Non‐linear structured population dynamics with co‐variates

Aubin, Jean‐Pierre; Bonneuil, Noël; Maurin, Franck

doi:10.1080/08898480009525493

Cited by 15 publications

(3 citation statements)

References 16 publications

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“…The ''tendency for diversity'' presented as a ''discovery'' in this book is in fact inherent in set-valued analysis, where sets of attainable sets are substituted to trajectories, differential inclusions to differential equations, viability kernels (the sets of states x for which there exists at least one trajectory remaining in a given set of constraints) to equilibria and attractors (Aubin et al, 2000). This exceeds by far ''a diffusion process'' (: 17), which is a restrictive view of the ''tendency to bio-diversity.''…”

Section: Book Reviewmentioning

confidence: 99%

A Review ofBiology's First Law. The Tendency for Diversity and Complexity to Increase in Evolutionary Systemsby Daniel W. McShea and Robert N. Brandon

Bonneuil¹

2011

Mathematical Population Studies

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Section: Book Reviewmentioning

confidence: 99%

A Review ofBiology's First Law. The Tendency for Diversity and Complexity to Increase in Evolutionary Systemsby Daniel W. McShea and Robert N. Brandon

Bonneuil¹

2011

Mathematical Population Studies

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“…This is actually a fixed-point problem that can be easily solved by the simple BanachPicard-Cacciopoli Fixed Point Theorem (see a presentation of basic linear model in [11] and in the last chapter of [8]). …”

Section: By Assuming That the Number B(t W) Of Births At Each Instanmentioning

confidence: 99%

Regulation of births for viability of populations governed by age-structured problems

Aubin

2011

J. Evol. Equ.

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International audienceWe assume that the evolution of the population is governed by a controlled McKendrick age-structured partial differential equation where the mortality rate and immigrated population levels are no longer known coefﬁcients, but contingent regulation parameters chosen for a given purpose, for instance, for requiring that the population satisﬁes prescribed viability constraints depending on time and age. The Lotka renewal equation relating the boundary condition (number of births) to the integral with respect to age of the population is replaced by the introduction of another regulation parameter in the boundary condition, regarded as a natality policy. We may control it by its derivative, regarded as a natality decision. We then construct a regulation map, associating with the population level, the time and the age the subset of natality policies, a mortality rates and an immigration levels needed for governing viable evolutions of the population

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“…The concept of the viable-capture basin also plays a fundamental role for solving first-order partial differential equations (see [12,13,14,15,16], chapter 8 of [2], [11] without boundary conditions, and [6,7,8,9] for the Dirichlet boundary-value problems for such systems). Finally, the viability kernel algorithm allows us to compute the viability kernel (see [31,47,50]).…”

Section: The Subsetmentioning

confidence: 99%

Viability Kernels and Capture Basins of Sets Under Differential Inclusions

Aubin

2001

SIAM J. Control Optim.

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This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions. It is based on a method devised by Hélène Frankowska for characterizing the value function of an optimal control problem as generalized (contingent or viscosity) solutions to Hamilton-Jacobi equations. These abstract results, interesting by themselves, can be applied to epigraphs of functions or graphs of maps and happen to be very efficient for solving other problems, such as stopping time problems, dynamical games, boundary-value problems for systems of partial differential equations, and impulse and hybrid control systems, which are the topics of other companion papers.

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Non‐linear structured population dynamics with co‐variates

Cited by 15 publications

References 16 publications

A Review ofBiology's First Law. The Tendency for Diversity and Complexity to Increase in Evolutionary Systemsby Daniel W. McShea and Robert N. Brandon

A Review ofBiology's First Law. The Tendency for Diversity and Complexity to Increase in Evolutionary Systemsby Daniel W. McShea and Robert N. Brandon

Regulation of births for viability of populations governed by age-structured problems

Viability Kernels and Capture Basins of Sets Under Differential Inclusions

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