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Optimal Harvesting in Stream Networks: Maximizing Biomass and Yield

Tung D. Nguyen, Zhisheng Shuai, Tingting Tang, Amy Veprauskas, Yixiang Wu, Ying Zhou

TL;DR

This work addresses optimal harvesting in stream networks by formulating a metapopulation model with logistic-type dynamics across $n$ patches connected by biased movement and a fixed total harvesting budget. It develops complete results for a two-patch system and asymptotic, network-wide insights for large growth rates, introducing an effective net-flow metric $I_i$ to rank patches and determine within-patch allocations. The main contributions show that maximizing biomass tends to concentrate effort on patches with high intraspecific competition, while maximizing yield concentrates on patches with low competition, with the optimal distribution within the selected subset guided by $I_i$; in homogeneous networks, a clear downstream/upstream dichotomy emerges depending on growth, movement, and budget parameters. The findings offer principled, spatially explicit guidance for harvest planning in fragmented river networks and highlight how transport asymmetries interact with density dependence to shape optimal control strategies.

Abstract

In this study, we develop a metapopulation model framework to identify optimal harvesting strategies for a population in a stream network. We consider two distinct optimization objectives: maximization of total biomass and maximization of total yield, under the constraint of a fixed total harvesting effort. We examine in detail the special case of a two-patch network and fully characterize the optimal strategies for each objective. We show that when the population growth rate exceeds a critical threshold, a single harvesting strategy can simultaneously maximize both objectives. For general $n$-patch networks with homogeneous growth rates across patches, we focus on the regime of large growth rates and demonstrate that the optimal harvesting strategy selects patches according to their intraspecific competition rates and an effective net flow metric determined by network connectivity parameters.

Optimal Harvesting in Stream Networks: Maximizing Biomass and Yield

TL;DR

This work addresses optimal harvesting in stream networks by formulating a metapopulation model with logistic-type dynamics across patches connected by biased movement and a fixed total harvesting budget. It develops complete results for a two-patch system and asymptotic, network-wide insights for large growth rates, introducing an effective net-flow metric to rank patches and determine within-patch allocations. The main contributions show that maximizing biomass tends to concentrate effort on patches with high intraspecific competition, while maximizing yield concentrates on patches with low competition, with the optimal distribution within the selected subset guided by ; in homogeneous networks, a clear downstream/upstream dichotomy emerges depending on growth, movement, and budget parameters. The findings offer principled, spatially explicit guidance for harvest planning in fragmented river networks and highlight how transport asymmetries interact with density dependence to shape optimal control strategies.

Abstract

In this study, we develop a metapopulation model framework to identify optimal harvesting strategies for a population in a stream network. We consider two distinct optimization objectives: maximization of total biomass and maximization of total yield, under the constraint of a fixed total harvesting effort. We examine in detail the special case of a two-patch network and fully characterize the optimal strategies for each objective. We show that when the population growth rate exceeds a critical threshold, a single harvesting strategy can simultaneously maximize both objectives. For general -patch networks with homogeneous growth rates across patches, we focus on the regime of large growth rates and demonstrate that the optimal harvesting strategy selects patches according to their intraspecific competition rates and an effective net flow metric determined by network connectivity parameters.
Paper Structure (14 sections, 21 theorems, 93 equations, 4 figures)

This paper contains 14 sections, 21 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model formulation and optimization problems
  3. Constrained harvesting in a two-patch system with biased movement
  4. Maximizing the total biomass when the patches are heterogeneous
  5. Maximizing the total biomass when the patches are homogeneous
  6. Maximizing the yield when the patches are homogeneous
  7. Constrained harvesting in $n-$patch systems
  8. Maximizing the total biomass
  9. Concentrating on the patches with the highest intraspecies competition rate
  10. Harvesting effort distribution among the patches with the highest intraspecies competition rate
  11. Maximizing the yield
  12. Concentrating on the patches with the smallest intraspecies competition rate
  13. Harvesting effort distribution among the patches with the smallest intraspecies competition rate
  14. Discussion

Key Result

Lemma 3.1

Suppose the intrinsic growth rates $r_1$ and $r_2$ satisfy Then there exists a unique positive equilibrium $\bm u^*$ which is asymptotically stable.

Figures (4)

  • Figure 1: In these two figures, $r_{tie}=2d+q+\frac{H}{2}$, $d = 1$ and $H=4$, the four critical curves $r_{crit}$, $r_m,r_M,r_{tie}$ in Figure \ref{['fig:left_figure_B']} separate the region to areas where the strategies for maximizing biomass are different. Four specific points are chosen with $q/H=1.75$ and different $r$ values. Figure \ref{['fig:right_figure_B']} shows the total biomass against the harvest parameter $\theta$ at these four points with $c=1$ and $q=7$.
  • Figure 2: In these two figures, $d = 1$ and $H=4$, four specific points are chosen with $q/H=0.75$ and different $r$ values. Figure \ref{['fig:right_Y']} shows the yield against the harvest parameter $\theta$ at these four points with $c=1$ and $q=3$.
  • Figure 3: A stream network with structure 3–1–1 (three patches on top, one in the middle, one below).
  • Figure 4: For a stream network with structure 3--1--1 (three patches on top, one in the middle, one below), the figure compares stabilized total biomass across five harvesting strategies as the intrinsic growth rate $r$ increases.

Theorems & Definitions (41)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Corollary 3.6
  • Lemma 3.7
  • ...and 31 more