System of nonlocal nonlinear differential equations and applications

首先，我们研究了增长率依赖于规模和时间的种群模型。得到了解对所有给定数据的连续依赖性。对于初值依赖性、年龄函数和出生函数，表现得比较简单，但对增长率的依赖性包含一个本质上的难题。求解方程的方法是所谓的特征线法，特征线由生长速率决定，因此当生长速率受到扰动时，特征线本身也会发生变化。这里的结果对于方程的稳定性很重要，同时可以用来研究具有非线性增长率的模型。接下来，我们研究了增长率取决于规模和总人口的规模依赖人口模型。最初，这些模型来自于描述森林或种植园中植物的种群动态。我们得到了更一般模型的存在唯一性结果，并在2002年香港国际会议上作了报告。肌肉收缩是肌球蛋白粗纤维和肌动蛋白细纤维之间相对滑动的结果。这种滑动发生在所谓的横桥将肌球蛋白连接到肌动蛋白上并充当弹簧时。肌肉收缩模型描述了附着的横桥密度的时间变化。在这项研究中，我们考虑了双曲型迁移方程组。我们的模型包含所谓的四态模型，其中有两个状态的附着和分离的跨桥分别。

First, we have investigated the population models with growth rate depending on size and time. Continuous dependence of solution on all given data is obtained. For the dependency on initial values, aging functions and birth functions, it is shown relatively simply, but the dependency on growth rate includes an essentially difficult problem. The way of solving equation is so-called the characteristic method and the characteristic curves are determined by the growth rate, and so when the growth rate is perturbed, the characteristic curves themselves change. The results here are important as the stability of equation, and at the same time, they can be used to investigate the models with nonlinear growth rate.Next, we have investigated the size-dependent population models with growth rate depending on size and the total population. Originally, the models come from describing the population dynamics of plants in forests or plantations. We obtained the existence and uniqueness results for more general models and we gave a presentation about this result at the international conference held at Hong Kong in 2002.Muscle contraction is a consequence of relative sliding between tha thick filament called myosin and the thin filament called actin. This sliding occurs when the so-called cross-bridges attach myosin to actin and act as spring. The muscle contraction models describe the temporal variation of the density of the attached cross-bridges. In this research, we have considered a system of hyperbolic transport equations. Our model contains the so-called four state models in which there are two state of attached and detached cross-bridges respectively.