Population dynamics in river networks: analysis of steady states
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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.
KeywordsRiver network Reaction–diffusion–advection Persistence Steady state
Mathematics Subject Classification35K57 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.