[Paper Review] Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system

Hélène Leman, Sylvie Méléard+1 more|arXiv (Cornell University)|Jan 6, 2014

Mathematical and Theoretical Epidemiology and Ecology Models7 references3 citations

TL;DR

This paper studies a nonlocal competitive Lotka-Volterra system with spatial diffusion and nonlocal competition, modeling populations structured by type and space. It establishes existence of steady states and proves that long-time behavior depends critically on competition rates and principal eigenvalues, showing that one type may dominate or coexistence may occur depending on parameter thresholds.

ABSTRACT

To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.

Motivation & Objective

• To understand how spatial structure and nonlocal competition affect long-term population dynamics in structured populations.
• To extend results from type-structured models to systems with both phenotypic and spatial structure.
• To analyze the existence and stability of steady states in a two-type (monomorphic/dimorphic) population model with diffusion and integral competition terms.
• To determine under what conditions one population type dominates or coexistence occurs over time.
• To investigate the impact of mutant invasion on spatial niche evolution through long-time behavior of the system.

Proposed method

• Models a dimorphic population using a system of two coupled parabolic PDEs with diffusion (m₁, m₂), spatially dependent growth rates (a₁(x), a₂(x)), and nonlocal competition via integral terms (I_ij).
• Imposes Neumann boundary conditions to model no-flux population movement at the domain boundary ∂X.
• Analyzes long-time behavior by studying the asymptotic limits of solutions using energy estimates and spectral analysis of associated linear operators.
• Applies the Hartman-Grobman theorem to assess local stability of steady states by linearizing the system around equilibrium points.
• Uses subspace reduction techniques by restricting initial conditions to eigenspaces of the principal eigenfunctions to simplify dynamics to a 2D Lotka-Volterra system.
• Employs L² and L∞ convergence results to establish convergence of solutions to steady states under specific parameter regimes.

Experimental results

Research questions

• RQ1 Under what conditions does a single population type dominate the other in the long-time limit?
• RQ2 How do the principal eigenvalues of the competition operators influence the stability of steady states?
• RQ3 Can the introduction of a new mutant type lead to long-term spatial niche evolution?
• RQ4 What determines whether coexistence of two types occurs or one type is excluded?
• RQ5 How does the interplay between nonlocal competition and spatial diffusion affect the asymptotic behavior of the system?

Key findings

• The system admits four non-negative steady states: (0,0), (ḡ₁,0), (0,ḡ₂), and (ḡ₁,ḡ₂), with (ḡ₁,0) and (0,ḡ₂) being monomorphic steady states.
• The steady state (ḡ₁,0) is globally asymptotically stable if H₁ > 0 and H₂μ₁₁ − H₁μ₂₁ < 0, meaning type 1 dominates when its intrinsic growth and competition parameters satisfy this inequality.
• The steady state (0,ḡ₂) is unstable if H₁ > 0 and H₂μ₁₁ − H₁μ₂₁ < 0, but some solutions still converge to it, indicating conditional stability.
• When H₁μ₂₂ − H₂μ₁₂ = 0 and H₂μ₁₁ − H₁μ₂₁ < 0, the system exhibits a mix of convergence to (ḡ₁,0) and non-convergence near (0,ḡ₂), confirming instability of the latter.
• The long-time limit of the solution is (ḡ₁,0) when ∫|g₂(t,x)|²dx → 0 and ∫|g₁(t,x)|²dx remains bounded away from zero, which occurs under the stated parameter conditions.
• Numerical illustrations confirm that mutant invasion can drive long-term evolution of spatial niches, highlighting the role of spatial structure in evolutionary dynamics.