Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.     

Analysis of a Lotka–Volterra food chain chemostat with converting time delays

A model of the food chain chemostat involving predator, prey and growth-limiting nutrients is considered. The model incorporates two discrete time delays in order to describe the time involved in converting processes. The Lotka–Volterra type increasing functions are used to describe the species uptakes. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) local stability and instability of the positive equilibrium and (ii) global stability of the non-negative equilibria. Numerical simulation suggests that the delays have both destabilizing and stabilizing effects, and the system can produce stable periodic solutions, quasi-periodic solutions and strange attractors.     

Dynamical behavior of a stochastic food chain chemostat model with Monod response functions

This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results.     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

一类未搅拌恒化器中食物网的共存态

本文讨论了一类在单种营养物输入的未搅拌恒化器中的简单食物网模型,该模型除了营养物以外,还包含有一个捕食者种群和两个竞争的食饵种群.应用Dancer不动点指数和度理论的知识,给出了该系统共存态存在的充分必要条件.     

Stationary distribution and extinction for a food chain chemostat model with random perturbation

In this paper, we study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results.     

Coexistence in a two species chemostat model with Markov switchings

This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.     

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.     

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.     

Bifurcation and chaos in a Monod type food chain chemostat with pulsed input and washout

In this paper, we introduce and study a model of a Monod type food chain chemostat with pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions 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, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

具有时滞和脉冲输入的一类双资源和两种微生物恒化器模型的分析

考虑一类双资源和两种微生物且具有时滞和脉冲输入的恒化器模型,证明了微生物灭绝周期解的存在性,并得到该周期解全局吸引性的临界条件和系统持久的充分条件,最后利用数值模拟结果说明本文的主要结论.     

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions 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, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

有限链上的左(右)零模

为了更好地描述信息的聚合,有限链上的左(右)nullnorm的概念被引入。然后,各种光滑的左(右)nullnorm的结构定理被给出,即给出了一个二元运算是光滑的左(右)nullnorm的充要条件。     

Blow-up in a three-species cooperating model

The properties of solutions for a parabolic system with homogeneous Dirichlet boundary conditions, which arises in a cooperating three-species food chain model, are investigated. It is shown that global solutions exist if the intraspecific competitions are strong whereas blowup solutions exist under certain conditions if the intra-specific competitions are weak.     

Dynamics of a non-autonomous ratio-dependent food chain model

A simple non-autonomous ratio-dependent food chain model is investigated. It is shown that the system is permanence, extinction, ultimate boundedness and globally asymptotic stability under some appropriate conditions. Moreover, by employing Mawhin’s coincidence degree theory, some easily applicable criteria are established for the global existence of positive periodic solution of this model.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

On the Effects of Risk Pooling in Supply Chain Management： Review and Extensions

The primary challenge in supply chain management （SCM） is matching supply with uncertain demand. Risk pooling is an efficient and promising strategy to meet this challenge by reducing the underlying demand uncertainty through aggregation. The main focus of this paper is to analyze the effects of risk pooling under different supply chain settings. There are two main contributions. First, we propose a mathematical framework which serves the multi-purpose of （1） unifying existing models on risk pooling in the literature, （2） providing new facets and insights of understanding existing results on risk pooling, and （3） setting up new ground for extending existing models and results. Second; we investigate one interesting effect of risk pooling, namely, the decreasing marginal return （or supermodularity）. We show that there are decreasing marginal returns in risk pooling practices under certain conditions, specifically when the demand is independent and identically distributed （I.I.D.） and normally distributed.     

带比例功能反应函数食物链交错扩散模型的整体解

本文研究了带有比例功能反应函数食物链交错扩散模型整体解的存在性和正平衡点的稳定性.利用能量方法和Gagliardo-Nirenberg型不等式,获得了该模型整体解的存在性和一致有界性,同时通过构造Lyapunov函数给出了该模型正平衡点全局渐近稳定的充分条件.     

目标回扣契约下供应链对突发事件的协调应对

本文的目的就是要研究当突发事件造成需求的市场分布发生变化时,供应链如何运用目标回扣契约进行协调应对.通过设置目标回扣契约参数,分别获得了需求突发下,集权供应链和分权供应链协调的条件.并且,研究结果表明,当突发事件造成需求的市场规模变化不是很大时,分权供应链仅通过调整目标回扣契约就能实现供应链的协调应对;当突发事件造成市场规模变化很大时,则需要采取调整目标回扣契约和改变生产计划两种策略来协调应对.这些结论,为供应链应急管理提供了一种新的应对策略.     

Fuzzy optimization for distribution of frozen food with imprecise times

Problems concerning the distribution routes for frozen products need to incorporate constraints that avoid breaks in the cold chain. The decision making process under uncertain environments is a common one in real logistics problems. The purpose of this study is to apply a fuzzy approach which will provide an optimal solution to the distribution of frozen food with uncertainty in its time values. A soft computing approach is used where fuzzy constraints are included in the modeling and the solution of the problem.