Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equations Quasimonotone systems of parabolic equations A competition model Appendix Bibliography.

Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith

On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity

On a Limiting System in the Lotka-Volterra Competition with Cross-Diffusion (非線形拡散系とその周辺 研究集会報告集)

On Lotka-Volterra competitive parabolic systems: Exclusion, coexistence and bistability

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.

Bifurcation and stability of a two-species reaction–diffusion–advection competition model

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

https://www.aimsciences.org/article/exportPdf?id=c0015032-617e-4be8-8cf0-83a806ebc6dc

Evolution of dispersal in advective homogeneous environments

Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments

In this paper, we consider a single species model and a two-species competition model arising in river ecology. An interesting feature of these models lies in the boundary conditions, where there may appear a net loss of individuals from the habitat ends, as measured by two parameters b u and b d . We first determine various sufficient conditions for the persistence of a single species, in terms of the critical habitat size and critical advection rate. Then we turn to study a competition system, where two populations are supposed to differ only in their advection speeds. By using the theory of monotone dynamical systems and principal spectrum, we achieve a complete understanding on the global dynamics, which suggests that weaker advective movement is more beneficial for species to win the competition, extending several existing work (Lou et al., 2016; Xu and Gan, 2019; Zhou and Zhao, 2018).

De Tang

Papers

On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity

On Lotka-Volterra competitive parabolic systems: Exclusion, coexistence and bistability

Evolution of dispersal in advective homogeneous environments

Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments

Population dynamics and evolution in river ecosystems