Ahmadjan Muhammadhaji

Global Attractivity of a Periodic Delayed -Species Model of Facultative Mutualism

Two classes of periodic -species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

PERMANENCE IN GENERAL NON-AUTONOMOUS LOTKA–VOLTERRA PREDATOR–PREY SYSTEMS WITH DISTRIBUTED DELAYS AND IMPULSES

In this paper, we study a general non-autonomous Lotka–Volterra predator–prey system with distributed delays and fixed time impulsive effects. Necessary and sufficient integral conditions on the permanence of species are established.

Global Stability of HIV-1 Infection Model with Two Time Delays

We investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two time delays describing time needed for infection of cell and CTLs generation. Our model admits three possible equilibria: infection-free equilibrium, CTL-absent infection equilibrium, and CTL-present infection equilibrium. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied.

On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models.

Periodic Solutions for -Species Lotka-Volterra Competitive Systems with Pure Delays

We study a class of periodic general -species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the -species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.

Global stability of an SI epidemic model with feedback controls in a patchy environment

Abstract In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n ( n  ≥ 2) patches. We derive the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable if R 0  ≤ 1. In the case of R 0  > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results.

Adaptive Control Strategy for Projective Synchronization of Neural Networks

In this paper, we studied the projective synchronization of a type of chaotic neural networks (NNs) by introducing a novel adaptive control strategy. We obtained some useful sufficient criteria for the projective synchronization of considered networks via designing novel adaptive controller and introducing a suitable Lyapunov function. In addition, we gave a numerical example to validate the feasibility of the obtained results. It is worth to mention that the projective synchronization is a very general and it includes chaos stabilization, anti-synchronization and complete synchronization as its special cases.

Finite-Time Synchronization of Complex Dynamical Networks with Time-Varying Delays and Nonidentical Nodes

In this paper, we investigated the finite-time synchronization (FTS) problem for a class of time-delayed complex networks with nonidentical nodes onto any uniformly smooth state. By employing the finite-time stability theorem and designing two types of novel controllers, we obtained some simple sufficient conditions for the FTS of addressed complex networks. Furthermore, we also analyzed the effects of control variables on synchronization performance. Finally, we showed the effectiveness and feasibility of our methods by giving two numerical examples.

Mathematical analysis of a nutrient–plankton system with delay

A mathematical model describing the interaction of nutrient–plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.

Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold value , then the susceptible pest eradication periodic solution is globally asymptotically stable and if , then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractive -periodic solution.