First-order hyperbolic partial differential equations with two internal variables have been used to model biological and epidemiological problems with two physiological structures, such as chronological age and infection age in epidemic models, age and another physiological character (maturation, size, stage) in population models, and cell-age and molecular content (cyclin content, maturity level, plasmid copies, telomere length) in cell population models. In this paper, we study nonlinear double physiologically structured population models with two internal variables by applying integrated semigroup theory and non-densely defined operators. We consider first a semilinear model and then a nonlinear model, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii’s fixed point theorem to obtain the existence of a steady state, and study the stability of the steady state by estimating the essential growth bound of the semigroup. Finally, we generalize the techniques to investigate a nonlinear age-size structured model with size-dependent growth rate.

具有两个内部变量的一阶双曲偏微分方程已被用来模拟具有两种生理结构的生物学和流行病学问题，例如流行模型中的年代年龄和感染年龄，年龄以及人群中的另一个生理特征（成熟度，大小，阶段）细胞群体模型中的细胞年龄和分子含量（细胞周期蛋白含量，成熟度，质粒拷贝，端粒长度）。在本文中，我们通过应用集成半群理论和非密集定义算子来研究具有两个内部变量的非线性双重生理结构人口模型。我们首先考虑一个半线性模型，然后考虑一个非线性模型，使用特征线的方法来找到无穷小生成器的分解和常数公式的变化，应用Krasnoselskii不动点定理获得稳态的存在，并通过估计半群的基本增长界来研究稳态的稳定性。最后，我们归纳了研究尺寸依赖增长率的非线性年龄尺寸结构模型的技术。