Scale-dependent erosional patterns in steady-state and transient-state landscapes

The role of lateral erosion in the evolution of nondendritic drainage networks to dendricity and the persistence of dynamic networks

Dendritic, i.e., tree-like, river networks are ubiquitous features on Earth's landscapes; however, how and why river networks organize themselves into this form are incompletely understood. A branching pattern has been argued to be an optimal state. Therefore, we should expect models of river evolution to drastically reorganize (suboptimal) purely nondendritic networks into (more optimal) dendritic networks. To date, current physically based models of river basin evolution are incapable of achieving this result without substantial allogenic forcing. Here, we present a model that does indeed accomplish massive drainage reorganization. The key feature in our model is basin-wide lateral incision of bedrock channels. The addition of this submodel allows for channels to laterally migrate, which generates river capture events and drainage migration. An important factor in the model that dictates the rate and frequency of drainage network reorganization is the ratio of two parameters, the lateral and vertical rock erodibility constants. In addition, our model is unique from others because its simulations approach a dynamic steady state. At a dynamic steady state, drainage networks persistently reorganize instead of approaching a stable configuration. Our model results suggest that lateral bedrock incision processes can drive major drainage reorganization and explain apparent long-lived transience in landscapes on Earth.

Information differences across spatial resolutions and scales for disease surveillance and analysis: The case of Visceral Leishmaniasis in Brazil

Nationwide disease surveillance at a high spatial resolution is desired for many infectious diseases, including Visceral Leishmaniasis. Statistical and mathematical models using data collected from surveillance activities often use a spatial resolution and scale either constrained by data availability or chosen arbitrarily. Sensitivity of model results to the choice of spatial resolution and scale is not, however, frequently evaluated. This study aims to determine if the choice of spatial resolution and scale are likely to impact statistical and mathematical analyses. Visceral Leishmaniasis in Brazil is used as a case study. Probabilistic characteristics of disease incidence, representing a likely outcome in a model, are compared across spatial resolutions and scales. Best fitting distributions were fit to annual incidence from 2004 to 2014 by municipality and by state. Best fits were defined as the distribution family and parameterization minimizing the sum of absolute error, evaluated through a simulated annealing algorithm. Gamma and Poisson distributions provided best fits for incidence, both among individual states and nationwide. Comparisons of distributions using Kullback-Leibler divergence shows that incidence by state and by municipality do not follow distributions that provide equivalent information. Few states with Gamma distributed incidence follow a distribution closely resembling that for national incidence. These results demonstrate empirically how choice of spatial resolution and scale can impact mathematical and statistical models.

Emergent spectral properties of river network topology: an optimal channel network approach

Characterization of river drainage networks has been a subject of research for many years. However, most previous studies have been limited to quantities which are loosely connected to the topological properties of these networks. In this work, through a graph-theoretic formulation of drainage river networks, we investigate the eigenvalue spectra of their adjacency matrix. First, we introduce a graph theory model for river networks and explore the properties of the network through its adjacency matrix. Next, we show that the eigenvalue spectra of such complex networks follow distinct patterns and exhibit striking features including a spectral gap in which no eigenvalue exists as well as a finite number of zero eigenvalues. We show that such spectral features are closely related to the branching topology of the associated river networks. In this regard, we find an empirical relation for the spectral gap and nullity in terms of the energy dissipation exponent of the drainage networks. In addition, the eigenvalue distribution is found to follow a finite-width probability density function with certain skewness which is related to the drainage pattern. Our results are based on optimal channel network simulations and validated through examples obtained from physical experiments on landscape evolution. These results suggest the potential of the spectral graph techniques in characterizing and modeling river networks.