Existence Of Order-1 Periodic Solution Research Articles

The state-dependent impulsive control for a general predator–prey model

In this paper, a general predator–prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.

Existence of order-1 periodic solutions for a viral infection model with state-dependent impulsive control

It is well known that the drug treatment is always combined with the injection of immune factors. In this paper, a virus infection model with state-dependent impulsive control is considered. Firstly, by deriving three categories of Bendixson domain and using the methods of geometry and successor function, we establish some criteria for the existence of positive order-1 periodic solution for a general model, which extends the existing results in the literature. Further, the criteria are used to obtain the existence of positive order-1 periodic solutions in the two cases that the positive equilibrium point is on the left or right side of the pulse line, respectively. Finally, an example is presented to illustrate our results.

Dynamic analysis of conversion from a drug-sensitivity strain to a drug-resistant strain

In this paper, a mathematical model of conversion from a drug-sensitivity strain to a drug-resistant strain is given to investigate how antibiotic usage may be optimized to preserve or restore antibiotic effectiveness. This novel theoretical framework could result in an optimal criterion on how to reduce the drug resistance to a reasonable range by using the antibiotic dressing strategy. The sufficient conditions of existence of order-1 periodic solution are obtained in view of the geometrical theory of the semi-continuous dynamical system and the qualitative properties of the corresponding continuous system. The stability of the order-1 periodic solution is proved by means of [H. J. Guo, L. S. Chen and X. Y. Song, Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property, Appl. Math. Comput. 271 (2015) 905–922]. Finally, our results are confirmed by means of numerical simulations.

State feedback impulsive therapy to SIS model of animal infectious diseases

Controlling animal infectious diseases and its related infra microbes is of great significance to public health, since a lot of infectious diseases originate from animal epidemics and they often threaten human health. A state feedback impulsive model is constructed to depict the transmission and treatment of animal epidemics. Basing on the impulsive model, the existence of order-1 periodic solution and its stability are proved with a novel method. The theoretical results indicate that the impulsive treatment triggered by the number of infectious is an efficient approach to control animal infectious disease from breakout. Numerical simulation is presented to support the theoretical conclusion in the end.

Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse

In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincaré criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions.

Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

In this paper, a Leslie predator–prey system with ratio-dependent and state impulsive feedback control is investigated by applying the geometry theory of differential equation. When the economic threshold level is under the positive equilibrium, the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system can be obtained. When the economic threshold level is above the positive equilibrium, and the positive equilibrium is a focus point, sufficient conditions of the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system are also acquired. Furthermore, when the positive equilibrium is an unstable focus point, the existence of order-1 periodic solution of the impulsive system can be obtained within limit cycle of the continuous system. The mathematical results can be verified by numerical simulations.

Homoclinic bifurcation of a ratio-dependent predator–prey system with impulsive harvesting

In this paper, we investigate a ratio-dependent prey–predator model with state-dependent impulsive harvesting where the prey growth rate is subject to a strong Allee effect. The existence of order-1 homoclinic cycle is obtained, and choosing $$\alpha $$ as a control parameter, the existence, uniqueness and stability of order-1 periodic solution of the system are discussed by means of the geometry theory of semi-continuous dynamic system. We also investigate that system exhibits the phenomenon of homoclinic bifurcation about parameter $$\alpha $$ . Moreover, the numerical simulations are provided to show the main results. The used methods are intuitive to prove the existence of order-1 periodic solution and homoclinic bifurcation.

A state feedback impulse model for computer worm control

Computer worm is a worldwide threat to the safety of Internet, which caused billions of dollars in damages over the past decade. Software patches have been widely used as one of approaches to protect computers against computer worms. In this study, an impulsive state feedback model was employed to study the transmission of computer worm and the preventive effect of operating system patching. The existence of order-1 periodic solution and its stability were proved with a novel method. The results demonstrated that the application of software patches is an effective approach to constrain the deluge of computer worm. Numerical simulation results were presented to support the theoretical analysis.

Dynamics Analysis and Biomass Productivity Optimisation of a Microbial Cultivation Process through Substrate Regulation

A microbial cultivation process model with variable biomass yield, control of substrate concentration, and biomass recycle is formulated, where the biochemical kinetics follows an extension of the Monod and Contois models. Control of substrate concentration allows for indirect monitoring of biomass and dissolved oxygen concentrations and consequently obtaining high yield and productivity of biomass. Dynamics analysis of the proposed model is carried out and the existence of order-1 periodic solution is deduced with a formulation of the period, which provides a theoretical possibility to convert the state-dependent control to a periodic one while keeping the dynamics unchanged. Moreover, the stability of the order-1 periodic solution is verified by a geometric method. The stability ensures a certain robustness of the adopted control; that is, even with an inaccurately detected substrate concentration or a deviation, the system will be always stable at the order-1 periodic solution under the control. The simulations are carried out to complement the theoretical results and optimisation of the biomass productivity is presented.

Qualitative analysis of a korean pine forest model with impulsive thinning measure

A korean pine forest model with impulsive thinning measure is presented by using impulsive state feedback system to investigate the periodicity of the regeneration process of the forest. Based on the qualitative properties of the corresponding continuous system, the existences of order-1 periodic solutions are discussed. If the positive equilibrium of the continuous system is globally stable, then the impulsive state feedback system has an order-1 periodic solution and no order-k(k⩾2) periodic solution. The conditions for the orbitally asymptotical stability of order-1 periodic solution are given and discussed by the analogue of the Poincaré criterion. For the case that the continuous system has a stable limit cycle, the existence of order-1 periodic solution of the impulsive state feedback system are also discussed, the results show that there are three kinds of order-1 periodic solutions. Finally, the mathematical results are verified by the numerical simulations. Moreover, the numerical results show that the impulsive state feedback system has order k(k⩾1) periodic solutions in the interior of the limit cycle of the continuous system for some parameters.

Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting

In this article, we investigate a prey– predator model with Allee effect and state-dependent impulsive harvesting. We obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (1.2) by means of the geometry theory of semicontinuous dynamic system and the method of successor function. We also obtain that system (1.2) exhibits the phenomenon of heteroclinic bifurcation about parameter \(\alpha \). The methods used in this article are novel and prove the existence of order-1 periodic solution and heteroclinic bifurcation.

Bifurcations and Periodic Solutions for an Algae-Fish Semicontinuous System

We propose an algae-fish semicontinuous system for the Zeya Reservoir to study the control of algae, including biological and chemical controls. The bifurcation and periodic solutions of the system were studied using a Poincaré map and a geometric method. The existence of order-1 periodic solution of the system is discussed. Based on previous analysis, we investigated the change in the location of the order-1 periodic solution with variable parameters and we described the transcritical bifurcation of the system. Finally, we provided a series of numerical results to illustrate the feasibility of the theoretical results. These results may help to facilitate a better understanding of algal control in the Zeya Reservoir.

HOMOCLINIC BIFURCATION IN SEMI-CONTINUOUS DYNAMIC SYSTEMS

In this paper, a class of homoclinic bifurcations in semi-continuous dynamic systems are investigated. On the basis of rotated vector fields theory, existence of order-1 periodic solution and the rotated vector fields of the semi-continuous dynamic system are discussed. Furthermore, homoclinic cycles and homoclinic bifurcations are described. Finally, an example is provided to show the validity of our theoretical results.