Geometric indicators of population persistence in branching continuous-space networks.

We study population persistence in branching tree networks emulating systems such as river basins, cave systems, organisms on vegetation surfaces, and vascular networks. Population dynamics are modeled using a reaction-diffusion-advection equation on a metric graph which provides a continuous, spatially explicit model of network habitat. A metric graph, in contrast to a standard graph, allows for population dynamics to occur within edges rather than just at graph vertices, subsequently adding a significant level of realism. Within this framework, we stochastically generate branching tree networks with a variety of geometric features and explore the effects of network geometry on the persistence of a population which advects toward a lethal outflow boundary. We identify a metric (CM), the distance from the lethal outflow point at which half of the habitable volume of the network lies upstream of, as a promising indicator of population persistence. This metric outperforms other metrics such as the maximum and minimum distances from the lethal outflow to an upstream boundary and the total habitable volume of the network. The strength of CM as a predictor of persistence suggests that it is a proper "system length" for the branching networks we examine here that generalizes the concept of habitat length in the classical linear space models.

Population persistence in river networks.

Organisms inhabiting river systems contend with downstream biased flow in a complex tree-like network. Differential equation models are often used to study population persistence, thus suggesting resolutions of the 'drift paradox', by considering the dependence of persistence on such variables as advection rate, dispersal characteristics, and domain size. Most previous models that explicitly considered network geometry artificially discretized river habitat into distinct patches. With the recent exception of Ramirez (J Math Biol 65:919-942, 2012), partial differential equation models have largely ignored the global geometry of river systems and the effects of tributary junctions by using intervals to describe the spatial domain. Taking advantage of recent developments in the analysis of eigenvalue problems on quantum graphs, we use a reaction-diffusion-advection equation on a metric tree graph to analyze persistence of a single population in terms of dispersal parameters and network geometry. The metric graph represents a continuous network where edges represent actual domain rather than connections among patches. Here, network geometry usually has a significant impact on persistence, and occasionally leads to dramatically altered predictions. This work ranges over such themes as model definition, reduction to a diffusion equation with the associated model features, numerical and analytical studies in radially symmetric geometries, and theoretical results for general domains. Notable in the model assumptions is that the zero-flux interior junction conditions are not restricted to conservation of hydrological discharge.