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The effect of "very fast" dispersal on two species competition with drift

Erin Ellefsen, Rana Parshad, Vaibhava Srivastava

TL;DR

The paper addresses how a faster dispersal mechanism, modeled by the $p$-Laplacian, competes with a slower species under downstream drift. It develops a degenerate two-species reaction-diffusion model combining regular diffusion and $p$-Laplacian diffusion, proves global existence of weak solutions for $ rac{3}{2} < p < 2$ via regularization and compactness arguments, and identifies regimes where the faster diffuser may lose to the slower one, contrasting with the classical $p=2$ results. Numerical simulations corroborate the theory, revealing rich dynamics, including initial-data-dependent outcomes and partial-diffusion scenarios. The findings have ecological and applied relevance for biodiversity, refuge design, and biological control in fragmented or changing habitats.

Abstract

Classical theory predicts that for two competing populations subject to a constant downstream drift, the faster disperser will competitively exclude the slower disperser. In the current work, we consider a novel model of a "much faster" dispersing species, modeled via a $p$-Laplacian operator, competing with a slower disperser. We prove global existence of weak solutions to this model for any positive initial condition, in the regime $\frac{3}{2} < p <2$. Counterintuitively, we show that while the faster disperser always wins - the "much faster" disperser could actually lose, for certain initial data. Several numerical simulations are conducted to confirm our analytical findings. Our results have implications for biodiversity, refuge design, and improved biological control, driven by habitat fragmentation and climate change.

The effect of "very fast" dispersal on two species competition with drift

TL;DR

The paper addresses how a faster dispersal mechanism, modeled by the -Laplacian, competes with a slower species under downstream drift. It develops a degenerate two-species reaction-diffusion model combining regular diffusion and -Laplacian diffusion, proves global existence of weak solutions for via regularization and compactness arguments, and identifies regimes where the faster diffuser may lose to the slower one, contrasting with the classical results. Numerical simulations corroborate the theory, revealing rich dynamics, including initial-data-dependent outcomes and partial-diffusion scenarios. The findings have ecological and applied relevance for biodiversity, refuge design, and biological control in fragmented or changing habitats.

Abstract

Classical theory predicts that for two competing populations subject to a constant downstream drift, the faster disperser will competitively exclude the slower disperser. In the current work, we consider a novel model of a "much faster" dispersing species, modeled via a -Laplacian operator, competing with a slower disperser. We prove global existence of weak solutions to this model for any positive initial condition, in the regime . Counterintuitively, we show that while the faster disperser always wins - the "much faster" disperser could actually lose, for certain initial data. Several numerical simulations are conducted to confirm our analytical findings. Our results have implications for biodiversity, refuge design, and improved biological control, driven by habitat fragmentation and climate change.

Paper Structure

This paper contains 14 sections, 21 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model formulation
  3. Movement Operator
  4. The model with a "much faster" diffuser
  5. The regularized system and preliminary estimates
  6. Alikakos-Moser iteration
  7. Global Existence of Weak Solutions
  8. Convergence of Non-linear term
  9. The faster diffuser "loses"
  10. The case: $\frac{3}{2} < p <2$
  11. The case: $\frac{4}{3} < p <\frac{3}{2}$
  12. Numerical Results
  13. Discussion and Conclusion
  14. Functional Preliminaries

Key Result

Lemma 2.1

Consider the system eq:pde_model. Then $\nabla a(x,v,\nabla v) \cdot \eta \iff \nabla v \cdot \eta = 0$.

Figures (4)

  • Figure 1: Solutions to system \ref{['eq:pde_modeld2']} with $q=0.5$, $d_1 = 0.2$, and $d_2 = 0.3$.
  • Figure 2: Solutions to system \ref{['eq:pde_modeld2-k']} with and $p=7/4$, $q=0.5$, $d_1 = 0.2$, and $d_2 = 0.3$, and $k=3/4$.
  • Figure 3: Solutions to equation \ref{['eq:pde_modeld2']}, with two species following $p-$Laplacian diffusion, $q=0.5$, $p=7/4$
  • Figure 4: Solutions to equation \ref{['eq:pde_modeld2']}, with two species following $p-$Laplacian diffusion, $q=0.5$, $d_1 = d_2 = 0.3$.

Theorems & Definitions (39)

  • Remark 1
  • Lemma 2.1
  • Theorem 2.2: Slower disperser wins
  • Definition 2.3: Weak solution
  • Theorem 2.4: Faster disperser wins
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Remark 2
  • Lemma 2.7
  • ...and 29 more