Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics

2007 ◽  
pp. 57-68 ◽  
Author(s):  
Guy Barles ◽  
Benoît Perthame
Keyword(s):  
Adaptive Dynamics ◽  
Jacobi Equations

scholarly journals A Hamilton–Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments

2017 ◽  
Vol 27 (13) ◽  
pp. 2425-2460 ◽  
Author(s):  
Sepideh Mirrahimi
Keyword(s):  
Evolutionary Biology ◽  
Gaussian Approximation ◽  
Adaptive Dynamics ◽  
Bimodal Distribution ◽  
Heterogeneous Environments ◽  
Structured Population ◽  
And Migration ◽  
Two Peaks ◽  
Jacobi Equations ◽  
General Method

In this work, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration between two habitats. Assuming that the effects of the mutations are small but nonzero, we show that the population’s phenotypical distribution has at most two peaks and we give explicit conditions under which the population will be monomorphic (unimodal distribution) or dimorphic (bimodal distribution). More importantly, we provide a general method to determine the dominant terms of the population’s distribution in each case. Our work, which is based on Hamilton–Jacobi equations with constraint, goes further than previous works where such tools were used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. The main elements for the computation of the dominant terms of the population’s distribution are the convergence of the logarithmic transform of the solution to the unique solution of a Hamilton–Jacobi equation and the computation of the correctors. This method allows indeed to go further than the Gaussian approximation commonly used by biologists and makes a connection between the theories of adaptive dynamics and quantitative genetics. Our work being motivated by biological questions, the objective of this paper is to provide the mathematical details which are necessary for our biological results [S. Mirrahimi and S. Gandon, The equilibrium between selection, mutation and migration in spatially heterogeneous environments, in preparation].

scholarly journals Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

2011 ◽  
Vol 23 (1) ◽  
pp. 377-394
Author(s):  
R. L. Foote ◽  
C. K. Han ◽  
J. W. Oh
Keyword(s):  
Jacobi Equations

scholarly journals Non-local Discrete $$\infty $$ ∞ -Poisson and Hamilton Jacobi Equations

2015 ◽  
Vol 55 (2) ◽  
pp. 229-241 ◽  
Author(s):  
Matthieu Toutain ◽  
Abderrahim Elmoataz ◽  
François Lozes ◽  
Alamin Mansouri
Keyword(s):  
Non Local ◽  
Jacobi Equations

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
Sign in / Sign up

Export Citation Format

Share Document