Optimizing Impulsive Releases: A Species Competition Model

This study focuses on optimizing species release $S_2$ to control species population $S_1$ through impulsive release strategies. We investigate the conditions required to remove species $S_1$, which is equivalent to the establishment of $S_2$. The research includes a theoretical analysis that examines the positivity, existence, and uniqueness of solutions, the conditions ensuring global stability, and a sufficient condition for controlling the $S_1$-free solution. In addition, we formulate an optimal control problem to maximize the effectiveness of $S_2$ releases, manage the population of $S_1$, and minimize the costs associated with this intervention strategy. Numerical simulations are conducted to validate the proposed theories and allow visualization of population dynamics under various release scenarios.