MIXING, FLOCKING AND COOPERATION: ANALYTICAL STUDIES OF TRANSPORT PHENOMENA IN BIOLOGY
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Transport behavior widely takes place in biological phenomena. In this thesis, I study the delicate interactions between transportation and other forces in biology, diffusion, alignment, cooperation included. The thesis focuses on three topics: (i) chemotaxis in moving fluid; (ii) flocking of birds and fish; (iii) multi-species chemotaxis and multi-species flocking dynamics.
We use the Patlak-Keller-Segel equations with additional advection to model the chemotaxis phenomena in the moving fluid. It is well-known that if there is no underlying fluid transport, the total number of cells in the environment determines the long-time behavior of the dynamics. If the number is large enough, cells will concentrate to form clusters and cause break-downs of the model. We discover that external strong fluid flow has the potential to suppress the possible blow-up in the system.
We use the Cucker-Smale model and the Motsch-Tadmor model to describe the flocking behavior of fish or birds. While the well-posedness theory of the PDE in one dimension is well-understood, little is known in dimension two. We give explicit sufficient condition to guarantee the existence of unique global strong solutions and clarify the role played by the stretching and vorticity.
In the last part of the thesis we introduce multi-species concepts into the chemotaxis models and flocking models. We discover new conditions to guarantee the well-posedness of the multi-species Patlak-Keller-Segel systems and the multi-species Cucker-Smale models.