Stationary solutions and spatial-temporal dynamics of a shadow system of LV competition models

Yuanyuan Zhang,Li Xia

doi:10.1186/s13662-017-1081-x

Abstract

The concern of the paper is nonconstant positive solutions of a class of Lotka-Volterra competition systems over 1D domains. We prove the existence of a positive monotonous solution to the shadow system for each small diffusion rate epsilon >0. Our theoretical results provide a foundation for further theoretical analysis on the shadow system and give insights on how diffusion and advection rates affect the pattern formation in the advective Lotka-Volterra competition systems. The second part of this paper includes numerical simulations of the nontrivial patterns to the shadow system and its original model. It is demonstrated that nontrivial patterns can develop from small perturbations of the homogeneous solution. Our numerics suggest that this system admits very interesting and complicated spatial-temporal dynamics even over 1D domains.

Highlights

This paper is concerned with the following boundary value problem with integral constraint: ⎧⎪⎪⎨ v + (a – b λ e–r (v ) – c v )v =, x ∈ (, L),⎪⎪⎩v L((xa) >– b, xλ∈e(– r, (v ) L); c v v )e–r ( ) =v dx (L) = = ( . )
We investigate the existence of a nonconstant positive solution to system ( . )
The global bifurcation is important especially when nonconstant positive solutions are concerned, and our results provide a foundation for further analysis on the shadow system ( . ), compared to the local branches, which have been investigated in detail in [ ]

Summary

Introduction

This paper is concerned with the following boundary value problem with integral constraint:. Where is a positive constant, v is a function of x, and λ is an unknown constant. Parameters ai, bi, ci, i = , , and r are positive constants, and is a smooth function. ) is the D shadow system to the following model:. X∈ , x∈∂ , Zhang and Xia Advances in Difference Equations (2017) 2017:25 which was proposed in [ ] to study the aggregation phenomenon of two competing species subject to Lotka-Volterra kinetics. See [ ] for the derivation of this model and biological justifications for the system parameters. To demonstrate our motivation for the study of ), we first recall the following results on Let (ui, vi) be positive solutions of with (d ,i, d ,i, χi).

Suppose that χi

If vk

Then we readily see that w satisfies