Asymptotic Stability Of Equilibrium Solution Research Articles

Spatiotemporal dynamics of a predation system with time delay and spatial diffusion

This paper investigates the spatiotemporal dynamics of a Monod–Haldane type predator–prey interaction system that incorporates: (1) a time delay in the predator response term in the predator equation; and (2) diffusion in both prey and predator. We provide rigorous results of our system including the asymptotic stability of equilibrium solutions and the existence and properties of Hopf bifurcations with or without time delay and diffusion. The effect of diffusion on bifurcated periodic solutions is investigated. We show that analytically and numerically at a certain value the carrying capacity or time-delay-driven stability or instability occurs when the corresponding system has either a unique interior equilibrium or two interior equilibria. Moreover, we illustrate the spatial patterns out of an initially nearly homogeneous state via numerical simulations, which show that the system dynamics exhibits complex pattern replication: spiral wave patterns and chaotic spiral patterns by increasing the control parameter K and time delay τ respectively. In addition, we obtain further spiral patterns with different initial conditions. These results indicate that the carrying capacity and time delay play an important role in pattern selection. Our results may provide useful biological insights on population management for predator–prey interaction systems.

Analysis of stochastic viral infection model with lytic and nonlytic immune responses

In this paper, we explore the stochastic viral infection model and show that this model has a unique global solution. We use the method of Lyapunov function to study the stochastic asymptotic stability of equilibrium solution for this model. Finally, the sufficient condition for persistence of the disease is established and our mathematical findings are illustrated.

Existence and Globally Asymptotic Stability of Equilibrium Solution for Fractional-Order Hybrid BAM Neural Networks with Distributed Delays and Impulses

This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the U-layer and V-layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.

Stochastic Asymptotic Stability of Nowak-May Model with Variable Diffusion Rates

We consider a stochastically perturbed Nowak-May model of virus dynamics within a host. We prove the global existence of unique strong solution. Using the Lyapunov method, we found sufficient conditions for the stochastic asymptotic stability of equilibrium solutions of this model.

On the existence and asymptotic stability of solutions for unsteady mixing-layer models

We introduce in this paper some elements for the mathematical analysis of turbulence models for oceanic surface mixing layers. We consider Richardson-number based vertical eddy diffusion models. We prove the existence of unsteady solutions if the initial condition is close to an equilibrium, via the inverse function theorem in Banach spaces. We use this result to prove the non-linear asymptotic stability of equilibrium solutions.

Global asymptotic stability analysis of nonlinear differential equations in hybrid bidirectional associative memory neural networks with distributed time-varying delays

In this paper, the problem on an unique equilibrium solution of the bidirectional associative memory neural networks with distributed time-varying delays is investigated via Lyapunov stability theory. By using topological degree theory and M-matrix theory, several sufficient conditions ensuring the existence, uniqueness and global asymptotic stability of equilibrium solution are derived. An illustrative example is given to demonstrate the effectiveness of the obtained results.

Stability of a Nonlinear Age-Dependent Population Model Containing a Control Variable

A very general nonlinear model of age-dependent population growth, which includes most of earlier presented extensions of the Lotka–von Foerster model, is considered. A criterion for local asymptotic stability of equilibrium solutions of the model is given. The conclusions of this paper are compared with analogous results for some important special cases considered by other authors.