On models of shared resource competition, coexistence and traveling waves

Wei Feng; Xin Lu; Karen Ward; Wei Feng; Xin Lu; Karen Ward

doi:10.3934/mbe.2026002

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 1: 22-39. doi: 10.3934/mbe.2026002

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Research article

On models of shared resource competition, coexistence and traveling waves

Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, NC 28403-5970, USA

Received: 16 September 2025 Revised: 01 November 2025 Accepted: 19 November 2025 Published: 24 November 2025

We investigate a two-species competition model in which both populations exploit a common standing resource. The dynamics are governed by a system of nonlinear differential equations admitting three equilibrium classes: extinction, competitive exclusion, and coexistence. Analytical conditions on the biological parameters ensuring the existence and asymptotic stability of these equilibria are derived, with particular emphasis on the coexistence equilibrium, representing the stable persistence of both species under shared-resource competition. In the corresponding reaction-diffusion model posed on an unbounded spatial domain, we further examine the stability of the coexistence equilibrium via traveling wavefronts. Using the upper–lower solution method, we establish the existence of traveling wave solutions connecting the extinction or single-dominance states to the coexistence state for a continuum of wave speeds exceeding a biologically determined minimal value, which depends explicitly on equilibrium magnitudes and other key parameters. Numerical simulations are provided to corroborate the theoretical results and to illustrate dynamic transitions from dominance to stable coexistence.
- reaction-diffusion systems,
- competition with shared resource,
- asymptotic stability,
- traveling wave flowing to coexistence state
Citation: Wei Feng, Xin Lu, Karen Ward. On models of shared resource competition, coexistence and traveling waves[J]. Mathematical Biosciences and Engineering, 2026, 23(1): 22-39. doi: 10.3934/mbe.2026002

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Abstract

We investigate a two-species competition model in which both populations exploit a common standing resource. The dynamics are governed by a system of nonlinear differential equations admitting three equilibrium classes: extinction, competitive exclusion, and coexistence. Analytical conditions on the biological parameters ensuring the existence and asymptotic stability of these equilibria are derived, with particular emphasis on the coexistence equilibrium, representing the stable persistence of both species under shared-resource competition. In the corresponding reaction-diffusion model posed on an unbounded spatial domain, we further examine the stability of the coexistence equilibrium via traveling wavefronts. Using the upper–lower solution method, we establish the existence of traveling wave solutions connecting the extinction or single-dominance states to the coexistence state for a continuum of wave speeds exceeding a biologically determined minimal value, which depends explicitly on equilibrium magnitudes and other key parameters. Numerical simulations are provided to corroborate the theoretical results and to illustrate dynamic transitions from dominance to stable coexistence.

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