This paper investigated the optimal development problem of a size-structured population model under a periodic environmental setting. The boundary condition of the novel model consists of a nonlinear recruitment process and a bounded input, which endow the model with more realistic and complex characteristics compared to traditional ones. First, we established the existence of a unique non-negative bounded solution and demonstrated the continuous dependence of the solutions on the control variable. Next, we showed that the adjoint system is also well-posed. Then, the Euler-Lagrange equations describing the exact structure of the optimal strategies were derived and the existence of a unique optimal policy was proved. Finally, some numerical results were presented. The obtained research results will contribute to the development of some renewable resources, such as fish resources.

This paper discusses the optimal contraception control problem for vermin. The novel model consists of a first-order partial differential equation for the age-dependent density of vermin and two ordinary differential equations for the amounts of female sterilant in the environment and in an individual. We first show that the hybrid system is well-posed by applying the fixed-point theorem. Then the structure of an optimal contraception policy is established by considering the normal cone and adjoint system. Moreover, there is a unique optimal policy by employing Ekeland's variational principle and fixed-point theory. The optimal policy that we have derived offers a rational deployment strategy for the use of sterilants as a means of efficacious pest control. These criteria guarantee that during the application of sterilants, the predetermined objectives are attained while simultaneously minimizing expenditure and environmental implications. Utilizing these optimality criteria facilitates the development of streamlined and economically viable pest management protocols.

This paper focused on a stochastic giving-up-smoking model with harmonic mean-type incidence rate, in which the population was divided into four types. Firstly, we showed that the model has a unique global positive solution. Then, stochastic permanence of the model was discussed, which means that the population described by the model will not grow wildly or disappear. Next, sufficient conditions for the elimination of smokers (including occasional smokers, chain smokers, and quit smokers) were established. Additionally, sufficient conditions for the existence of an ergodic stationary distribution were derived, meaning that all types of smokers can be persistent. Moreover, we discussed how to control the size of the smoker population from the perspective of economics. Finally, some numerical simulations were introduced.