Research On Biological Mathematical Model With State - Dependent Feedback Control Strategy

The state feedback control strategy (SFCS) has been widely applied in many sciences including fields of ecology, life science and medicine, which can be defined as the control strategy only been implemented once a state of the system reaches a prescribed threshold level. Therefore, it is very important to establish and inves-tigate the corresponding mathematical models for this ubiquitous threshold strat-egy. Moreover, the state dependent impulsive differential equations or impulsive semi-dynamical systems can be used to provide a natural descriptions and charac-terizations for numerous of the SFCS. Recently, significant progress has been made in the theories of impulsive semi-dynamical system, which enriched its applications. Based on the economic threshold of integrated pest control, the threshold poten-tial of neurons, the blood glucose concentration and the threshold of tumor cells, we propose and develop several mathematical models including pest-natural enemy model, quadratic integrate-and-fire neuron model, glucose-insulin monitor model and tumour-immune dynamic model concerning SFCS, then provides comprehen-sively qualitative analysis and reveals important biological conclusions of the models by using the theories of impulsive semi-dynamical systems and novel analysis tech-niques.For pest-natural enemy model and quadratic integrate-and-fire neuron model with SFCS, the domains of impulsive and phase sets under different cases are defined according to the different positions among the threshold levels, reset values and the equilibria, and then the Poincare map is constructed in the phase sets. By using the analogue of the Poincare criterion, the threshold conditions for the existence and stability of the semi-trivial periodic solution are provided and subsequently the transcritical bifurcation near the semi-trivial periodic solution is discussed in detail. Further, the different parameter spaces for the existence and stability of an order-1 periodic solution are investigated with the help of the definitions and properties of the Poincare map. In addition, the multistability and the existence of order-k(k≥2) periodic solutions have also been studied theoretically. In pest con-trol, the results show that the pest population can be maintained below a prescribed threshold level without predators, and the pest and natural enemy populations can persist and oscillate periodically with the maximal amplitude of pest population less than the given threshold when the natural enemies are introduced. The main results suggest that the dose of pesticide and number of natural enemies or the fre-quency of pesticide applications and natural enemies released should be increased when resources are limited in order to control the density of pest population below the threshold. In neuron system, the numerical simulations are carried out to sub-stantiate our results, which reveal that the system has rich dynamical behaviors. It clarifies that the parameters should be chosen carefully in order to control the number of spikes per burst as regular spiking corresponds to an order-1 periodic solution and bursting corresponds to an order-k(k≥2) periodic solutions, which indicates that the analytical methods can be used as the basis for understanding dynamical behaviors of neuron models.For glucose-insulin monitor model and tumour-immune dynamic model with SFCS, we consider two cases based on the different frequencies of medication:adop-tive cellular immunotherapy (ACI) is applied more frequent than inputs of interleukin-2 (IL-2), and inputs of IL-2 are more frequent than applications of ACI. By using the theories of impulsive differential equations, the permanence and global stability of the positive periodic solution for glucose-insulin regulatory system are studied, and then the conditions for the existence and stability of the tumour-free periodic solution are provided. The numerical results reveal that the times, the dose and the frequency of glucose infusions and insulin injections are crucial for insulin ther-apies, and it suggests that insulin injections before glucose infusions is proved to be very effective for controlling the blood glucose level. Besides, the effects of period, dosage and times of drug deliveries on the amplitudes of the tumour-free periodic solution are discussed in detail. Numerical studies show that the different treatment strategies may have a great influence on outcomes. In particular, the applications of ACI alone can lead to the tumour eradication and the use of only IL-2 cannot result in a tumour-free body; when ACI treatment and inputs of IL-2 are applied at the same time, the results of such a combination would not be different from that of treating with ACI alone; it is concluded that the immunotherapy of case 1 represents a better way to administer cancer immunotherapy. For the model with SFCS, the results suggest that the maximum number at which the blood glucose concentration reaches the threshold and its relatively duration largely depend on the injection dose of insulin, glucose infusion rate and glucose infusion period, and the blood glucose level never reaches or exceeds threshold. Meanwhile, it indicates that the initial densities, the effector cell:tumour cell ratios, the periods and a given critical number of tumour cells are crucial for the applications of successful chemotherapy control strategies.