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.

翻译：我们提出一个时间依赖的对流-反应-扩散（ARD）$N$物种竞争模型，以研究放养与捕捞（SH）效应对种群动态的影响。在持续分析中，我们探讨了无通量边界条件（即无个体可跨越边界）下，异质环境中两个竞争物种的竞争结果。我们建立了关于解的存在性、唯一性和正性的结论。作为后续研究，我们针对带有SH作用的非线性耦合ARD $N$物种竞争模型，提出、分析并测试了两种新型全离散解耦线性化算法。这些时间步进算法在时间上具有一阶和二阶精度，在空间上具有最优精度。我们严格证明了解耦格式的稳定性和最优收敛定理。通过数值实验和合成数据，我们验证了分析所预测的收敛速率及算法的有效性。我们还数值研究了捕捞或放养以及扩散参数对物种种群密度演化的影响，并观察了在最优放养或捕捞条件下的共存情景。