Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.     

Permanence and stability of a predator–prey system with stage structure for predator

A stage-structured predator–prey system with delays for prey and predator, respectively, is proposed and analyzed. Mathematical analysis of the model equations with regard to boundedness of solutions, permanence and stability are analyzed. Some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium, respectively, are obtained.     

Analysis of competitive chemostat models with the Beddington–DeAngelis functional response and impulsive effect

In this paper, we introduce and study a competitive system with Beddington–DeAngelis type functional response in periodic pulsed chemostat conditions. We investigate the subsystem with substrate and one of the microorganisms and study the stability of the periodic solutions, which are the boundary periodic solution of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate and one of the microorganism. Further, we prove that the system is permanent if the impulsive period less than some critical value. Therefore, our results are valuable for the manufacture of products by genetically altered organisms.     

Synchronization of the unified chaotic system

This paper studies the synchronization problem of the unified chaotic system. Three different methods, linear feedback method, nonlinear feedback method and impulsive control method are used to control synchronization of the unified chaotic systems. Based on the Lyapunov stability theory and impulsive control method, the conditions of synchronization are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the three different methods.     

Note on permanence and global stability in delayed ratio-dependent predator-prey models with monotonic functional responses

Sufficient conditions of the permanence and global stability for the general delayed ratio-dependent predator-prey model

     

Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models

The theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time-delayed reaction and diffusion population models.     

Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei [Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].     

Periodicity and stability for a Lotka-Volterra type competition system with feedback controls and deviating arguments

This paper deals with the general periodic Lotka-Volterra type competition systems with feedback controls and deviating arguments. By employing fixed point index theory on cone, an explicit necessary and sufficient condition for the global existence of the positive periodic solution of the systems is proved. By constructing a suitable Lyapunov functional, a set of easily verifiable sufficient conditions for the global asymptotic stability of the positive periodic solution of the systems is given.     

Partial functional differential equation with an integral condition and applications to population dynamics

In this work we consider a semilinear functional partial differential equation with an integral condition. We apply the method of semidiscretization in time, to establish the existence and uniqueness of solutions. We also study the continuation of the solution to the maximal interval of existence. Finally we give examples to demonstrate the applications of our results.     

Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay

This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique.     

Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays

A diffusive Lotka-Volterra type model with nonlocal delays for two competitive species is considered. The existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four-dimensional system without delay. Furthermore, in order to prove the asymptotic stability (up to translation) of bistable wavefronts of the system, the existence, regularity and comparison theorem of solutions of the corresponding Cauchy problem are first established for the systems on R by appealing to the theory of abstract functional differential equations. The asymptotic stability (up to translation) of bistable wavefronts are then proved by spectral methods. In particular, we also prove that the spreading speed is unique by upper and lower solutions technique. From the point of view of ecology, our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation.     

Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage

This paper is concerned with the monotonicity and uniqueness of traveling waves for a reaction-diffusion model with quiescent stage. We first obtain the exponential decay rate of wave profiles, and then we show that any profile is strictly monotone by using the strong comparison principle. Furthermore, we prove the uniqueness (up to translation) of all traveling waves including even the waves with minimal speed.     

Note on “Bifurcation of a predator-prey model with undercrowding effect”

Inexistence and existence of limit cycles in a predator-prey model with undercrowding effect are studied. The number of limit cycles obtained by Zheng Weiwei in 2000 is corrected.     

Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity

This paper is concerned with the global stability of wavefronts with minimal speeds for degenerate nonlocal dispersal equations. By the method of super- and subsolutions and a squeezing technique, the wavefront with minimal speed is proved to be globally asymptotically stable with phase shift.     

Global-stability problem for coupled systems of differential equations on networks

The global-stability problem of equilibria is investigated for coupled systems of differential equations on networks. Using results from graph theory, we develop a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

Traveling waves in a nonlocal dispersal population model with age-structure

This paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age-structure. If the birth function is monotone, then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness. It is proved that the traveling wavefront is strictly increasing and unique up to a translation. The asymptotic behavior of traveling wavefronts is also obtained. If the birth function is not monotone, the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi-monotonicity.