Predator–prey systems in streams and rivers

Hilker, Frank M.; Lewis, Mark A.

doi:10.1007/s12080-009-0062-4

Cited by 53 publications

(33 citation statements)

References 78 publications

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“…A key idea is that populations must have the ability to invade upstream in order to persist (Lutscher et al 2010). Once that necessary condition is met, persistence is determined by the interplay of the size of the habitable river domain, population dispersal characteristics, and ecological interactions (see also Lutscher et al 2006Lutscher et al , 2007Hilker and Lewis 2010;Kolpas and Nisbet 2010).…”

Section: Introductionmentioning

confidence: 99%

Modeling population persistence in continuous aquatic networks using metric graphs

Sarhad¹,

Anderson²

2015

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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

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Section: Introductionmentioning

confidence: 99%

Modeling population persistence in continuous aquatic networks using metric graphs

Sarhad¹,

Anderson²

2015

fal

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“…This steady state is incompatible with the zero flux boundary condition, and the predators are gradually washed out of system (Fig. 5;Hilker and Lewis 2010). This occurs via a transition front moving across the domain, so that the prey-only steady state appears to "invade" the co-existence steady state.…”

Section: An Illustrative Examplementioning

confidence: 99%

“…In the context of that application, a Dirichlet boundary condition corresponds to a hostile boundary such as a waterfall or a region containing toxic waste water, while a zero-flux condition is of Robin type. A zero Neumann boundary condition is known as a Danckwert boundary condition and corresponds to a long river in which the downstream boundary has little influence (Lutscher et al 2006;Nauman 2008 §9.3.1;Hilker and Lewis 2010). For most equation systems, a convectively unstable steady state is stable on a finite domain with separated boundary conditions, as is the case for (1a), (1b).…”

Section: Absolute and Convective Stabilitymentioning

confidence: 99%

“…Most predator-prey studies do not include advection terms, but allowing c = 0 enables a clearer illustration of some of the concepts we will be discussing. Advection of this type arises naturally in river-based predator-prey systems (Hilker and Lewis 2010). The prey consumption rate per predator is an increasing saturating function of the Numerical simulation of predators invading prey using the Rosenzweig-MacArthur (1963) model (1a), (1b) without advection.…”

Section: Introductionmentioning

confidence: 99%

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A Mathematical Biologist’s Guide to Absolute and Convective Instability

2013

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Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability.They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator-prey systems.

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“…These could include additional spatial heterogeneity in vital and movement rates as well as the inclusion of species interactions (e.g., competition or predation). Previous single-segment models have found both within-segment spatial variation (Lutscher et al 2006(Lutscher et al , 2007Jin and Lewis 2011;Lam et al 2016) and species interactions (Lutscher et al 2007;Hilker and Lewis 2010) can dramatically alter persistence outcomes. These issues have not been explored as extensively in the framework of branching networks (but see Goldberg et al 2010;Auerbach and Poff 2011).…”

Section: Another Look At CMmentioning

confidence: 99%

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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Predator–prey systems in streams and rivers

Cited by 53 publications

References 78 publications

Modeling population persistence in continuous aquatic networks using metric graphs

Modeling population persistence in continuous aquatic networks using metric graphs

A Mathematical Biologist’s Guide to Absolute and Convective Instability

Geometric indicators of population persistence in branching continuous-space networks

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