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In this paper, we propose an epidemic model of SIR type with ratio-dependent impulse control and Beddington–DeAngelis (B–D) incidence. According to the magnitude of the basic reproductive number ℛ 0 and the relation of the endemic equilibrium (S ∗ , I ∗) and the ratio threshold h , dynamical analysis of the controlled system is conducted. Under the control strategy, if ℛ 0 ≤ 1 , the solutions converge to the disease-free equilibrium. If ℛ 0 > 1 and I ∗ S ∗ > h , the impulsive system has periodic solution that is orbitally asymptotically stable, and order- k   (k > 2) periodic solution does not exist. Furthermore, if ℛ 0 > 1 and I ∗ S ∗ < h , the solution converges either to the endemic equilibrium or to a periodic solution, which is proved to be determined by the initial value. Finally, numerical simulations are performed to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

To describe non-continuous control strategy triggered by a predefined ratio of the concerned population in disease progression, an SIR epidemic model with non-monotone incidence is established, dealing with the integrated interventions of vaccination and isolation as ratio-dependent impulsive control. Based on the analysis of multiple equilibria and stability for the continuous subsystem, complex property of the impulsive controlled system is conducted theoretically and numerically. It is found that the ratio magnitude as well as the initial value plays an important role in the threshold dynamics, determining the trajectory convergence to the order-k(k < 3 ) periodic solution, the endemic equilibria or the disease free equilibrium under certain conditions. Moreover, numerical examples are provided to demonstrate the theoretical results, and in particular, the case of converging to an order-2 periodic solution is presented in simulation, revealing the influence of the ratio threshold control on disease transmission. [ABSTRACT FROM AUTHOR]

Research on agricultural pest control data indicates that the actual effectiveness of insecticides may exhibit slight variations, influenced by factors such as the type and quantity of pests. Additionally, the development of resistance by pests, as a result of adaptation to the environment, can impact the precision of pest control strategies. Hence we suggest implementing a nonlinear pulse state-dependent feedback control system, considering resource limitations, to investigate the influence of varying pesticide fatality rates on pest outbreaks. This paper takes into account the fear effect and anti-predator behavior to accurately portray the farmland ecosystem. We assessed the phase set and Poincaré map under the conditions for the existence of order-k (where k = 1 , 2 , 3 ) periodic solutions and examined their stability behavior. Remarkably, the system exhibited diverse bifurcation phenomena associated with pesticide-related parameters. Furthermore, the system demonstrated a multistability phenomenon involving the coexistence of the order-1 periodic solution and the limit cycle. This suggests that altering control strategies can disturb the initial coexistence status of pests and natural enemies. It also underscores that resource limitations can indeed impact the outbreak patterns and frequencies of pests. In addition to the variability in pesticide fatality rates, the inclusion of natural enemy releases in the nonlinear pulse strategy contributes to the model exhibiting complex dynamic characteristics. All these findings are substantiated by numerical simulations. [ABSTRACT FROM AUTHOR]

A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist. [ABSTRACT FROM AUTHOR]

A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist.

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k (k≥2) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

As the disease spreads, it will inevitably cause important damage to the life and health of the population, resulting in changes in the population quantity. In addition, in some economically underdeveloped areas, limited medical resources will also have an important impact on the prevention and control of diseases. Based on these, a susceptible-infected-recovered-susceptible (SIRS) epidemic model is established, where state-dependent pulse control strategy, varying total population and limited medical resources are introduced. By using the qualitative theory of ordinary differential equation, differential inequality techniques, Poincaré map, and other methods, some sufficient conditions of the existence and orbital asymptotical stability of positive order-1 or order-2 periodic solution are obtained in various situations. Theoretical results imply that the proportion of infected class can be controlled at a desired low level for a long time and disease will not break out among population. Finally, based on realistic parameters of brucellosis in ruminants, numerical simulations have been performed to expalin/extend our analytical results and the feasibility of the state-dependent feedback control strategy. [ABSTRACT FROM AUTHOR]

In this paper, a pest management predator–prey model with weak Allee effect on predator and state feedback impulsive control on prey is introduced and analysed, where the yield of predator released and intensity of pesticide sprayed are assumed to be linearly dependent on the selected pest control level. For the proposed model, the existence and stability of the order-1 periodic orbit of the control system are discussed. Meanwhile, with the aim of minimizing the input cost in practice, an optimization model is constructed to determine the optimal quantity of the predator released and the intensity of pesticide sprayed. The theoretical results and numerical simulations indicated that the number of pests can be limited to below an economic threshold and displays periodic variation under the proposed control strategy. In addition, it indicated in numerical simulations that an order-2 periodic orbit exists for some certain parameters.

In this paper, a pest management predator–prey model with weak Allee effect on predator and state feedback impulsive control on prey is introduced and analysed, where the yield of predator released and intensity of pesticide sprayed are assumed to be linearly dependent on the selected pest control level. For the proposed model, the existence and stability of the order-1 periodic orbit of the control system are discussed. Meanwhile, with the aim of minimizing the input cost in practice, an optimization model is constructed to determine the optimal quantity of the predator released and the intensity of pesticide sprayed. The theoretical results and numerical simulations indicated that the number of pests can be limited to below an economic threshold and displays periodic variation under the proposed control strategy. In addition, it indicated in numerical simulations that an order-2 periodic orbit exists for some certain parameters. [ABSTRACT FROM AUTHOR]

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model's global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order- k (k ≥ 2) periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed. [ABSTRACT FROM AUTHOR]

In recent decades, West Nile virus (WNV) has become a substantial public health concern in many subtropical and tropical countries throughout the world. WNV is considered significant effects on host populations, but to date some effects have been difficult to measure. Based on a simple and easy operating statistical method, a simpler model describing the transmission of WNV between mosquitoes and birds is proposed. By the linearization method and Bendixson–Dulac theorem of differential equation, we obtain the basic reproductive number R 0 for this disease, which illustrates the global asymptotical stability of the disease-free equilibrium and endemic equilibrium. Further, in order to control the spread of WNV, the model is extended to a control model, where state-dependent pulse control strategies are introduced. By the Poincaré map, differential inequality techniques and qualitative theory of ordinary differential equation, the existence and orbital stability of order-1 or order-2 periodic solutions for this control model are obtained. Finally, numerical simulations clearly illustrate the main theoretical results and feasibility of state-dependent pulse control strategies, and also discuss the roles of control parameters in the course of WNV control. Theoretical results and numerical simulation also show that the quantity of infected birds can be kept within a lower level. This provides a new control strategy to against the spread of WNV. [ABSTRACT FROM AUTHOR]

A planar predator–prey impacting system model with a nonmonotonic functional response function is proposed and analyzed. The existence and stability of a boundary order-1 periodic solution were investigated and the threshold conditions for a transcritical bifurcation and stable switching were obtained, and also the definition and properties of the Poincaré map are discussed. The main results indicate that multiple discontinuous points of the Poincaré map could induce the coexistence of multiple order-1 periodic solutions. Numerical analyses reveal the complex dynamics of the model including periodic adding and halving bifurcations, which could result in multiple active phases, among them rapid spiking and quiescence phases which can switch from one to another and consequently create complex bursting patterns. The main results reveal that it is beneficial to restore the stability and balance of a ecosystem for species with group defence by moderately reducing population densities and the group defence capacity. [ABSTRACT FROM AUTHOR]

With the consideration of mechanism of prevention and control for the spread of infectious diseases, we propose, in this paper, a state dependent pulse vaccination and medication control strategy for a SIRS type epidemic dynamic system. The sufficient conditions on the existence and orbital stability of positive order-1 or order-2 periodic solution are presented. Numerical simulations are carried out to illustrate the main results and compare numerically the state dependent vaccination strategy and the fixed time pulse vaccination strategy. [ABSTRACT FROM AUTHOR]