Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.     

Global stability of a generalized epidemic model

The competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology. We show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average Lyapunov function theorem and some dynamical system theory. This result is anticipated by H.J. Bremermann and H.R. Thieme (1989) [6] where they showed that the equilibrium is locally stable — the global result has not been established previously.     

Global stability of an SAIRS epidemic model with vaccinations,transient immunity and treatment

We consider an SAIRS epidemic model with vaccinations and treatment, where asymptomatic and symptomatic infectious individuals are considered in the transmission of the disease. We found the basic reproduction number, ${ℛ}_0$ and using ${ℛ}_0$, we conducted global stability analysis. We proved when ${ℛ}_0<1$, the disease-free equilibrium is globally stable. If ${ℛ}_0>1$, the disease-free equilibrium in unstable and a unique endemic equilibrium exists. We explored the global stability of the endemic equilibrium and noticed it is globally stable under certain conditions. Moreover, we then considered a special case of the SAIRS model, the SAIR model. We proved the disease-free equilibrium is globally stability when ${ℛ}_0<1$ and the endemic equilibrium is globally stable when ${ℛ}_0>1$. Next, we numerically simulated our analytical results and plotted these for various cases. Finally, we performed sensitivity analysis to tell us how each parameter in the system affects disease transmission.     

Global stability of a multiple infected compartments model for waterborne diseases

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number $R_0$, we prove that, if $R_0⩽1$, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if $R_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.     

Global exponential stability for shunting inhibitory CNNs with delays

In this paper, the global exponential stability for shunting inhibitory cellular neural networks (SICNNs) with delays is studied by constructing suitable Lyapunov functionals and applying some critical analysis techniques. The sufficient conditions guaranteeing the network’s global exponential stability are obtained. Our results impose less restrictive conditions than those in the references. Moreover an example is given to illustrate the feasibility of the conditions in our results.     

Positive solutions of a competition model for two resources in the unstirred chemostat

This paper deals with a competition model between two species for two growth-limiting and perfectly complementary resources in the unstirred chemostat. The main purpose is to determine the exact range of the parameters of two species so that the system possesses positive solutions, and to investigate multiple positive steady states of the system. The main tools used here include the monotone methods and the topological fixed point theory developed by Amann.     

一个带毒生化反应竞争模型正平衡解的全局稳定性

研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数γi(S)(i=1,2)为一般的非减可导函数时的生化反应模型,指出两篇文献中存在的问题,分析系统平衡点的稳定性,三维系统经历Hopf分支的条件和由此产生的周期解的稳定性.证明系统中某一微生物物种处于竞争劣势而趋于灭绝时另一微生物物种和营养在二维稳定流形上极限环的存在性.并用具体的实例验证文中所得结论.     

Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population

In this paper, we propose a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population, which is derived from the continuous case by using the well-known backward Euler method and by applying a Lyapunov function technique, which is a discrete version of that in the paper by Prüss et al. [J. Prüss, L. Pujo-Menjouet, G.F. Webb, R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 225-235]. It is shown that the global dynamics of this discrete epidemic model with latency are fully determined by a single threshold parameter.     

Global stability for a class of mass action systems allowing for latency in tuberculosis

A very general compartmental model of the spread of an infectious disease with mass action incidence is given. The global stability of this system is completely determined using Lyapunov functions. The general system exhibits the traditional threshold behaviour. The dimension of the system is arbitrary, allowing, in particular, for detailed modelling of the distribution of latency times for tuberculosis.     

A chemostat model of resource competition and allelopathy

Presented is a chemostat model in which one microbial population excretes a poison that increases the mortality of another, with toxin production increasing as the growth rate of the toxic species decreases. The model is intended to explore the role of allelopathy in blooms of harmful algae, such as red tide (Karenia brevis) and golden algae (Prymnesium parvum). This study introduces the model and its biological basis, and proceeds to the analysis of its asymptotic states. All theoretical results are supported by a set of numerical simulations. A discussion of biological conclusions and similarities to other mathematical models is also presented.     

Global analysis for a delayed SIV model with direct and environmental transmissions

In this paper, we propose a new SIV epidemic model with time delay, which also involves both direct and environmental transmissions. For such model, we first introduce the basic reproduction number $\mathscr{R}$ by using the next generation matrix. And then global stability of the equilibria is discussed by means of Lyapunov functionals and LaSalle''s invariance principle for delay differential equations, which shows that the infection-free equilibrium of the system is globally asymptotically stable if $\mathscr{R}<1$ and the epidemic equilibrium of the system is globally asymptotically stable for $\m     

A competition model for two resources in un-stirred chemostat

This paper studies a un-stirred chemostat with two species competing for two growth-limiting, non-reproducing resources. We determine the conditions for positive steady states of the two species, and then consider the global attractors of the model. In addition, we obtain the conditions under which the two populations uniformly strongly persist or go to extinction. Since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species, it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species. In particular, it is found that both instability and bi-stability subcases of the two semitrivial steady states are included in the coexistence region. The two populations will go to extinction when both possess large diffusion rates. If just one of them spreads faster with the other one diffusing slower, then the related semitrivial steady state will be globally attracting. The techniques used for the above results consist of the degree theory, the semigroup theory, and the maximum principle.     

Global exponential stability for a class of retarded functional differential equations with applications in neural networks

In this paper, the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time-varying delays are studied by employing several Lyapunov functionals. A number of sufficient conditions for these types of stability are presented. Our results show that these conditions are milder and more general than previously known criteria, and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Furthermore, the results obtained for neural networks with time-varying delays do not assume symmetry of the connection matrix.     

Global asymptotical stability for a fishery model with seasonal harvesting

A new fishery model is proposed by using the strategy of seasonal harvesting. Sufficient and necessary conditions are established to ensure the existence of a unique equilibrium or a periodic solution by the approach of Poincar′e maps. It is shown that the equilibrium or the periodic solution is globally asymptotically stable. Numerical examples are provided to demonstrate the model dynamics and some biological implications are given as well.     

A delayed chemostat model with general nonmonotone response functions and differential removal rates

A chemostat model with general nonmonotone response functions is considered. The nutrient conversion process involves time delay. We show that under certain conditions, when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, the competitive exclusion principle holds. In the case of insignificant death rates, the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times (see [H.M. El-Owaidy, M. Ismail, Asymptotic behavior of the chemostat model with delayed response in growth, Chaos Solitons Fractals 13 (2002) 787-795; H.M. El-Owaidy, A.A. Moniem, Asymptotic behavior of a chemostat model with delayed response growth, Appl. Math. Comput. 147 (2004) 147-161; S. Yuan, M. Han, Z. Ma, Competition in the chemostat: convergence of a model with delayed response in growth, Chaos Solitons Fractals 17 (2003) 659-667]). However, the proofs are all incorrect. In this paper, we provide a correct proof that also applies in the case of differential death rates. In addition, we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium, thus showing the outcome can be initial condition dependent. Moreover, we show that when the species specific death rates are included, damped oscillations may occur even when there is no delay. Thus, the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments.     

Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

In this paper, we investigate a disease transmission model of SIRS type with latent period τ?0 and the specific nonmonotone incidence rate, namely, . For the basic reproduction number R₀>1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.     

Delay versus age-of-infection - Global stability

We consider an SIR model of disease transmission which has been formulated with delay in order to give a fixed duration of infectiousness. A Lyapunov functional that has worked for similar models does not work here. Instead, the model is shown to be a consequence of an age-of-infection model for which the same class of Lyapunov functionals has worked, resolving the global stability.