Narrow Results

In this paper, a new sufficient condition is given for the global asymptotic stability and global exponential output stability of a unique equilibrium points of delayed cellular neural networks (DCNNs) by using Lyapunov method. This condition imposes constraints on the feedback matrices and delayed feedback matrices of DCNNs and is independent of the delay. The obtained results extend and improve upon those in the earlier literature, and this condition is also less restrictive than those given in the earlier references. Two examples compared with the previous results in the literatures are presented and a simulation result is also given.

This paper aims at studying the basic properties and chaotic synchronization of complex Lorenz system:

Numerically we show that this system is a chaotic system and exhibits chaotic attractors. The necessary conditions for system (⋆) to generate chaos are obtained. Analytical and numerical calculations are presented to achieve synchronization. Active control technique is used to synchronize chaotic attractors of equations (⋆).

We consider a diffusion equation on a domain Ω with a cubic reaction at the boundary. It is known that there are no patterns when the domain Ω is a ball, but the existence of such patterns is still unkown in the more general case in which Ω is convex. The goal of this paper is to present numerical evidence of the existence of nonconstant stable equilibria when Ω is the unit square. These patterns are found by continuation of families of unstable equilibria that bifurcate from constant solutions.

Interesting dynamical features such as periodic solutions of binary cellular automata are rare and therefore difficult to find, in general. In this paper, we illustrate an effective method in identifying fixed and periodic points of traditional one- and two-dimensional $p$-valued cellular automaton systems, using cycle graphs. We also show that when the binary or $p$-valued cellular states are extended to real-values and when defined as doubly-infinite vectors, there exists a continuum of periodic solutions for every period, even when the governing local rules are simply linear. In addition, we demonstrate the important effect that boundary conditions have on systems’ dynamical structures.

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

The importance of exogenous reinfection versus endogenous reactivation for the resurgence of tuberculosis (TB) has been a matter of ongoing debate. Previous mathematical models of TB give conflicting results on the possibility of multiple stable equilibria in the presence of reinfection, and hence the failure to control the disease even when the basic reproductive number is less than unity. The present study reconsiders the effect of exogenous reinfection, by extending previous studies to incorporate a generalized rate of reinfection as a function of the number of actively infected individuals. A mathematical model is developed to include all possible routes to the development of active TB (progressive primary infection, endogenous reactivation, and exogenous reinfection). The model is qualitatively analyzed to show the existence of multiple equilibria under realistic assumptions and plausible range of parameter values. Two examples, of unbounded and saturated incidence rates of reinfection, are given, and simulation results using estimated parameter values are presented. The results reflect exogenous reinfection as a major cause of TB emergence, especially in high prevalence areas, with important public health implications for controlling its spread.

This paper deals with problems of stability and travelling waves for a class of recurrent neural networks with arbitrary exponents k and m. A novel model which is not only nonlinear but also coupled is proposed. This paper makes the following contributions: (1) Conditions for local stablility of 1-D networks and 2-D networks are established with a series of mathematical arguments. (2) Completely convergence of 1-D neural networks is proved by constructing a suitable energy function. (3) The nonuniform solution of the networks is obtained when the connectivity is Gaussian profile. (4) Travelling waves of the networks are analyzed with the connectivity profile. Finally, simulation examples are employed to illustrate the obtained results.

The new idea of group defense as recently introduced by the author in the context of two interacting populations is in this paper applied to communities subject also to a disease. The system is formulated with the bare minimum of interactions among all the populations involved in order to highlight the effects of the nonlinearity describing the defense mechanism. A key parameter identified in the purely demographic model, which completely describes its outcomes, is seen here to have an important role also, in that it is dropping below a threshold prevents the disease from invading the environment and causes the healthy prey and predators to coexist via persistent oscillations.

A new deterministic model for the transmission dynamics of human papillomavirus (HPV) and related cancers, in the presence of the Gardasil vaccine (which targets four HPV types), is presented. In the absence of routine vaccination in the community, the model is shown to undergo the phenomenon of backward bifurcation. This phenomenon, which has important consequences on the feasibility of effective disease control in the community, arises due to the re-infection of recovered individuals. For the special case when backward bifurcation does not occur, the disease-free equilibrium (DFE) of the model is shown to be globally-asymptotically stable (GAS) if the associated reproduction number is less than unity. The model with vaccination is also rigorously analyzed. Numerical simulations of the model with vaccination show that, with the assumed 90% efficacy of the Gardasil vaccine, the effective community-wide control of the four Gardasil-preventable HPV types is feasible if the Gardasil coverage rate is high enough (in the range 78–88%).

In this paper, a predator–prey model with prey refuge was developed to investigate how prey refuge affect the dynamics of predator–prey interaction. We studied the existence and stability of equilibria, and then derived the sufficient conditions for the bifurcation such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcation. In addition, a series of numerical simulations were carried out to illustrate the theoretical analysis, and the numerical results are consistent with the analytical results. Our results demonstrate that prey refuge has a great impact on the predator–prey dynamics.

In this paper, a mathematical model to study the simultaneous effect of two toxicants (one is more toxic than the other) on the growth and survival of a biological species is proposed. The cases of instantaneous spill, constant and periodic emissions of each of the toxicant into the environment are considered. It is shown that in the case of an instantaneous spill of each of the toxicant into the environment, the species after its initial decrease in density may recover to its original level after a period of time, the magnitude of which depends on the toxicity and washout rate of each of the toxicant. However, if both the toxicants are emitted with constant rates, the species in the habitat is doomed to extinction sooner than the case of a single toxicant having the same influx and washout rates as one of them, the extinction rate becoming faster with the increase in toxicity and emission rate of the other toxicant. It is also shown that for a small amplitude periodic emission of the toxicant with a constant mean, the stability behavior of the system is same as that of the case of the constant emission. It is found further through the model study that if suitable efforts are made to reduce the emission rate of each of the toxicant at the source and its concentration in the environment by some removal mechanism, an appropriate level of species density can be maintained.

Smoking can cause numerous diseases like lung cancer, mouth diseases, bronchitis, stroke, heart diseases, etc. To overcome such and other related diseases, we need to avoid the smoking habit from the population at large scale. The key purpose of this work is to discuss and analyze different kinds of smoking phases which are very common in our society. The model is formulated in the form of a system of ordinary differential equations keeping in view the existing literature. The model is investigated for possible equilibria besides positivity and boundedness of solution. The model exhibits six equilibrium points and the stability analysis of each fixed point is investigated. The local stability analysis of each equilibrium point is carried out with the help of linearization approach and it is shown that these equilibria are locally asymptotically stable under certain conditions. We also checked the fixed points for global analysis with the help of Lyapunov stability theory and stability criteria were obtain therein. We also perform the local sensitivity analysis. To verify the theoretical results, a scheme is developed using non-standard finite difference method. Simulations suggest that the obtained graphical results support our analytical findings with a high degree of accuracy.

We model and analyze strategic interaction over time in a duopoly. Each period the firms independently and simultaneously take two sequential decisions. First, they decide whether or not to advertise, then they set prices for goods which are imperfect substitutes. Not only the own, but also the other firm's past advertisement efforts affect the current "sales potential" of each firm. How much of this potential materializes as immediate sales, depends on current advertisement decisions. If both firms advertise, "sales potential" turns into demand, otherwise part of it "evaporates" and does not materialize. We determine feasible rewards and equilibria for the limiting average reward criterion. Uniqueness of equilibrium is by no means guaranteed, but Pareto efficiency may serve very well as a refinement criterion for wide ranges of the advertisement costs.

We develop a two-stage (four component) model for youths with serious drinking problems and their treatment. The youths with alcohol problems are split into two classes, namely those who admit to having a problem and those who do not. It is shown that the model possesses two steady states, one where people have no alcohol problems and one where there is an endemic state involving those with an alcohol problem. The stability of these states is analyzed and a threshold established such that each state will be stable depending on whether the incidence rate is above or below the threshold. The model is analyzed in the context of actual data.

A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if $R_0\le 1$; while the drug spread equilibrium is also globally asymptotically stable if $R_0>1$. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.

In this paper, we revisit the model by Guedj et al. [J. Guedj, R. Thibaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, Biometrics63 (2007) 198–206; J. Guedj, R. Thibaut and D. Commenges, Practical identifiability of HIV dynamics models, Bull. Math. Biol.69 (2007) 2493–2513] which describes the effect of treatment with reverse transcriptase (RT) inhibitors and incorporates the class of quiescent cells. We prove that there is a threshold value $\overline{\eta }$ of drug efficiency $\eta$ such that if $\eta >\overline{\eta }$, the basic reproduction number $R_0<1$ and the infection is cleared and if $\eta <\overline{\eta }$, the infectious equilibrium is globally asymptotically stable. When the drug efficiency function $\eta (t)$ is periodic and of the bang–bang type we establish a threshold, in terms of spectral radius of some matrix, between the clearance and the persistence of the disease. As stated in related works [L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol.69 (2007) 2027–2060; P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bull. Math. Biol.71 (2009) 189–210.], we prove that the increase of the drug efficiency or the active duration of drug must clear the infection more quickly. We illustrate our results by some numerical simulations.

It is well known that the cotton aphid is the major pest in cotton fields of Northwest China, and seven-spot ladybird is an important natural enemy among the various possible natural enemies of cotton aphid. In order to increase the applications of population dynamics in integrated pest management and control the cotton aphids biologically, we need to understand the population dynamics of cotton aphid and their natural enemies. A delay predator–prey system on cotton aphid and seven-spot ladybird beetle are proposed in this paper. Based on the comparison theorem and an iterative method, we investigate the global attractivity of the equilibrium points which have important biological meanings. Furthermore, some numerical simulations were carried out to illustrate and expand our theoretical results, in which a conjecture to generalize the well-known Theorem 16.4 in H. R. Thiemes book was put forward, which was taken as the open problem. The numerical simulations show coexistence of periodic solution, confirming the theoretical prediction.

In this paper, we are concerned with the stability for a model in the form of system of integro-partial differential equations, which governs the evolution of two competing age-structured populations. The age-specified environment is incorporated in the vital rates, which displays the hierarchy of ages. By a non-zero fixed-point result, we show the existence of positive equilibria. Some conditions for the stability of steady states are derived by means of semigroup method. Furthermore, numerical experiments are also presented.

In clinical practice, many cirrhosis scores based on alanine aminotransferase (ALT) levels exist. Although the most recent direct acting antivirals (DAAs) reduce fibrosis and ALT levels, the Hepatitis C virus (HCV) is not always removed. In this paper, we study a mathematical model of the HCV virus, which takes into account the role of the immune system, to investigate the ALT behavior during therapy. We find five equilibrium points and analyze their stability. A sufficient condition for global asymptotical stability of the infection-free equilibrium is obtained and local asymptotical stability conditions are given for the immune-free infection and cytotoxic T lymphocytes (CTL) response equilibria. The stability of the infection equilibrium with the full immune response is numerically performed.

We model and analyze strategic interaction over time in a duopolistic market. Each period the firms independently and simultaneously choose whether to advertise or not. Advertising increases the own immediate sales, but may also cause an externality, e.g., increase or decrease the immediate sales of the other firm ceteris paribus.

There exists also an effect of past advertisement efforts on current sales. The ‘market potential’ of each firm is determined by its own but also by its opponent's past efforts. A higher effort of either firm leads to an increase of the market potential, however the impact of the own past efforts is always stronger than the impact of the opponent's past efforts. How much of the market potential materializes as immediate sales, then depends on the current advertisement decisions.

We determine feasible rewards and (subgame perfect) equilibria for the limiting average reward criterion using methods inspired by the repeated-games literature. Uniqueness of equilibrium is by no means guaranteed, but Pareto efficiency may serve very well as a refinement criterion for wide ranges of the advertisement costs.