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
Mots-clés : Selection-mutation dynamics, evolutionary stable distribution, energy dissipation
@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},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016058},
zbl = {1378.92047},
language = {en},
url = {http://archive.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 - http://archive.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 http://archive.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. http://archive.numdam.org/articles/10.1051/m2an/2016058/
and , Explicit solutions for replicator-mutator equation: extinction vs. acceleration. SIAM J. Appl. Math. 74 (2014) 1919–1934. | DOI | Zbl
and , When the exception becomes the rule: the disappearance of limiting similarity in the Lotka-Volterra model. J. Math. Biol. 258 (2009) 89–94. | Zbl
, and , Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result. Methods Appl. Anal. 16 (2009) 321–340. | DOI | Zbl
, , and , The non-local Fisher–KPP equation: traveling waves and steady states. Nonlinearity 22 (2009) 2813–2844. | DOI | Zbl
and , Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52 (1997) 179–197. | DOI | Zbl
R. Bürger, The mathematical theory of selection, recombination and mutation. Wiley, New York (2000). | Zbl
, and , Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69 (2006) 297–231. | DOI | Zbl
, and , From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models 24 (2008) 2–44. | DOI | Zbl
and , Evolutionary distributions and competition by way of reaction-diffusion and by way of convolution. Bull. Math. Biol. 75 (2013) 2305–2323. | DOI | Zbl
and , On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds. Math. Program. 67 (1994) 189–224. | DOI | Zbl
and , An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6 (1996) 418–445. | DOI | Zbl
, , and , On selection dynamics for continuous structured populations. Commun. Math. Sci. 6 (2008) 729–747. | DOI | Zbl
and , A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004) 1880–1919. | DOI | Zbl
D. Furihata and T. Matsuo, Discrete variational derivative method: a structure-preserving numerical method for partial differential equations. Chapman and Hall/CRC numerical analysis and scientific computation series. ISBN 978-1-4200-9445-9 (2011). | Zbl
and , Dynamics of Nonlocal Fisher concentration points: a nonlinear analysis of Turing patterns. Math. Model. Nat. Phenomenon 2 (2007) 135–151.
, and , Adaptive dynamics: modeling Darwin’s divergence principle. C. R. Biol. 329 (2006) 876–879. | DOI
P.E. Gill, W. Murray and M.H. Wright, Practical Optimization. London, Academic Press (1981). | Zbl
, Traveling front of a nonlocal Fisher equation. J. Math. Biol. 41 (2000) 272–284. | DOI | Zbl
, A globally convergent method for nonlinear programming. J. Optimiz. Theory Appl. 22 (1977) 297–309. | DOI | Zbl
and , On selection dynamics for competitive interactions. J. Math. Biol. 63 (2011) 493–517. | DOI | Zbl
, Some extensions of Liapunov’s second method. IRE Trans. Circuit Theory 7 (1960) 520–527. | DOI
, and , Entropy satisfying schemes for computing selection dynamics in competitive interactions. SIAM J. Numer. Anal. 53 (2015) 1393–1417. | DOI | Zbl
and , Entropy/energy stable schemes for evolutionary dispersal models. J. Comput. Phys. 256 (2014) 656–677. | DOI | Zbl
, Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations. Math. Comput. 84 (2015) 2263–2295. | DOI | Zbl
and , The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8 (2010) 541–564. | DOI | Zbl
, and , Dirac mass dynamics in multidimensional nonlocal parabolic equations. Commun. Partial Differ. Eq. 36 (2011) 1071–1098. | DOI | Zbl
S. Mirrahimi, B. Perthame, E. Bouin and P. Millien, Population formulation of adaptative meso-evolution: theory and numerics. Mathematics and Biosciences in Interaction (2011) 159–174.
, and , Direct competition results from strong competition for limited resource. J. Math. Biol. 68 (2014) 931–949. | DOI | Zbl
J. Nocedal and S.J. Wright, Numerical Optimization, 2nd edn. Springer Series in Operations Research. Springer Verlag (2006).
and , Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57 (2008) 3275–3301. | DOI | Zbl
, Long time evolution of populations under selection and vanishing mutations. Acta Appl. Math. 114 (2011) 1–14. | DOI | Zbl
, Local stability of evolutionary attractors for continuous structured populations. Monatsh. Math. 165 (2012) 117–144. | DOI | Zbl
and , Two fish species competition model with nonlinear interactions and equilibrium catches. Res. Popul. Ecol. 20 (1978) 123–140. | DOI
, , and , An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program. 107 (2006) 391–408. | DOI | Zbl
Cité par Sources :
