Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 53, Issue 3, Pages 1393-1417Publisher
SIAM PUBLICATIONS
DOI: 10.1137/140965739
Keywords
selection dynamics; evolutionary stable distribution; relative entropy; positivity
Categories
Funding
- National Science Foundation [DMS13-12636]
- National Science Foundation of China [11271218]
- China Scholarship Council
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1312636] Funding Source: National Science Foundation
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In this paper, we present entropy satisfying schemes for solving an integro-differential equation that describes the evolution of a population structured with respect to a continuous trait. In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493-517] solutions are shown to converge toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming problem and can be computed by any well-established nonlinear programing algorithm. The schemes are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm both accuracy and the entropy satisfying property and to underline the efficiency of capturing the large time asymptotic behavior of numerical solutions in various settings.
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