Vorschau

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Abstract

Processes containing multistability and switching play an important role in cell signalling. Coupled to cell-to-cell communication of diffusing ligands such processes may give rise to spatial pattern in biological systems. This leads to a new type of mathematical models consisting of nonlinear partial differential equations of diffusion, transport and reactions coupled with dynamical systems controlling the transitions.

In this thesis we propose a model consisting of one reaction-diffusion equation with homogeneous Neumann boundary conditions coupled to one ordinary differential equation containing bistability in the kinetic functions. We analyse the ability of our model to produce patterns. Therefore, we compare two cases of the model, where one does include the hysteresis effect and the second one does not. We show that the model without hysteresis in the kinetic functions is not able to describe pattern formation, because all spatially inhomogeneous stationary solutions are unstable. Furthermore, we prove that the model including hysteresis possesses an infinite number of stationary solutions. There are monotone and periodic solutions. Moreover, we prove the existence of irregular solutions, which, restricted to certain intervals, consist of different monotone ones. All stationary solutions are discontinuous in one component. Furthermore, we show under which conditions on the parameters a plurality of these solutions is stable.

Since the mechanism for pattern formation in our model is different from the usual Turing mechanism, patterns do not evolve spontaneously from small perturbations, but they need a sufficiently strong external signal for their emergence. In terms of our model we prove that there is coexistence of different patterns for the same set of parameters, with the final pattern strongly depending on the initial perturbation.