The Emergence of Range Limits in Advective Environments

In this paper, we study the asymptotic profile of the steady state of a reaction-diffusion-advection model in ecology proposed in [E. Pachepsky et al., Theoret. Popul. Biol., 67 (2005), pp. 61--73; D. Speirs and W. Gurney, Ecology, 82 (2001), pp. 1219--1237]. The model describes the population dynamics of a single species experiencing a unidirectional flow. We show the existence of one or more internal transition layers and determine their locations. Such locations can be understood as the upstream invasion limits of the species. It turns out that these invasion limits are connected to the upstream spreading speed of the species and are sometimes subject to the effect of migration from upstream source patches.

[1]  M A Lewis,et al.  Persistence, spread and the drift paradox. , 2005, Theoretical population biology.

[2]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[3]  William Gurney,et al.  POPULATION PERSISTENCE IN RIVERS AND ESTUARIES , 2001 .

[4]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[5]  P. Hess,et al.  3.—A Criterion for the Existence of Solutions of Non-linear Elliptic Boundary Value Problems , 1976, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Hans F. Weinberger,et al.  Long-Time Behavior of a Class of Biological Models , 1982 .

[7]  G. Minshall,et al.  The River Continuum Concept , 1980 .

[8]  Mark L. Taper,et al.  Theoretical models of species' borders: single species approaches , 2005 .

[9]  Yuan Lou,et al.  Evolution of dispersal in open advective environments , 2013, Journal of Mathematical Biology.

[10]  Judith L. Li,et al.  From continua to patches: examining stream community structure over large environmental gradients , 2002 .

[11]  Chris Cosner,et al.  Diffusive logistic equations with indefinite weights: population models in disrupted environments II , 1991 .

[12]  Yihong Du,et al.  Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol. 1: Maximum Principles and Applications , 2006 .

[13]  G. Minshall,et al.  Regional patterns in periphyton accrual and diatom assemblage structure in a heterogeneous nutrient landscape , 2002 .

[14]  Frithjof Lutscher,et al.  Spatial patterns and coexistence mechanisms in systems with unidirectional flow. , 2007, Theoretical population biology.

[15]  Frithjof Lutscher,et al.  Effects of Heterogeneity on Spread and Persistence in Rivers , 2006, Bulletin of mathematical biology.

[16]  Yuan Lou,et al.  Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions , 2015 .

[17]  F. Lutscher,et al.  POPULATION DYNAMICS IN RIVERS: ANALYSIS OF STEADY STATES , 2022 .

[18]  Yihong Du,et al.  Spreading speed revisited: Analysis of a free boundary model , 2012, Networks Heterog. Media.

[19]  P. Hess,et al.  Periodic-Parabolic Boundary Value Problems and Positivity , 1991 .