| Abstract: |
| \begin{abstract}
A variety of population models are analyzed via optimal control theory, some of these models, which are determined ordinary differential equations, represent population dynamics. One of them is species augmentation, which is a very common model, is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. The model of species augmentation is formulated with two coupled first-order ordinary differential equations. The purpose of this work, by using optimal control characterization of this model is to investigate corresponding four coupled first-order ordinary differential equations to Pontyagin`s maximum principle. The first, via the Hamiltonian of the model of species augmentation, first-order conditions of maximum principle are determined, then we focus on aforementioned a system of four coupled first-order ordinary differential equations. The second, we give our attention application of some related analytic methods, which are Lie symmetry, $\lambda$-symmetry and Jacobi last multiplier to this four-coupled system. Also, the results of the extended Prelle-Singer method for coupled systems are evaluated for the corresponding system of species augmentation.
\end{abstract} |
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