Modeling population persistence in continuous aquatic networks using metric graphs

Sarhad, Jonathan J.; Anderson, Kurt E.

doi:10.1127/fal/2015/0635

Cited by 7 publications

(18 citation statements)

References 25 publications

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“…Previous studies (Sarhad et al 2014;Sarhad and Anderson 2015) have identified geometric indicators of population persistence in highly symmetric networks which can be described using relatively few parameters. However, actual networks such as rivers or vegetation branches are decidedly not symmetric and can differ greatly in size and shape.…”

Section: Aims and Outlinementioning

confidence: 99%

“…The network models introduced in Sarhad et al (2014), Sarhad and Anderson (2015) and used here extend the case of the finite segment with boundary conditions to continuous tree networks. The extended model features boundary conditions as well as junction conditions (Fig.…”

Section: Extension To Networkmentioning

confidence: 99%

“…An additional departure from the single segment model which is featured in networks is that Λ 1 is influenced by the interaction of geometric parameters and dispersal parameters at a junction (see (2.7); see also Sarhad et al 2014;Sarhad and Anderson 2015). The computation of Λ 1 in bounded networks is less trivial than in the single segment case but, analogously, Λ 1 is numerically calculated as smallest positive root of the determinant of a 2n × 2n matrix, for a network with n edges.…”

Section: Persistence Criteria In Networkmentioning

confidence: 99%

“…Here, we sought to advance the application of metric graphs in biology by investigating how global geometric features of branching networks modeled as metric graphs characterize population persistence. The particular case study was of a population dispersing by diffusion and advection toward a lethal downstream boundary, which has received extensive previous attention (Speirs and Gurney 2001;Lutscher et al 2005Lutscher et al , 2007Ramirez 2012;Sarhad et al 2014;Sarhad and Anderson 2015). Towards this end, we examined various metrics related to network geometry and tested their robustness as indicators of population persistence in stochastically generated branching networks.…”

Section: Another Look At CMmentioning

confidence: 99%

“…In contrast, many biological systems such as river basins, cave systems, organisms on vegetation surfaces, and vascular networks have a spatial structure which may be more appropriately modeled by metric graphs, which allow for dynamics to occur within edges (e.g., Ramirez 2012;Sarhad et al 2014;Sarhad and Anderson 2015). Unlike standard graphs, metric graphs encode spatially continuous networks where edges represent actual habitat rather than merely connections among discrete nodes.…”

Section: Introductionmentioning

confidence: 99%

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Geometric indicators of population persistence in branching continuous-space networks

2016

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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.Keywords Population dynamics · Branching networks · Reaction-diffusionadvection · Partial differential equations · (Metric) graph theory · Principal eigenvalue analysis Mathematics Subject Classification 05C12 (distance in graphs) · 35K57 (reactiondiffusion) · 58C40 (spectral theory; eigenvalue problems) · 35K20 (initial-boundary value problems for second-order parabolic equations) · 35J25 (boundary value problems for second-order elliptic equations) · 92D40 (ecology)

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Section: Aims and Outlinementioning

confidence: 99%

Section: Extension To Networkmentioning

confidence: 99%

Section: Persistence Criteria In Networkmentioning

confidence: 99%

Section: Another Look At CMmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geometric indicators of population persistence in branching continuous-space networks

2016

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Metacommunity theory meets restoration: isolation may mediate how ecological communities respond to stream restoration

Swan

Brown

2017

Ecological Applications

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An often-cited benefit of river restoration is an increase in biodiversity or shift in composition to more desirable taxa. Yet, hard manipulations of habitat structure often fail to elicit a significant response in terms of biodiversity patterns. In contrast to conventional wisdom, the dispersal of organisms may have as large an influence on biodiversity patterns as environmental conditions. This influence of dispersal may be particularly influential in river networks that are linear branching, or dendritic, and thus constrain most dispersal to the river corridor. As such, some locations in river networks, such as isolated headwaters, are expected to respond less to environmental factors and less by dispersal than more well-connected downstream reaches. We applied this metacommunity framework to study how restoration drives biodiversity patterns in river networks. By comparing assemblage structure in headwater vs. more well-connected mainstem sites, we learned that headwater restoration efforts supported higher biodiversity and exhibited more stable ecological communities compared with adjacent, unrestored reaches. Such differences were not evident in mainstem reaches. Consistent with theory and mounting empirical evidence, we attribute this finding to a relatively higher influence of dispersal-driven factors on assemblage structure in more well-connected, higher order reaches. An implication of this work is that, if biodiversity is to be a goal of restoration activity, such local manipulations of habitat should elicit a more profound response in small, isolated streams than in larger downstream reaches. These results offer another significant finding supporting the notion that restoration activity cannot proceed in isolation of larger-scale, catchment-level degradation.

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Population Dynamics in River Networks

2019

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Natural rivers connect to each other to form networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydrological, physical, and biological factors, especially the structure of the river network, on population persistence. physical and hydrological features in a river network, can greatly influence the spatial distribution of the flow profile (including the flow velocity, water depth, etc.) in branches of the network. Moreover, the population dispersal vectors may be constrained by the network configuration and the flow profile, and species life history traits may depend on varying habitat conditions in the network. As a result, population distribution and persistence in river systems can be significantly affected by the network topology or structure, see e.g., [8,11,12,[14][15][16]46]. Then there arise interesting questions such as whether a population can persist in the desert streams of the southwestern United State while the streams are experiencing substantial natural drying trends [11], or whether dendritic geometry enhances dynamic stability of ecological systems [15] etc. Furthermore, other related dynamics in the network, such as the dynamics of water-born infectious diseases like cholera may also be greatly affected by the river network geometry (see e.g., [6]).Branches in a river network have been modeled as point nodes in a network of habitats in individual based models [11,16] and matrix population models for stage-structured populations [14,39]. However this oversimplifies the spatial heterogeneity of river networks. In a real river ecosystem, organisms mainly live in the branches of the network and the connections between branches (e.g., the network nodes) are mainly for population transitions from one branch to another. To take into account this realistic situation, in recent works [48-50], integro-differential equations and reactiondiffusion-advection equations were used to model population dynamics in river networks where the network branches, instead of the network nodes, are the main habitat for organisms. Here the river networks are modeled under the framework of metric graphs (or metric networks). A metric graph is a graph G = (V, E) with a set V of vertices and a set E of edges, such that each edge e ∈ E is associated with either a closed bounded interval. Mathematic notion of metric graphs was first introduced in the context of wave propagation on thin graph-like domains [5,25], and they are also called quantum graphs.The theories of parabolic and elliptic equations as well as the corr...

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Modeling population persistence in continuous aquatic networks using metric graphs

Cited by 7 publications

References 25 publications

Geometric indicators of population persistence in branching continuous-space networks

Geometric indicators of population persistence in branching continuous-space networks

Metacommunity theory meets restoration: isolation may mediate how ecological communities respond to stream restoration

Population Dynamics in River Networks

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