Experience

Publications

• Population persistence in river networks

  Journal of Mathematical Biology

  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.

• Population persistence in river networks

  Journal of Mathematical Biology

  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.

• The nonlinear Poisson equation via a Newton-embedding procedure

  Complex Variables and Elliptic Equations

  This article considers the semilinear boundary value problem given by the Poisson equation, Du1⁄4f(u) in a bounded domain Rn (n42) with a smooth boundary. For the zero boundary value case, we approximate a solution using the Newton-embedding procedure. With the assumptions that f, f 0, and f 00 are bounded functions on R, with f 050, and R3, the Newton-embedding procedure yields a continuous solution. This study is in response to an independent work which applies the same procedure, but assuming that f 0 maps the Sobolev space H1( ) to the space of Ho ̈ lder continuous functions C ð Þ, and f(u), f0(u), and f00(u) have uniform bounds. In the first part of this article, we prove that these assumptions force f to be a constant function. In the remainder of the article, we prove the existence, uniqueness, and H2-regularity in the linear elliptic problem given by each iteration of Newton’s method. We then use the regularity estimate to achieve convergence.

• Population persistence in river networks

  Journal of Mathematical Biology

  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.

• The nonlinear Poisson equation via a Newton-embedding procedure

  Complex Variables and Elliptic Equations

  This article considers the semilinear boundary value problem given by the Poisson equation, Du1⁄4f(u) in a bounded domain Rn (n42) with a smooth boundary. For the zero boundary value case, we approximate a solution using the Newton-embedding procedure. With the assumptions that f, f 0, and f 00 are bounded functions on R, with f 050, and R3, the Newton-embedding procedure yields a continuous solution. This study is in response to an independent work which applies the same procedure, but assuming that f 0 maps the Sobolev space H1( ) to the space of Ho ̈ lder continuous functions C ð Þ, and f(u), f0(u), and f00(u) have uniform bounds. In the first part of this article, we prove that these assumptions force f to be a constant function. In the remainder of the article, we prove the existence, uniqueness, and H2-regularity in the linear elliptic problem given by each iteration of Newton’s method. We then use the regularity estimate to achieve convergence.

• Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets

  Journal of Noncommutive Geometry

  We construct Dirac operators and spectral triples for certain, not necessarily self- similar, fractal sets built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics. As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami’s geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.

• Population persistence in river networks

  Journal of Mathematical Biology

  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.

• The nonlinear Poisson equation via a Newton-embedding procedure

  Complex Variables and Elliptic Equations

  This article considers the semilinear boundary value problem given by the Poisson equation, Du1⁄4f(u) in a bounded domain Rn (n42) with a smooth boundary. For the zero boundary value case, we approximate a solution using the Newton-embedding procedure. With the assumptions that f, f 0, and f 00 are bounded functions on R, with f 050, and R3, the Newton-embedding procedure yields a continuous solution. This study is in response to an independent work which applies the same procedure, but assuming that f 0 maps the Sobolev space H1( ) to the space of Ho ̈ lder continuous functions C ð Þ, and f(u), f0(u), and f00(u) have uniform bounds. In the first part of this article, we prove that these assumptions force f to be a constant function. In the remainder of the article, we prove the existence, uniqueness, and H2-regularity in the linear elliptic problem given by each iteration of Newton’s method. We then use the regularity estimate to achieve convergence.

• Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets

  Journal of Noncommutive Geometry

  We construct Dirac operators and spectral triples for certain, not necessarily self- similar, fractal sets built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics. As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami’s geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.

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

  Journal of Mathematical Biology

  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

  Journal of Mathematical Biology

  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.

• The nonlinear Poisson equation via a Newton-embedding procedure

  Complex Variables and Elliptic Equations

  This article considers the semilinear boundary value problem given by the Poisson equation, Du1⁄4f(u) in a bounded domain Rn (n42) with a smooth boundary. For the zero boundary value case, we approximate a solution using the Newton-embedding procedure. With the assumptions that f, f 0, and f 00 are bounded functions on R, with f 050, and R3, the Newton-embedding procedure yields a continuous solution. This study is in response to an independent work which applies the same procedure, but assuming that f 0 maps the Sobolev space H1( ) to the space of Ho ̈ lder continuous functions C ð Þ, and f(u), f0(u), and f00(u) have uniform bounds. In the first part of this article, we prove that these assumptions force f to be a constant function. In the remainder of the article, we prove the existence, uniqueness, and H2-regularity in the linear elliptic problem given by each iteration of Newton’s method. We then use the regularity estimate to achieve convergence.

• Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets

  Journal of Noncommutive Geometry

  We construct Dirac operators and spectral triples for certain, not necessarily self- similar, fractal sets built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics. As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami’s geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.

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

  Journal of Mathematical Biology

  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.

• Modeling population persistence in continuous aquatic networks using metric graphs

  Fundamental and Applied Limnology

  Branching structure of aquatic networks can have substantial ecological consequences. Recent advances have greatly improved our ability to analyze ecological data in the context of river networks, yet we still lack a well-integrated body of theory for predicting and explaining emerging patterns. One hindrance to achieving this goal is the absence of an appropriate framework for modeling ecological processes in aquatic networks. Previous attempts to model the effects of branching network structure on ecological dynamics have typically treated river networks as a set of interconnected patches, artificially discretizing an essentially continuous system. Recent reviews have highlighted the shortcomings of such an approach and called for alternative methods for modeling river habitat in a more natural way. Here, we introduce a framework for modeling branching river networks as continuous systems using dynamic, spatially-explicit models linked to metric graphs. Unlike traditional graphs, metric graphs encode a continuous branching system where edges represent actual domain rather than simple connections among discrete nodes. Graph edges are connected by junction conditions that represent branch confluences. Using the metric graph framework, we model the effects of movement, network geometry, and the distribution of habitat within the network on population persistence for three different types of hypothetical systems. Via numerical simu- lations, we found that movement rates, habitat length, and the distribution of habitable area all play large roles in determining persistence potential. In particular, movement behaviors and habitat distributions that reduce the encounter rate between individuals and lethal boundaries increase population persistence across all model types. We conclude by describing extensions and other potential applications of our framework, including suggested models for populations with in- and out-of-network movement modes and species interactions.