Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Dynamic analysis of a turbidostat model with the feedback control

In this paper, a mathematical model with impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of positive order one periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentration of microorganism and substrate. By investigating the periodic solution, the period and the initial point of the periodic solution are given. The results show that turbidostat with impulsive state feedback control tends to an order one periodic solution.     

Dynamical behaviors of a prey-predator system with impulsive control

In this paper, we study dynamics of a prey-predator system under the impulsive control. Sufficient conditions of the existence and the stability of semi-trivial periodic solutions are obtained by using the analogue of the Poincaré criterion. It is shown that the positive periodic solution bifurcates from the semi-trivial periodic solution through a transcritical bifurcation. A strategy of impulsive state feedback control is suggested to ensure the persistence of two species. Furthermore, a steady positive period-2 solution bifurcates from the positive periodic solution by the flip bifurcation, and the chaotic solution is generated via a cascade of flip bifurcations. Numerical simulations are also illustrated which agree well with our theoretical analysis.     

非线性脉冲状态依赖捕食-被捕食模型的定性分析

由于资源的有限性以及害虫群体对杀虫剂的抗性发展等因素,使得杀虫剂对害虫的杀死率具有饱和效应．因此,当害虫的数量达到经济阈值时, 杀虫剂对害虫的杀死率与经济阈值有关．为了刻画上述饱和效应,建立了一类非线性脉冲状态依赖捕食被捕食模型．利用Lambert W函数和脉冲半动力系统的相关技巧,分析了模型阶1正周期解的存在性和稳定性, 得到了相应的充分条件．进而讨论了非线性脉冲与线性脉冲对阶1周期解存在性的影响.     

Geometrical analysis of a pest management model in food-limited environments with nonlinear impulsive state feedback control

In this paper, a nonlinear impulsive state feedback control system is proposed to model an integrated pest management in food-limited environments. In the system, impulsive feedback control measures are implemented to control pests on the basis of the quantitative state of pests. Mathematically, an intuitive geometric analysis is used to indicate the existence of periodic solutions. The stability of periodic solutions is investigated by using Analogue of Poincar\''{e} Criterion. At last, numerical simulations are given to verify the theoretical analysis.     

Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solution of an impulsive delay differential equation with Allee effect. The results of this paper improve and generalize noticeably the known theorems in the literature.     

具有状态反馈脉冲控制的种群互惠动力系统的研究

研究具有状态反馈脉冲控制的种群互惠动力系统.首先利用微分方程几何理论和后继函数的方法得到一般系统阶1周期解的存在条件;然后研究了一类特殊系统,说明了该系统在一定条件下存在唯一的阶1周期解,并且给出了该阶1周期解轨道渐近稳定的条什,此外还探讨了该系统阶2周期解的存在性问题.     

The Dynamics of a Prey-dependent Consumption Model Concerning Integrated Pest Management

A mathematical model for the dynamics of a prey-dependent consumption model concerning integrated pest management is proposed and analyzed. We show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than some critical values. Furthermore, the conditions for the permanence of the system are given. By using bifurcation theory, we show the existence of a nontrival periodic solution if the pest-eradication periodic solution loses its stability. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamics, which implies that dynamical behaviors of prey-dependent consumption concerning integrated pest management are very complex, including period-doubling cascades, chaotic bands with periodic windows, crises, symmetry-breaking bifurcations and supertransients.     

Homoclinic bifurcation for a general state-dependent Kolmogorov type predator–prey model with harvesting

In this paper, a general Kolmogorov type predator–prey model is considered. Together with a constant-yield predator harvesting, the state dependent feedback control strategies which take into account the impulsive harvesting on predators as well as the impulsive stocking on the prey are incorporated in the process of population interactions. We firstly study the existence of an order-1 homoclinic cycle for the system. It is shown that an order-1 positive periodic solution bifurcates from the order-1 homoclinic cycle through a homoclinic bifurcation as the impulsive predator harvesting rate crosses some critical value. The uniqueness and stability of the order-1 positive periodic solution are derived by applying the geometry theory of differential equations and the method of successor function. Finally, some numerical examples are provided to illustrate the main results. These results indicate that careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.     

Feasibility of time-limited control of a competition system with impulsive harvest

A kind of time-limited control model on a competition system with impulsive harvest, described by impulsive differential equation with the initial and boundary value problem, is presented. The existence of solution of the model, corresponding to the feasibility of the short-term control, is discussed. By the comparison principle, the conditions under which the model has a solution are found by a series of the upper solutions, and the conditions under which the model has no solution are also given by a series of the lower solutions. Finally, the practical meanings of those conditions are explained. As an example, if other parameters are given, the times of the impulsive control is estimated and the theoretical results are verified by numerical simulations.     

Existence and stability of periodic solution of a stage‐structured model with state‐dependent impulsive effects

A single population growth model with stage‐structured and state‐dependent impulsive control is proposed. By using the Poincar'e map and the analogue of Poincaré's criterion, we prove the existence and the stability of positive order‐1 or order‐2 periodic solution. Moreover, we show that there is no periodic solution with order greater than or equal to three. Numerical results are carried out to illustrate the feasibility of our main results and the superiority of state feedback control strategy is also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Persistence and propagation of a discrete-time map and PDE hybrid model with strong Allee effect

Persistence and propagation of species are fundamental questions in spatial ecology. This paper focuses on the impact of Allee effect on the persistence and propagation of a population with birth pulse. We investigate the threshold dynamics of an impulsive reaction–diffusion model and provide the existence of bistable traveling waves connecting two stable equilibria. To prove the existence of bistable waves, we extend the method of monotone semiflows to impulsive reaction–diffusion systems. We use the methods of upper and lower solutions and the convergence theorem for monotone semiflows to prove the global stability of traveling waves and their uniqueness up to translation. Then we enhance the stability of bistable traveling waves to global exponential stability. Numerical simulations illustrate our theoretical results.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

Transitions in a logistic population model incorporating an Allee effect

In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.     

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

A turbidostat is an apparatus with feedback control system used to continuously culturing microorganisms. The dilution rate of the turbidostat can be regulated by the control system when the concentration of microorganism, detected by photoelectricity system or other devices, reaches a preset value. Based on the design ideas of the turbidostat, a differential equation with impulsive state feedback control is proposed for a kind of turbidostat system in this paper. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of periodic solution of order one are obtained according to the preset value and the types of the positive equilibrium of the corresponding system without impulsive control. Furthermore, it is pointed out that the system either tends to a stable state or has a periodic solution. Finally, the theoretical results are verified by numerical simulations.     

一类状态反馈脉冲控制的捕食者-食饵动力系统

本文研究了具有状态反馈脉冲控制的一类捕食者-食饵动力系统.我们首先利用微分方程几何理论和后继函数的方法得到该系统阶1周期解的存在性、唯一性和轨道渐近稳定性;然后说明了该系统不存在阶k(k=2,3,…)周期解,最后简单分析了相关结论在实践中的应用.     

The dynamics of a stage-structured predator-prey system with impulsive effect and Holling mass defence

In this paper, a stage-structured Holling mass defence predator-prey model with impulsive effect is investigated. By using comparison theorem and the stroboscopic technique, sufficient conditions for the global attractivity of mature prey-extinction periodic solution and permanence of the system are obtained. Furthermore, the numerical analysis is also inserted to illuminate the feasibility of the theoretical results.     

Dynamic behavior of an eco-epidemic system with impulsive birth

In the paper, we investigate an eco-epidemic system with impulsive birth. The conditions for the stability of infection-free periodic solution are given by applying Floquet theory of linear periodic impulsive equation. And we give the conditions of persistence by constructing a consequence of some abstract monotone iterative schemes. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive periodic solution. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.