Dynamical analysis of a two-species competitive system with state feedback impulsive control

In this paper,three competitive systems with different kinds of state-dependent control are presented and investigated.The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory,and the stability of the order-1 periodic solution of each system is also given.Besides,sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincare criterion,respectively.Finally,numerical simulations are carried out to verify the theoretical results.

DYNAMICS ON A HOLLING II PREDATOR-PREY MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL

to biological and chemical control strategy for pest control, a Holling II func- tional response predator-prey system concerning state-dependent impulsive control is investigated. We define the successor functions of semi-continuous dynamic system and give an existence theorem of order 1 periodic solution of such a system. By means of sequence convergence rules and quMitative analysis, we successfully get the conditions of existence and attractiveness of order 1 periodic solution. Our results show that our method used in this paper is more efficient and easier than the existing methods to prove the existence and attractiveness of order 1 periodic solution.

HETEROCLINIC BIFURCATIONS OF A PREY-PREDATOR FISHERY MODEL WITH IMPULSIVE HARVESTING

In this paper, we consider a prey-predator fishery model with Allee effect and state- dependent impulsive harvesting. First, we investigate the existence of order-1 heteroclinic cycle. Second, choosing p as a control parameter, we obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system （2.3） by using the geometry theory of semi-continuous dynamic systems. Finally, on the basis of the theory of rotated vector fields, heteroclinic bifurcation to perturbed system of system （2.3） is also studied. The methods used in this paper are novel to prove the existence of order-1 heteroclinic cycle and heteroclinic bifurcations.

Modeling and qualitative analysis of diabetes therapies with state feedback control

For the therapies of diabetes mellitus, a uovel mathematical model with two state impulses： impulsive injection of insulin and impulsive injection of glucagon, is proposed. To avoid hypoglycemia and hyperglycemia, the injections of insulin and glucagon are determined by closely monitoring the plasma glucose level of the patients. By using differential equation geometry theory, the existence of periodic solution and the attrac- tion region of the system have been obtained, which ensures that injections in such an automated way can keep the blood glucose concentration under control. The simula- tion results verify that the better insulin injection strategy in closed-loop control is a larger dose but longer interval rather than a smaller dose but shorter interval. Besides, our numerical analysis reveals that medicine studies and practice that slow down the insulin degradation are helpful for the plasma glucose control. Our findings can provide significant guidance in both design of artificial pancreas and clinical treatment.

A pest control model with state-dependent impulses

In this paper, a pest control model with state-dependent impulses is firstly established, which relies on releasing of natural enemies, together with spraying pesticides. By using the successor function of differential equation geometry rules, the existence of order one periodic solution is discussed. According to the Analogue of Poincare＇s Criterion, the orbitally asymptotic stability of the order one periodic solution is obtained. Furthermore, we investigated the global attractor of the system. From a biological point of view, our results indicate that： （1） the pest population can be controlled below some threshold; （2） compared to single measure, it is more efficient to take two measures for reducing the level of the pests.

Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar

In this paper,the system with impulsive state feedback control corresponding to the sulphitation reaction in process of manufacturo of sugar is considered.By means of square approximation and a series of switched systems,the periodic solution is approximated by a series of continuous hybrid limit cycles.Similar to the analysis of limit cycles of continuous systems,the existence and stability criteria of the order-1 periodic solution are obtained.Further,numerical analysis and discussion are given.

Dynamics of bilateral control system with state feedback for price adjustment strategy

The price fluctuation of commodities affects their demand,and it strongly influences the development of industry and the stability of the international economy.A price-output model with two price-dependent impulses is presented in this paper to describe their mutual restrictions and the price adjustment strategies.The equilibria and dynamic properties of the impulsive-free price-output model are qualitatively analyzed,and the results show the hazard of free price fluctuation.For the price-output model with two price-dependent impulses,conditions for the existence of the order-2 periodic solution are obtained,and the orbital asymptotic stability is proved.Those results reveal the positive role of price adjustment strategies.Finally,some numerical simulations are carried out to verify the theoretical results.