Modeling the relationship between domain size and population persistence in branching river networks

Background/Question/Methods

Ecologists often model the relationship between river length and population persistence by treating rivers as a one dimensional domain.  However, changing the length of a one dimensional domain ignores that, at larger scales, a river system's geometry resembles a branching network rather than a line. Furthermore, habitable cross sectional areas may vary throughout the system.  Our question is how and when increasing habitat in a branching river network affects persistence differently than increasing the length of habitat in linear system.  We model population dynamics using reaction-diffusion-advection equations on a metric graph, which represents a continuous branching system where edges represent actual domain rather than simple connections among nodes.  Linearized growth rates around the zero steady state are used to identify when the population will grow at low density. Via principal eigenvalue analysis, we link growths rates to habitat length and model parameters.  Habitat is added to the river network by adding branching levels, with various assumptions on boundary conditions and on how cross section and branch length decay going upstream in the system.

Results/Conclusions

Our results indicate that the persistence in the river network and one dimensional habitat show the same scaling relationship with model parameters when cross-sectional areas are reduced by half at each upstream branching level. In other cases, increasing habitat in the one dimensional case can either underestimate or overestimate ranges of parameter values allowing for persistence.  For example, when there is no advection and all boundaries are absorbing (lethal), increasing the size of the network may not notably increase persistence. This is in contrast to the corresponding one dimensional problem where persistence is often guaranteed by increasing habitat length. We show that most conditions produce positive relationships between river network size and population persistence.  However, increasing habitat length at certain branching levels may provide only marginal benefits in terms of persistence given realistic constraints on the number of branching levels.  Our modeling framework provides a general way to explore ecological dynamics in river networks and other habitats with branching structure.