ANALYSIS OF SYSTEMS OF DIFFERENTIAL EQUATIONS HAVING SEVERAL NONLOCAL TERMS AND APPLICATIONS

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First, we have investigated size structured population models where the growth rate, the mortality rate and the fertility rate depend on size and total population weighted differently each other. We obtained the existence, the representation, and the uniqueness of solutions. By using Schauder's fixed point theorem, we obtained the existence of solutions in a general model. For the uniqueness, we imposed some stronger conditions on the mortality rate and the fertility rate. Unfortunately, it is not known whether the stronger conditions can be removed or not. The results are published in the journal, J.Math.Anal.Appl.Next, we have investigated a general time-dependent model of size structured population dynamics, where the growth rate depends on the size and time. We obtained the existence, positivity, and the uniqueness of solutions. The study of the time dependent model is needed to consider nonlinear harvesting problems.In fact, we applied the results to a nonlinear harvesting problem of size structured population models. Our model has the harvesting rate depending only on time, but the growth rate depends on the size and time and the fertility rate depends on the weighted total population.We gave a presentation about this result at the international conference held at Florida in 2004.Further, we have developed the theory of nonlinear transport equations with nonlocal terms prescribed by a multi-valued function and applied the results to nonlinear four state models of muscle contraction.Here, we obtained not only the existence and uniqueness results, but also the results of the support and the boundedness of the solution. The results will be published in the journal, Adv.Math.Sci.Anal.