In this paper, we design, analyze and numerically validate energy dissipating finite volume schemes for a competition-mutation equation with a gradient flow structure. The model describes the evolution of a population structured with respect to a continuous trait. Both semi-discrete and fully discrete schemes are demonstrated to satisfy the two desired properties: positivity of numerical solutions and energy dissipation. These ensure that the positive steady state is asymptotically stable. Moreover, the discrete steady state is proven to be the same as the minimizer of a discrete energy function. As a comparison, the positive steady state can also be produced by a nonlinear programming solver. Finally, a series of numerical tests is provided to demonstrate both accuracy and the energy dissipation property of the numerical schemes. The numerical solutions of the model with small mutation are shown to be close to those of the corresponding model with linear competition.
Accepté le :
DOI : 10.1051/m2an/2016058
Keywords: Selection-mutation dynamics, evolutionary stable distribution, energy dissipation
Cai, Wenli 1 ; Liu, Hailiang 2
@article{M2AN_2017__51_4_1223_0,
author = {Cai, Wenli and Liu, Hailiang},
title = {A finite volume method for nonlocal competition-mutation equations with a gradient flow structure},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1223--1243},
year = {2017},
publisher = {EDP Sciences},
volume = {51},
number = {4},
doi = {10.1051/m2an/2016058},
zbl = {1378.92047},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2016058/}
}
TY - JOUR AU - Cai, Wenli AU - Liu, Hailiang TI - A finite volume method for nonlocal competition-mutation equations with a gradient flow structure JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1223 EP - 1243 VL - 51 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2016058/ DO - 10.1051/m2an/2016058 LA - en ID - M2AN_2017__51_4_1223_0 ER -
%0 Journal Article %A Cai, Wenli %A Liu, Hailiang %T A finite volume method for nonlocal competition-mutation equations with a gradient flow structure %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1223-1243 %V 51 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2016058/ %R 10.1051/m2an/2016058 %G en %F M2AN_2017__51_4_1223_0
Cai, Wenli; Liu, Hailiang. A finite volume method for nonlocal competition-mutation equations with a gradient flow structure. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1223-1243. doi: 10.1051/m2an/2016058
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