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

TL;DR

This work develops a time-dependent ARD model for $N$-species competition with stocking and harvesting in heterogeneous environments under no-flux boundaries, establishing existence, uniqueness, and positivity for $N=1$ and $2$. It introduces two fully discrete decoupled linearized finite element schemes, BE and DBDF-2, proving stability and convergence with first- and second-order time accuracy and optimal spatial accuracy. Numerical experiments with manufactured solutions verify convergence rates and illustrate how advection, diffusion, and SH shape coexistence, extinction, and spatial patterns. The results offer a computationally efficient decoupled framework for SH-driven population dynamics and point to future work on traveling waves and enhanced FE implementations.

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.

Figures (11)

• Figure 5.1: Average density of each species: (a) short-, and (b) long-range with the advection coefficients $\beta_1=0.001$, and $\beta_2=0.01$, intrinsic growth rates $r_i=1$, and harvesting coefficients $\gamma_i=0.0$, for $i=\overline{1,2}$.
• Figure 5.2: Average density of each species: (a) short-, and (b) long-range with the advection coefficients $\beta_1=0.001$, and $\beta_2=0.01$, intrinsic growth rates $r_i=1$, harvesting coefficients $\gamma_i=0.0$, for $i=\overline{1,2}$, and stationary carrying capacity.
• Figure 5.3: Average density of each species: (a) short-, and (b) long-range with the harvesting coefficients $\gamma_1=0.001$, and $\gamma_2=0.01$ for fixed and equal growth rate and advection coefficients.
• Figure 5.4: Average density of each species: (a) short-, and (b) long-range with the harvesting coefficients $\gamma_1=0.001$, and $\gamma_2=0.0$ (no harvesting) for fixed and equal growth rate and advection coefficients.
• Figure 5.5: Average density of each species: (a) short-, and (b) long-range with the harvesting coefficients $\gamma_1=0.001$, and $\gamma_2=0.0$, intrinsic growth rates $r_i=1$, advection coefficients $\beta_i=0.0$, for $i=\overline{1,2}$, and stationary carrying capacity.

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Theorem 1
• proof