Stocking and Harvesting Effects in Advection-Reaction-Diffusion Model: Exploring Decoupled Algorithms and Analysis

TL;DR

This paper introduces and analyzes novel decoupled algorithms for a complex advection-reaction-diffusion model to study the impact of stocking and harvesting on multi-species population dynamics, supported by rigorous proofs and numerical validation.

Contribution

It develops and rigorously analyzes two new fully discrete decoupled algorithms for a nonlinear ARD N-species competition model with SH effects, including stability and convergence proofs.

Findings

Algorithms achieve first and second order accuracy in time

Numerical experiments confirm theoretical convergence rates

Stocking and harvesting significantly influence species coexistence

Abstract

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.