion. The natural enemies of herbivores are attracted by such induced volatile. This type of mechanism is known as “indirect defense mechanism” for plants. A tritrophic interaction among the plants, herbivores and their carnivorous enemies is considered along with “indirect defense mechanism” of plants in recruitment rate of carnivorous enemies. A systematic analysis of the dynamics of plant–herbivore–carnivore system with Holling type II functional response is done in this paper. The conditions for existence and feasibility of boundary and interior equilibrium points of the system are also determined. The conditions for the global stability of positive equilibrium point is derived. Persistence of the system is studied. Here, different types of bifurcations are analyzed with respect to many parameters. A key role is played by immigration rate of carnivores in determining the eventual state of the ecosystem. Finally, numerical simulations are performed to validate the obtained results.

Neimark–Sacker bifurcation and chaos control in discrete-time predator–prey model with parasites

In this paper, we discuss the qualitative behavior of a four-dimensional discrete-time predator–prey model with parasites. We investigate existence and uniqueness of positive steady state and find parametric conditions for local asymptotic stability of positive equilibrium point of given system. It is also proved that the system undergoes Neimark–Sacker bifurcation (NSB) at positive equilibrium point with the help of an explicit criterion for NSB. The system shows chaotic dynamics at increasing values of bifurcation parameter. Chaos control is also discussed through implementation of hybrid control strategy, which is based on feedback control methodology and parameter perturbation. Finally, numerical simulations are conducted to illustrate theoretical results.

Analysis of a Delayed Predator–Prey System with Harvesting

Abstract This article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

Dynamic analysis of mathematical model of ethanol fermentation with gas stripping

In this paper, a mathematical model for ethanol fermentation with gas stripping is investigated. Firstly, the model with continuous substrate input is taken. We study the existence and local stability of two equilibrium points. According to Poincare–Bendixson Theorem, the sufficient condition for the globally asymptotical stability of positive equilibrium point is obtained, which implies that we can get stable ethanol product. Secondly, we study the model with impulsive substrate input and obtain the sufficient condition for the local stability of cell-free periodic solution by using the Floquet’s theory of impulsive differential equation and small-amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical (subcritical) bifurcation. Finally, our results are confirmed by means of numerical simulation.