Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions

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• 作者：À ; ngel Calsina^a ; ^{acalsina@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; Sí ; lvia Cuadrado^a ; ^{silvia@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; Laurent Desvillettes^b ; ^{desvillettes@math.univ-paris-diderot.fr" class="auth_mail" title="E-mail the corresponding author} ; Gaë ; l Raoul^c ; ^{raoul@cmap.polytechnique.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Selection&ndash ; mutation equations ; Asymptotic behaviour ; Spectral theory ; Population dynamics
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：444
• 期：2
• 页码：1515-1541
• 全文大小：1079 K

文摘

In this paper, we study the asymptotic (large time) behaviour of a selection–mutation–competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε   of mutations. We show that depending on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303523&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303523&_rdoc=1&_issn=0022247X&md5=7ea1364a4f84806727f36082e9289c84" title="Click to view the MathML source">α>0class="mathContainer hidden">class="mathCode">$\alpha >0$, the limit class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303523&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303523&_rdoc=1&_issn=0022247X&md5=8dea560db55b8b01399e60ccf0ea98dc" title="Click to view the MathML source">ε→0class="mathContainer hidden">class="mathCode">$\epsilon \to 0$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303523&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303523&_rdoc=1&_issn=0022247X&md5=e84b638d5ce3042cf0091038473ef79d" title="Click to view the MathML source">t=ε^−αclass="mathContainer hidden">class="mathCode">$t={\epsilon }^{-\alpha }$ can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).