Optimal control for a size-structured predator-prey model in a polluted environment

. In this paper, we deal with an optimal harvesting problem for a periodic predator-prey hybrid system dependent on size-structure in a polluted environment. In other words, a size-dependent model in an environment with a small toxicant content has been established. The well-posedness of state system is proved by using the fixed point theorem. The necessary optimality conditions are derived by tangent-normal cone technique in nonlinear functional analysis. The existence of a unique optimal harvesting policy is verified via the Ekeland’s variational principle. The optimal harvesting problem has an optimal harvesting policy, which has a Bang-Bang structure and provides a threshold for the optimal harvesting problem. Using the optimization theories and methods in mathematics to control phenomena of life. The objective function represents the total economic profit from the harvested population. Some theoretical results obtained in this paper provide a scientific theoretical basis for the practical application of the model.


Introduction
With the rapid development of the global economy, many new technologies have been applied to industrial and agricultural production activities. The invasion of toxicants is very easy to happen, and environmental pollution has become more and more serious. Bush fires in Australia, SARS, Ebola virus, AIV, H1N1 influenza, and COVID-19, etc., these phenomena are threatening the ecological balance and causing serious harm to the survival of human beings and other creatures. Therefore, it is necessary to study the effects of toxicants on biological populations. Hallam et al. proposed an idea of using dynamics methods to study ecotoxicology in the literature [1][2][3]. They established a toxicants-population model and studied the persistence and extinction of a population in a polluted environment. Since the 1980s, people have conducted in-depth research on the topic of ecotoxicology. Nowadays, there is a large number of literature on ecotoxicology problems [4][5][6], but sizestructured factors are not considered in these models.
For many populations, the individual size determines the life parameters of the individual to a large extent, such as reproductive rate, mortality, metabolic capacity and predation capacity, etc., thereby affecting the dynamic behavior of the populations. In addition, the individual size-structure can explain some phenomena that other structures cannot explain, such as the selfthinning of plants and the elastic growth of individuals. Furthermore, the individual size has good operability, such as easy measurement, convenient scientific management of ecological resources, and easy access to size-based statistical data. It is more convenient in practical applications. This kind of models has achieved remarkable results through theory, numerical calculation and experimental methods, which can be found in the literature [7][8][9][10]. Due to the influence of seasonal changes and other factors, the living environment of the populations often experience periodic changes. For the research on the optimal harvest problems dependent on individual size-structured models in a periodic environment can be referred to [11][12][13], in which the literature [11] discussed harvesting problem for nonlinear size-dependent population model in periodic environments. At present, only a few papers have focused on the optimal control problems of population models with size-structure and periodic effects in a polluted environment. Inspired by the above, this paper discusses the optimal harvesting problem of a periodic predator-prey system dependent on size-structure in a polluted environment.
The remaining part of this paper is as follows. The problem is described and the main methods of proof are given in the next section. The well-posedness is proved, the optimality conditions for the harvesting problem are derived, the existence of a unique optimal policy is obtained and the problem is discussed in Sect. 3. Finally, we give a short conclusion in the last section.
Next, we will prove the existence and uniqueness of non-negative solution of the toxicant-population model by the fixed point theorem, deduct the optimality conditions of the optimal harvesting problem by employing tangent-normal cone techniques, and obtain the existence of optimal harvesting policy by the Ekeland's variational principle.

Well-posedness of the state system
In this section, we discuss the well-posedness of state system (1), and first give the following definition.  [14] and satisfies: hold, then the state system (1) has unique non-negative solution that satisfies the following conditions: where By Bellman's lemma, we get that We define a mapping: Similarly, we have     Next, we will prove that F is a contracting operator.
Let   where Similarly, we have where    Similarly, where   We define an equivalent norm in space X by The proof process of lemma 3.1.3 is similar to that of Theorem 3.1.2, which is omitted here.

Optimality conditions
In this section, we consider the adjoint system of (1) and establish first-order necessary conditions for optimal harvesting of (2).  , , , , z z z z z is the solution of the following system be an optimal policy for the optimal harvesting problem of (1)- (2). Then       , , ,  , , , , q q q q q is the solution of the following adjoint system (12):

Existence of optimal policy
In this section, the existence of optimal policy will be established. First, define the embedding mapping  