Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment

De Tang

doi:10.3934/dcdsb.2019037

Abstract

We consider a two-species Lotka-Volterra weak competition model in a one-dimensional advective homogeneous environment, where individuals are exposed to unidirectional flow. It is assumed that two species have the same population dynamics but different diffusion rates, advection rates and intensities of competition. We study the following useful scenarios: (1) if one species disperses by random diffusion only and the other assumes both random and unidirectional movements, two species will coexist; (2) if two species are drifting along the different direction, two species will coexist; (3) if the intensities of inter-specific competition are small enough, two species will coexist; (4) if the intensities of inter-specific competition are close to 1, the competitive exclusion principle holds. These results provide a new mechanism for the coexistence of competing species. Finally, we apply a perturbation argument to illustrate that two species will converge to the unique coexistence steady state.

Highlights

Logistic equations have long been used as models in population dynamics and their use dates back at least to the work of Verhulst: ut = u(r − u), t > 0, (1)where u(t) denotes the total population of a species and r is a positive constant representing the carrying capacity of the environment
When considering two aquatic species which are competing for the same resources in the water column, we turn to the following Lotka-Volterra competition model including advection forces:
Taking the intensity of competition into consideration, we study two aquatic species which are competing for the same resources in the water column, as described by the following Lotka-Volterra weak competition model including advection forces:

Summary

Introduction

The authors [29] study a classical two species LotkaVolterra competition-diffusion-advection system (which includes weak competition) and present a complete classification on all possible long-time dynamical behaviours under the special conditions. The authors [12] study the dynamics of system (6) when α1 = α2 = 0 and r = r(x) is not a constant Their results suggest that system (6) has a unique coexistence steady state (U ∗, V ∗) which is globally asymptotically stable when d1 and d2 are close enough. Theorem 1.4 implies that for any fixed diffusion rates and advection rates, if the intensities of inter-specific competition are both small, two species will coexist . Remark 5. (i) By some standard discussion (See, e.g. [3, 9, 12]), we know that θd,α(x) → r in L∞(0, L) as d → ∞ or |α| → 0. (ii) System (6) has two semi-trivial steady states (θd1,α1 , 0) and (0, θd2,α2 )

We present an estimate about

Setting U

This shows that

Then there exist a neighborhood