scholarly journals The Canonical Equation of Adaptive Dynamics: A Mathematical View

Selection ◽  
2002 ◽  
Vol 2 (1-2) ◽  
pp. 73-83 ◽  
Author(s):  
N. Champagnat ◽  
R. Ferričre ◽  
G. Ben Arous4
Keyword(s):  
Adaptive Dynamics ◽  
Canonical Equation

scholarly journals The Branching Bifurcation of Adaptive Dynamics

2015 ◽  
Vol 25 (07) ◽  
pp. 1540001 ◽  
Author(s):  
Fabio Della Rossa ◽  
Fabio Dercole ◽  
Pietro Landi
Keyword(s):  
Adaptive Dynamics ◽  
Stationary Regime ◽  
Joint Effect ◽  
Evolutionary Stability ◽  
Canonical Equation ◽  
Mutant Type ◽  
Invasion Fitness ◽  
Individual Traits ◽  
Classical Setting ◽  
Intermediate Traits

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits — phenotypes or strategies — of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present — resident — types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits — the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching — the joint effect of resident-mutant coexistence and evolutionary instability — have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

scholarly journals The canonical equation of adaptive dynamics for Mendelian diploids and haplo-diploids

2013 ◽  
Vol 3 (6) ◽  
pp. 20130025 ◽  
Author(s):  
Johan A. J. Metz ◽  
Carolien G. F. de Kovel
Keyword(s):  
Time Scale ◽  
Life Histories ◽  
Adaptive Dynamics ◽  
Canonical Equation ◽  
Small Step ◽  
Evolutionary Time ◽  
Scale Separation ◽  
Slowing Down ◽  
Time Scale Separation ◽  
Simplifying Assumptions

One of the powerful tools of adaptive dynamics is its so-called canonical equation (CE), a differential equation describing how the prevailing trait vector changes over evolutionary time. The derivation of the CE is based on two simplifying assumptions, separation of population dynamical and mutational time scales and small mutational steps. (It may appear that these two conditions rarely go together. However, for small step sizes the time-scale separation need not be very strict.) The CE was derived in 1996, with mathematical rigour being added in 2003. Both papers consider only well-mixed clonal populations with the simplest possible life histories. In 2008, the CE's reach was heuristically extended to locally well-mixed populations with general life histories. We, again heuristically, extend it further to Mendelian diploids and haplo-diploids. Away from strict time-scale separation the CE does an even better approximation job in the Mendelian than in the clonal case owing to gene substitutions occurring effectively in parallel, which obviates slowing down by clonal interference.

scholarly journals Adaptive dynamics of unstable cancer populations: The canonical equation

2018 ◽  
Vol 11 (8) ◽  
pp. 1283-1292 ◽  
Author(s):  
Guim Aguadé-Gorgorió ◽  
Ricard Solé
Keyword(s):  
Adaptive Dynamics ◽  
Canonical Equation

scholarly journals Adaptive dynamics of unstable cancer populations: the canonical equation

2017 ◽  
Author(s):  
Guim Aguadé-Gorgorió ◽  
Ricard Solé
Keyword(s):  
Immune Responses ◽  
Tumor Progression ◽  
Cancer Cell ◽  
Genetic Instability ◽  
Adaptive Dynamics ◽  
Canonical Equation ◽  
Cell Populations ◽  
Simple Version ◽  
Physical Constraints ◽  
Environment Modelling

In most instances of tumour development, genetic instability plays a role in allowing cancer cell populations to respond to selection barriers, such as physical constraints or immune responses, and rapidly adapt to an always changing environment. Modelling instability is a nontrivial task, since by definition evolving changing instability leads to changes in the underlying landscape. In this paper we explore mathematically a simple version of unstable tumor progression using the formalism of Adaptive Dynamics (AD) where selection and mutation are explicitly coupled. Using a set of basic fitness landscapes, the so called canonical equation for the evolution of genetic instability on a minimal scenario associated to a population of unstable cells is derived. The implications and potential extensions of this model are discussed.

scholarly journals Function-valued adaptive dynamics and the calculus of variations

2005 ◽  
Vol 52 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Kalle Parvinen ◽  
Ulf Dieckmann ◽  
Mikko Heino
Keyword(s):  
Calculus Of Variations ◽  
Adaptive Dynamics
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