论文标题
按年龄和表型性状结构的人口模型中浓度的动力学
Dynamics of concentration in a population model structured by age and a phenotypical trait
论文作者
论文摘要
我们研究了一个数学模型，描述了按年龄和表型性状结构的人群的生长过程，受衰老，个人之间的竞争和罕见突变的竞争。我们的目标是描述解决方案对更新类型方程的渐近行为，然后得出说明这种人群自适应动态的特性。我们从简化的模型开始，通过丢弃突变的效果，这使我们能够介绍主要思想并陈述全部结果。然后，我们讨论一般模型及其局限性。我们的方法使用正式限制操作员的特征元素，该元素取决于模型的结构变量并定义有效的适应性。然后，我们引入了一种新方法，将收敛证明减少到熵估计值，而不是对受约束的汉密尔顿 - 雅各比方程的估计。数值测试说明了理论，并根据有效的适应性表明了最合适性状的选择。对于突变的问题，出现了异常的哈密顿量，其指数增长出现，为此，我们使用不常见的先验估计值和新的唯一性结果显示了全球粘度解决方案的存在。
We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behaviour of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations. Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we show existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result.
