Population dynamics in river networks: analysis of steady states

  • Olga VasilyevaEmail author


We study the population dynamics of an aquatic species in a river network. The habitat is viewed as a binary tree-like metric graph with the population density satisfying a reaction–diffusion–advection equation on each edge, along with the appropriate junction and boundary conditions. In the case of a linear reaction term and hostile downstream boundary condition, the question of persistence in such models was studied by Sarhad, Carlson and Anderson. We focus on the case of a nonlinear (logistic) reaction term and use an outflow downstream boundary condition. We obtain necessary and sufficient conditions for the existence and uniqueness of a positive steady state solution for a simple Y-shaped river network (with a single junction). We show that the existence of a positive steady state is equivalent to the persistence condition for the linearized model. The method can be generalized to a binary tree-like river network with an arbitrary number of segments.


River network Reaction–diffusion–advection Persistence Steady state 

Mathematics Subject Classification

35K57 92B05 35R02 



The author would like to thank Frithjof Lutscher and Mark Lewis for valuable discussions, and two anonymous referees for helpful feedback. The author was supported by NSERC Discovery Grant.


  1. Fagan WF (2002) Connectivity, fragmentation, and extinction risk in dendritic metapopulations. Ecology 83:3243–3249CrossRefGoogle Scholar
  2. Guysinsky M, Hasselblatt B, Rayskin V (2003) Differentiability of the Hartman–Grobman linearization. Discrete Contin Dyn Syst 9(4):979–984MathSciNetCrossRefzbMATHGoogle Scholar
  3. Kolokolnikov T, Ou C, Yuan Y (2009) Profiles of self-shading, sinking phytoplankton with finite depth. J Math Biol 59(1):105–122MathSciNetCrossRefzbMATHGoogle Scholar
  4. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  5. Lam KY, Lou Y, Lutscher F (2015) Evolution of dispersal in closed advective environments. J Biol Dyn 9(sup 1):188–212MathSciNetCrossRefGoogle Scholar
  6. Lam KY, Lou Y, Lutscher F (2016) The emergence of range limits in advective environments. SIAM J Appl Math 76(2):641–662MathSciNetCrossRefzbMATHGoogle Scholar
  7. Lou Y, Lutscher F (2013) Evolution of dispersal in open advective environments. J Math Biol 69:1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  8. Lou Y, Zhou P (2015) Evolution of dispersal in advective homogeneous environment: the effect of boundary conditions. J Differ Equ 259(1):141–171MathSciNetCrossRefzbMATHGoogle Scholar
  9. Lou Y, Zhao X-Q, Zhou P (2019) Global dynamics of a Lotka–Volterra competition–diffusion–advection system in heterogeneous environments. J Math Pures Appl 121:47–82MathSciNetCrossRefzbMATHGoogle Scholar
  10. Lutscher F, Pachepsky E, Lewis MA (2005) The effect of dispersal patterns on stream populations. SIAM Rev 47(4):749–772MathSciNetCrossRefzbMATHGoogle Scholar
  11. Lutscher F, Nisbet R, Pachepsky E (2010) Population persistence in the face of advection. Theor Ecol 3:271–284CrossRefGoogle Scholar
  12. McKenzie HW, Jin Y, Jacobsen J, Lewis MA (2012) \(R_0\) Analysis of of spatiotemporal model for a stream population. SIAM J Appl Dyn Syst 11(2):567–596MathSciNetCrossRefzbMATHGoogle Scholar
  13. Pachepsky E, Lutscher F, Nisbet R, Lewis MA (2005) Persistence, spread and the drift paradox. Theor Popul Biol 67:61–73CrossRefzbMATHGoogle Scholar
  14. Perko L (2000) Differential equations and dynamical systems. Springer, BerlinzbMATHGoogle Scholar
  15. Ramirez JM (2012) Population persistence under advection–diffusion in river networks. J Math Biol 65(5):919–942MathSciNetCrossRefzbMATHGoogle Scholar
  16. Sarhad J, Carlson R, Anderson KE (2014) Population persistence in river networks. J Math Biol 69(2):401–448MathSciNetCrossRefzbMATHGoogle Scholar
  17. Speirs DC, Gurney WSC (2001) Population persistence in rivers and estuaries. Ecology 82(5):1219–1237CrossRefGoogle Scholar
  18. Vasilyeva O, Lutscher F (2010) Population dynamics in rivers: analysis of steady states. Can Appl Math Q 18(4):439–469MathSciNetzbMATHGoogle Scholar
  19. Zhou P, Xiao D (2018) Global dynamics of a classical Lotka–Volterra competition–diffusion–advection system. J Funct Anal 275:356–380MathSciNetCrossRefzbMATHGoogle Scholar
  20. Zhou P, Zhao X-Q (2018) Evolution of passive movement in advective environment: general boundary condition. J Differ Equ 264:4176–4198MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Grenfell Campus, Memorial University of NewfoundlandCorner BrookCanada

Personalised recommendations