Global dynamics of a two-species competition patch model in a Y-shaped river network

TL;DR

This work analyzes a two-species competition patch model on a Y-shaped river network, where identical growth is modulated by different random and directed movements. The authors prove a nonexistence result for positive equilibria within a region defined by $S=\mathcal{S}_1\cup\mathcal{S}_2$, and, using principal eigenvalue analysis and monotone dynamical systems, establish global competition exclusion: the semi-trivial equilibrium corresponding to the species with favorable movement is globally stable depending on the region ($S_1$ or $S_2$). A key conclusion is that when random dispersal rates are equal, the species with the smaller drift rate is driven to extinction, highlighting drift-rate selection in branched river networks. The results rely on auxiliary sign-estimates of stationary balance terms and a careful case analysis across the river branches, providing a rigorous link between network topology, movement strategies, and persistence outcomes.

Abstract

In this paper, we investigate a two-species Lotka-Volterra competition patch model in a Y-shaped river network, where the two species are assumed to be identical except for their random and directed movements. We show that competition exclusion can occur under certain conditions, i.e., one of the semi-trivial equilibrium is globally asymptotically stable. Especially, if the random dispersal rates of the two species are equal, the species with a smaller drift rate will drive the other species to extinction, which suggests that small drift rates are favored.

Figures (8)

• Figure 1: A stream with $n$ patches, where patch $n$ is the upstream end, and patch $1$ is the downstream end. Here $d$ is the random movement rate and $q$ is the directed drift rate for one species.
• Figure 2: A Y-shaped river network 45-Vasilyeva-2019.
• Figure 3: A Y-shaped river network. Here $m_1=2$, $m_2=3$, $m_3=3$, and $d$ and $q$ are the dispersal rate and the drift rate of one species, respectively.
• Figure 4: Illustration of $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$.
• Figure 5: The graphs of $\tilde{f}^1_1$, $\tilde{f}^1_2$ and $\tilde{f}^1_3$ with respect to $q$. Here $d=200$ and $r=3$.

Theorems & Definitions (30)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Remark 3.5
• Lemma 3.6
• proof