Global dynamics of controlled competitive diffusion in heterogeneous environments

Xinyu Wang; Lipu Zhang; Xinyuan Jin; Yao Tong; Xinyu Wang; Lipu Zhang; Xinyuan Jin; Yao Tong

doi:10.3934/era.2026044

Electronic Research Archive

2026, Volume 34, Issue 2: 938-961. doi: 10.3934/era.2026044

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Research article

Global dynamics of controlled competitive diffusion in heterogeneous environments

1.
School of Media Engineering, Communication University of Zhejiang, Hangzhou 310018, China
2.
Key Lab of Film and TV Media Technology of Zhejiang Province, Hangzhou 310018, China

Received: 10 December 2025 Revised: 13 January 2026 Accepted: 20 January 2026 Published: 29 January 2026

We establish global well-posedness for a two-species competitive reaction-diffusion system in bounded two-dimensional domains under Neumann boundary conditions. The system incorporates spatially heterogeneous growth continuous over the closed domain and time-dependent bounded asymmetric intervention. A sharp threshold condition relating the target species' promotion strength to its natural decay rate is necessary and sufficient to maintain the carrying-capacity constraint globally in time, with the two species' total density not exceeding 1. Meanwhile, the unit square with both species' densities ranging between 0 and 1 remains positively invariant, regardless of the target species' promotion strength. Solutions are classical and unique, uniformly bounded, and Lipschitz-continuous with respect to initial data. Numerical simulations show that asymptotically applied interventions that satisfy the threshold condition result in a near-complete eradication of the competitor species at full efficiency. The promoted species establishes a heterogeneous spatial distribution shaped by growth heterogeneity. This framework extends classical diffusive Lotka-Volterra models and provides rigorous theoretical support for ecological management and misinformation suppression strategies.
- reaction-diffusion system,
- asymptotic intervention,
- global well-posedness,
- two-component dynamics,
- heterogeneous propagation efficiency
Citation: Xinyu Wang, Lipu Zhang, Xinyuan Jin, Yao Tong. Global dynamics of controlled competitive diffusion in heterogeneous environments[J]. Electronic Research Archive, 2026, 34(2): 938-961. doi: 10.3934/era.2026044

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Abstract

We establish global well-posedness for a two-species competitive reaction-diffusion system in bounded two-dimensional domains under Neumann boundary conditions. The system incorporates spatially heterogeneous growth continuous over the closed domain and time-dependent bounded asymmetric intervention. A sharp threshold condition relating the target species' promotion strength to its natural decay rate is necessary and sufficient to maintain the carrying-capacity constraint globally in time, with the two species' total density not exceeding 1. Meanwhile, the unit square with both species' densities ranging between 0 and 1 remains positively invariant, regardless of the target species' promotion strength. Solutions are classical and unique, uniformly bounded, and Lipschitz-continuous with respect to initial data. Numerical simulations show that asymptotically applied interventions that satisfy the threshold condition result in a near-complete eradication of the competitor species at full efficiency. The promoted species establishes a heterogeneous spatial distribution shaped by growth heterogeneity. This framework extends classical diffusive Lotka-Volterra models and provides rigorous theoretical support for ecological management and misinformation suppression strategies.

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