  We propose a time-dependent Advection Reaction Diffusion (ARD) $N$-species
competition model to investigate the Stocking and Harvesting (SH) effect on
population dynamics. For ongoing analysis, we explore the outcomes of a
competition between two competing species in a heterogeneous environment under
no-flux boundary conditions, meaning no individual can cross the boundaries. We
establish results concerning the existence, uniqueness, and positivity of the
solution. As a continuation, we propose, analyze, and test two novel fully
discrete decoupled linearized algorithms for a nonlinearly coupled ARD
$N$-species competition model with SH effort. The time-stepping algorithms are
first and second order accurate in time and optimally accurate in space.
Stability and optimal convergence theorems of the decoupled schemes are proved
rigorously. We verify the predicted convergence rates of our analysis and the
efficacy of the algorithms using numerical experiments and synthetic data for
analytical test problems. We also study the effect of harvesting or stocking
and diffusion parameters on the evolution of species population density
numerically and observe the coexistence scenario subject to optimal stocking or
harvesting.
