The role of harvesting and growth rate for spatially heterogeneous populations

This paper investigates the competition of two species in a heterogeneous environment subject to the effect of harvesting. The most realistic harvesting case is connected with the intrinsic growth rate, and the harvesting functions are developed based on this clause instead of random choice. We prove the existence and uniqueness of the solution to the model we consider. Theoretically, we state that when species coexist, one may drive the other to die out, and both species extinct, considering all possible rational values of parameters. These results highlight a comparative study between two harvesting coefficients. Finally, we solve the model using a backward-Euler, decoupled, and linearized time-stepping fully discrete algorithm and observe a match between the theoretical and numerical findings.