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Systems of the real world can be observed to reach an understanding of their inner structure and behavior. The collection of experimental data may be organized into a mathematical model to obtain a formal description of the behavior of the observed system.

One of the most interesting and challenging research perspectives for applied mathematicians is the description of the collective behavior of large populations of interacting entities whose microscopic state is described not only by mechanical variables, typically position and velocity, but also by a biological state related to an organized, and maybe even intelligent, behavior.

The analysis developed in Chapter 2 has provided a general mathematical framework which can be used as a background to model specific biological phenomena related to complex multicellular systems. Mathematical models can be designed by identifying the cell populations which participate, and then, according to the specific biological phenomena, by modelling pair interactions at the microscopic level between cells of the same or different populations.

This chapter develops a qualitative analysis of the initial value problem for the various mathematical models proposed in Chapter 3. The problem is stated by linking the evolution equations to suitable initial conditions. The analysis is developed with classical methods of functional analysis (see Zeidler 1995 ), and provides the background for the simulations which will be proposed in Chapter 5.

A qualitative analysis of the initial value problem for various models of the immune competition against an aggressive host was developed in Chapter 4. The analysis showed that the problem is locally well posed, while special attention was devoted to identifying the output of the competition and, in particular, the influence of the parameters of the model over the abovementioned asymptotic behavior.

The various mathematical models proposed and analyzed in the preceding chapters describe multicellular systems in the spatially homogeneous case. As we have seen, these models are able to describe several interesting phenomena related to several aspects of the immune competition. On the other hand, a space structure is needed to model cellular motion, as well as to recover macroscopic models from the underlying microscopic description.

A general mathematical approach to the modelling of multicellular systems in view of applications to the mathematical description of complex biological systems has been developed in this book.