Dynamical behavior for an eco-epidemiological model with discrete and distributed delay

Stability and bifurcation analysis on a Logistic model with discrete and distributed delays

Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

In this paper, a class of three-neuron network with discrete and distributed delays is introduced. We first give a detailed Hopf bifurcation analysis for the proposed network. Choosing the discrete time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Moreover, by using the normal form theory and center manifold theorem, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, numerical simulations are presented to demonstrate the effectiveness of our theoretical results.

Bifurcation and global periodic solutions in a delayed facultative mutualism system

Necessary and Sufficient Conditions for Delay-Dependent Asymptotic Stability of Linear Discrete Time Delay Autonomous Systems

In this paper, the robustness of a new visual servoing scheme with respect to camera calibration errors is analyzed. This scheme, called 2D 1/2 visual servoing, is based on the estimation of the partial camera displacement from the current to the desired camera poses at each iteration of the control law. Visual features and data extracted from the partial displacement allow as to design a decoupled control law controlling the six camera DOF. The necessary and sufficient conditions for local asymptotic stability in presence of camera calibration errors are easily obtained. Then, thanks to the simple structure of the system, sufficient conditions for global asymptotic stability are proposed. Finally, experimental results show the validity of our approach and its robustness not only with respect camera calibration errors but also to robot calibration errors.

Analysis of stability and Hopf bifurcation for a viral infectious model with delay

<p style='text-indent:20px;'>The spread of infectious diseases is often accompanied by a rise in the awareness programs to educate the general public about the infection risk and suggest necessary preventive practices. In the present paper we propose to study the impact of awareness on the dynamics of the classical <inline-formula><tex-math id="M2">\begin{document}$ SEIR $\end{document}</tex-math></inline-formula> by considering the budget allocation to warn people as a new dynamic variable. In the model formulation, it is assumed that the susceptible individuals contract the infection via a direct contact with infected individuals, and that the transmission rate is presented by a general decreasing function of the availability of funds. We further introduced a time delay in the growth rate of the budget allocation related to the number of reported infected cases. The existence and the stability criteria of the equilibrium states are obtained in terms of the basic reproduction number <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_{a} $\end{document}</tex-math></inline-formula>. It is shown that <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{a} \leq 1 $\end{document}</tex-math></inline-formula> is a necessary and sufficient condition for the global stability of the disease-free equilibrium, and by application of the geometric approach based on the third additive compound matrix we derived sufficient conditions for the global stability of the positive equilibrium state in the absence of delay. Our analysis reveals that awareness programs have the ability to reduce the infection prevalence. However, delay in providing funds destabilizes the system and give rise to periodic oscillations through Hopf-bifurcation. The direction and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. Numerical simulations and sensitive analysis are provided to illustrate the theoretical findings..

A general system of difference equations is presented for multispecies communities with density dependent population growth and delayed maturity. Interspecific competition, mutualism, predation, commensalism, and amensalism are accommodated. A sufficient condition for the local asymptotic stability of a coexistence equilibrium in this system is then proven. Using this system, the generalisation of the Beverton-Holt and Leslie-Gower models of competition to multispecies systems with possible maturation delays is presented and shown to yield interesting stability properties. The stability of coexistence depends on the relative abundances of the species at the unique interior equilibrium. A sufficient condition for local stability is derived that only requires intraspecific competition to outweigh interspecific competition. The condition does not depend on maturation delays. The derived stability properties are used to develop a novel estimation approach for the coefficients of interspecific competition. This approach finds an optimal configuration given two conjectures. First, coexisting species strive to outcompete competitors. Second, persisting species are more likely in stable systems with strong dampening of perturbations and high ecological resilience. The optimal solution is compared to estimates of niche overlap using an empirical example of malaria mosquito vectors with delayed maturity in the Anopheles gambiae sensu lato species complex.

Mathematical study of stage-structured pests control through impulsively released natural enemies with discrete and distributed delays

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

In this paper, the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays is studied by using a hybrid control strategy. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are established. In addition, the onset of an inherent bifurcation is delayed. Based on the normal form theory and the center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Numerical simulation results confirm that the hybrid controller is efficient in controlling Hopf bifurcation.

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane. The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species. The first Manuscript was on the study of the following equation: [Mathematical equations can not be displayed here, refer to PDF] Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation: [Mathematical equations can not be displayed here, refer to PDF] The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the di erence equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium. In my second Manuscript, I considered the difference equation: [Mathematical equations can not be displayed here, refer to PDF] where C;D and F are positive numbers and the initial conditions x-1 and x0 are non-negative numbers. Equation (2) which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium. The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts: Theorem

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

A remark on stabilization of nonlinear systems with discrete and distributed delays