Introduction to Numerical Continuation Methods

Allgower, Eugene L.; Georg, Kurt

doi:10.1137/1.9780898719154

Cited by 637 publications

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“…From this stock-fishery model, we derive the stock's basic reproduction ratio in dependence of the adaptive trait, and from this, the evolutionary dynamics of maturation. Using 44 bifurcation analysis (Kuznetsov, 2004) and numerical continuation techniques (Allgower and Georg, 2003), we study the selective pressures exerted on the stock by different levels of fishing mortality and by different levels of selectivity for size 46 and/or maturity. In this way, we assess the potential for fish stocks to experience disruptive selection and thus potentially undergo maturation diversification ( Figure 2).…”

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confidence: 99%

Fisheries-induced disruptive selection

Landi

Hui

Dieckmann

2015

Journal of Theoretical Biology

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Commercial harvesting is recognized to induce adaptive responses of life-history traits in fish populations, in particular by shifting the age and size at maturation through directional selection. In addition to such evolution of a target stock, the corresponding fishery itself may adapt, in terms of fishing policy, technological progress, fleet dynamics, and adaptive harvest. The aim of this study is to assess how the interplay between natural and artificial selection, in the simplest setting in which a fishery and a target stock coevolve, can lead to disruptive selection, which in turn may cause trait diversification. To this end, we build an eco-evolutionary model for a size-structured population, in which both the stock's maturation schedule and the fishery's harvest rate are adaptive, while fishing may be subject to a selective policy based on fish size and/or maturity stage. Using numerical bifurcation analysis, we study how the potential for disruptive selection changes with fishing policy, fishing mortality, harvest specialization, life-history tradeoffs associated with early maturation, and other demographic and environmental parameters. We report the following findings. First, fisheriesinduced disruptive selection is readily caused by commonly used fishing policies, and occurs even for policies that are not specific for fish size or maturity, provided that the harvest is sufficiently adaptive and large individuals are targeted intensively. Second, disruptive selection is more likely in stocks in which the selective pressure for early maturation is naturally strong, provided life-history tradeoffs are sufficiently consequential. Third, when a fish stock is overexploited, fisheries targeting only large individuals might slightly increase sustainable yield by causing trait diversification (even though the resultant yield always remains lower than the maximum sustainable yield that could be obtained under low fishing mortality, without causing disruptive selection). We discuss the broader implications of our results and highlight how these can be taken into account for designing evolutionarily informed fisheries-management regimes.

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confidence: 99%

Fisheries-induced disruptive selection

Landi

Hui

Dieckmann

2015

Journal of Theoretical Biology

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“…Under p-variation, assuming certain regularity conditions (see e.g. Chapter 1 of Allgower and Georg 2003), an equilibrium defines a curve in the equilibrium-parameter space. We consider particular equilibrium branches for the types defined in Section 3.…”

Section: One Parameter Variation: Equilibrium Branches and Bifurcatiomentioning

confidence: 99%

“…., such that for a given tolerance tol, f (u i ) < tol for all i (see e.g. Allgower and Georg 2003). We here use the pseudo-arclength continuation, which is a predictor-corrector method that computes the approximation u i+1 to the curve from the approximation u i in two steps:…”

Section: Numerical Continuationmentioning

confidence: 99%

“…We here use the pseudo code language established in Allgower and Georg (2003). Before the continuation of an equilibrium or a bifurcation, Algorithm 1 reduces the dimension of u 0 to obtainû 0 .…”

Section: Appendixmentioning

confidence: 99%

“…These packages use continuation methods (see e.g. Allgower and Georg 2003) to compute points that approximate an implicitly defined curve. Assuming that an initial point of the curve is known, the methods predict the next point of the curve by computing its tangent, and then correct the predicted point by applying a modified Newton method (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Sanz

Getto

2016

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With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally we apply the method to the above models using existing data for Daphnia magna consuming Algae, and for Perca fluviatilis feeding on Daphnia magna. In particular we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.

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Unidimensional Scaling

Leeuw¹

2005

Encyclopedia of Statistics in Behavioral Science

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This is an entry for The Encyclopedia of Statistics in Behavioral Science, to be published by Wiley in 2005. Unidimensional scaling is the special one-dimensional case of multidimensional scaling [5]. It is often discussed separately, because the unidimensional case is quite different from the general multidimensional case. It is applied in situations where we have a strong reason to believe there is only one interesting underlying dimension, such as time, ability, or preference. And unidimensional scaling techniques are very different from multidimensional scaling techniques, because they use very different algorithms to minimize their loss functions. The classical form of unidimensional scaling starts with a symmetric and non-negative matrix = {δ i j } of dissimilarities and another symmetric and non-negative matrix W = {w i j } of weights. Both W and have a zero diagonal. Unidimensional scaling finds coordinates x i for n points on the Date: March 27, 2004.

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Introduction to Numerical Continuation Methods

Cited by 637 publications

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Fisheries-induced disruptive selection

Fisheries-induced disruptive selection

Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models

Unidimensional Scaling

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