Stability Of Interior Equilibrium Point Research Articles

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

<abstract><p>We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.</p></abstract>

Disease control prey–predator model incorporating prey refuge under fuzzy uncertainty

Ecological system depends on prey predator interaction. Sometimes diseases are spread among prey or predator or both species. In this paper, a disease control prey–predator model with Holling type-II functional response incorporating refuge in the susceptible prey is proposed. The impact of additional food for predator species in the system is investigated. Using some assumptions, the crisp model is formulated and it is converted into fuzzy model and defuzzified models by using basic defuzzification method. In the theoretical section, the condition of boundedness, existence of equilibrium points, global stability of interior equilibrium point and Hopf bifurcation on a refuge parameter are investigated. Theoretical results are verified in our numerical simulation section. Using MATLAB package we examined the behaviour of the species in the presence and absence of additional food in both crisp and fuzzy environments. The phase trajectories for different initial conditions in both environments and variations in population of the species are presented. Using MATCONT package, we present the bifurcation scenarios when the additional food parameter and refuge parameter vary. We compute the existence of Hopf point (H) and branch point (BP) in the model for suitable supply of additional food and intensity of refuge parameter. Finally the sensitivities of the parameters are plotted graphically. This theoretical and numerical study may be useful to control the infectious diseases in real world ecological systems.

Allee effect can simplify the dynamics of a prey-predator model

In this work, we investigate a prey-predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and ratio dependent functional response. we fulfill a comprehensive bifurcation analysis, constructing the two-parametric bifurcation diagrams which describes the effect of density dependent death rate parameter, and also show possible phase portraits. We have also investigated the model in the absence of Allee effect and corresponding bifurcation diagram has been presented to show the dynamical changes in the system. Then we compare the properties of the ratio dependent prey-predator model with and without the Allee effect and show that Allee effect has a significant role in the dynamics. Allee effect can preserve local extinction of populations and suppress the stability of interior equilibrium point. Finally, all the analytical results are validated with the help of numerical simulations.

Spatiotemporal Dynamics of a Diffusive Predator–Prey System with Allee Effect and Threshold Hunting

In this paper, we study a diffusive predator–prey system with the Allee effect and threshold hunting. First, the number of interior equilibrium points is determined by discussing the relation of parameters. Then, preliminary analysis on the local asymptotic stability and bifurcations of non-spatial system based on ordinary differential equations is presented. It is noted that four stable equilibrium points coexist due to the Allee effect and threshold hunting. The stability of interior equilibrium points and the existence of Turing instability induced by the diffusion, spatially homogeneous and inhomogeneous Hopf bifurcation, Turing–Hopf bifurcation are studied by analyzing the corresponding characteristic equation for spatial system. By constructing generalized Jacobian matrix, we analyze the stability of interior equilibrium point where u-component is equal to the threshold of functional response. These results show that the Allee effect, threshold hunting and diffusion have significant impacts on the dynamics. Last, we present some numerical simulations that supplement the analytic results.

Asymptotic stability of one prey and two predators model with two functional responses

We formulate a mathematical model to study the complex dynamical behavior of a three dimensional model consisting of one prey and two predators involving Beddington–DeAngelis and Crowley–Martin functional responses. The existence and stability conditions of the equilibrium points are analyzed. The global asymptotic stability of the interior equilibrium point, if exists, is proved by considering Lyapunov function. Several numerical simulations are performed to illustrate the theoretical analysis. The multiple states of stability are observed in one example whereas another example exhibits the global stability of interior equilibrium point.

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.

EFFECT OF DELAY ON TWO COMPETING ORGANISM IN A POLLUTED ENVIRONMENT

This paper concentrates on the study of delay effect on a chemostat type model of two competing organisms in the presence of toxicant. It is assumed that one of the competitors is affected by the toxicant while other detoxifies (absorbs) the toxicant. Uniform persistence and the stability of interior equilibrium point is discussed. Delay analysis has been carried out when the organisms are in coexistence, bistable and dominance respectively. Lastly the results are verified through computer simulation. Numerical simulation suggests that delay has a destabilizing effect (may lead to stability switches).