This paper investigated impulsive insulin therapy within an extended glucose-insulin regulatory system that included insulin-degrading enzyme dynamics. Two treatment modes were examined: periodic delivery and state-dependent feedback. For the periodic model, we established solution positivity and boundedness, proved the global asymptotic stability of a unique positive periodic solution for type I diabetes, and derived permanence bounds along with a safe dosing region for type II diabetes. For the state-dependent model, under mild conditions, we demonstrated the existence and global orbital asymptotic stability of a unique order-1 periodic orbit within a physiologically relevant glucose range. Numerical simulations supported the theoretical results and illustrated how the choice of dosing period, bolus size, and threshold level influenced long-term glucose control. The results provided a unified theoretical framework for comparing and optimizing impulsive insulin therapies.

This paper proposes and studies a switched interactive model of wild and sterile mosquitoes with stage and sex structure. Sterile males are released periodically and impulsively and remain sexually active for time $\overline{T}$. We investigate the dynamical behavior of the system when the release period $T$ is shorter than the sexual lifespan $\overline{T}$, corresponding to a relatively frequent release. We first determine two important thresholds, $m_1^{\ast }$ and $m_2^{\ast }$, for the release amount $m$ and prove the exponential asymptotic stability of the extinction equilibrium. Using fixed point theory, we establish the existence of positive periodic solutions for $0<m<m_1^{\ast }$ and $m_1^{\ast }\le m<m_2^{\ast }$. Furthermore, by applying the comparison theorem of monotone systems, we demonstrate that the extinction equilibrium is globally asymptotically stable when $m\ge m_2^{\ast }$. Finally, numerical examples are presented to confirm our theoretical results.

This paper focuses on the key issues of mosquito population control, particularly exploring the impact of periodic releases of sterile males in the population model with a stage structure. We construct and analyze a model that includes only sexually active sterile mosquitoes in the dynamic interaction system. We focus on the system's dynamical behaviors under two scenarios: when the sexual lifespan $\overline{T}$ equals the release period $T$ of sterile mosquitoes, and when $\overline{T}$ is less than $T$. In the first scenario, we explore the existence and stability of equilibria, identifying a pivotal threshold $m^{\ast }$ that determines the requisite release amount. In the second scenario, we convert the problem into an impulsive switched system and derive sufficient conditions for the local asymptotic stability of the extinction equilibrium. We also establish the existence of positive periodic solutions using the geometric method of differential equations and the fixed point theorem. Our conclusions show that the relationship between the sexual lifespan and release period of sterile mosquitoes significantly impacts the stability of the mosquito population. Additionally, our numerical simulations not only corroborate but they also complement our theoretical findings.

The current research presents a predator-prey model that incorporates both a Gilpin-Ayala growth function and a Holling type Ⅲ functional response. Two Lyapunov functions are established to confirm the global asymptotic stability of the positive equilibrium $P^{\ast }$ and the predator extinction equilibrium $P_k$. Considering ecological protection and commercial incentives, we also incorporated a weighted harvesting strategy and pulse control into the model. We investigated intricate dynamical problems instigated by the weighting harvesting and pulse effects, and affirmed the existence and local asymptotic stability of both predator-extinction periodic solution and positive order-1 periodic solution. In the end, a suite of numerical simulations were carried out using MATLAB, aiming to corroborate the theoretical findings and deliver conclusions rooted in a biological context.