System of nonlocal nonlinear differential equations and applications

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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.