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Pulse injection of insulin analogues is an important strategy to control glucose concentrations and can be combined with α$$ \alpha $$‐glucosidase inhibitors to reduce adverse effects to improve glucose control. To elucidate this combination therapy strategy, we assumed dietary intake in the form of pulse glucose injection and proposed a novel mathematical model incorporating pulse injection insulin and α$$ \alpha $$‐glucosidase inhibitors. In type 1 diabetes, the existence and uniqueness of the positive periodic solution is confirmed utilizing the Lambert W function. The global asymptotic stability of the positive periodic solution is achieved through the application of Floquet multiplier theory and the comparison principle. Furthermore, in type 2 diabetes, the permanence of the system is also confirmed through the comparison theorem. Numerical analysis validated the theoretical calculations, highlighting the significance of insulin injection dosage and frequency, as well as α$$ \alpha $$‐glucosidase inhibitor therapy. Additionally, we systematically assessed a rational approach for diabetes treatment combined with α$$ \alpha $$‐glucosidase inhibitors, providing more practical clinical strategies. This facilitates the extension of our model to encompass other drugs that may be utilized in future clinical interventions. [ABSTRACT FROM AUTHOR]

Due to resource limitation, nonlinear impulsive control tactics related to integrated pest management have been proposed in a generalized pest-natural enemy model, which allows us to address the effects of nonlinear pulse control on the dynamics and successful pest control. The threshold conditions for the existence and global stability of pest-free periodic solution are provided by Floquet theorem and analytic methods. The existence of a nontrivial periodic solution is confirmed by showing the existence of nontrivial fixed point of the stroboscopic mapping determined by time snapshot, which equals to the common impulsive period. In order to address the applications of generalized results and to reveal how the nonlinear impulses affect the successful pest control, as an example the model with Holling II functional response function is investigated carefully. The main results reveal that the pest free periodic solution and a stable interior positive periodic solution can coexist for a wide range of parameters, which indicates that the local stability does not imply the global stability of the pest free periodic solution when nonlinear impulsive control is considered, and consequently the resource limitation (i.e. nonlinear control) may result in difficulties for successful pest control.

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey (λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey (a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed. [ABSTRACT FROM AUTHOR]