Thomas Hillen: Open and solved problems in chemotaxis

Since the early papers about chemotaxis (Patlak 1953, Keller-Segel 1970) the mathematical investigation of chemosensitive movement has flourished. In particular, after the ground breaking results on blow-up solutions by Jaeger and Luckhaus (1990) a whole industry for mathematical analysis of chemotaxis models has developed. Many interesting and challenging problems could be solved up to date.

At the last SIAM Dyn. Syst. meeting in Snwobird (2003) we discussed among colleagues what are the interesting open problems in this area which are worthwhile to study. This is a very good question. In my point of view the following items are well understood and give only little room for improvement. Obviously, one could sharpen an estimate here, proof existence under slightly weaker assumptions there etc. but this is getting tedious.

Well understood:

• The derivation of classical chemotaxis models is well understood. I have seen derivations from specific random walk descriptions (Patlak 1953), from Fickian-assumptions (Keller-Segel), from continuous transport equations (Alt, Dickinson, Tranquillo, Hillen, Othmer etc.) rigorous derivations from reinforced random walks (Stevens), derivations from Langevin equations (Erdmann, Hillen, Hadeler, Lutscher), and lately a derivation from fluid dynamical principles (Byrne and Owen). The model appears again and again and it is justified to study this model in detail

• Blow-up is well understood. It is almost impossible to keep up with the vast literature on blow-up solutoins for chemotaxis. It was very timely that Dirk Horstmann wrote two reviews about this area (2003, 2004 Jahresberichte DMV). The first review summarizes all known results on finite time blow-up and describes the historical development. The second review deals with traveling wave solutions.   

• Applications to Dictyostelium discoideum, Salmonella typhimurium,  Eschirichia coli. The chemotaxis model and variations of it have been proven to be capable to describe chemotaxis for the above mentioned species (Dallon, Othmer, Ford, Lauffenburger, Rivero et al, Tyson et al, Dolak, Hillen). The chemotaxis models need new challanges! 

Open questions:

1. Pattern formation. Pattern formation for chemosensitive moevemtn models has not nearly been investigated in such a great detail as blow-up. Although, if these models have been used to describe experiments (e.g. Rivero, Tranquillo, Buettner, Lauffenburger, or Ford et al, Tyson et al, Dolak), then always  models are used whose solutions do not blow up in finite time. Various mechanisms are able to prevent blow-up: (i) saturation effects, (ii) volume filling effects, (iii) quorum sensing effects, (iv) finite sampling radius (see Hillen+Schmeiser, in preparation). The corresponding models for chemosensitive movement show interesting properties of pattern formation (as shown in Hillen, Painter, and Painter, Hillen, and Dolak+Hillen). It is a wide open field to understand these patterns in detail, understand underlying bifurcations, find the global attractor, study appropriate asymptotic scalings, compare to experiments etc. etc. In all those questions preliminary results have appeard recently (Potapov, Hillen, Osaki, Yagi, Wrzosek).

2. Global existence for the quorum sensing model. No more details, sorry, because we are working on this right now.

3. Hopf-bifurcations in the quorum sensing model (again no details).

4. Other applications, for example in cancer modeling, e.g. angiogenesis, (Levine, Sleeman, Chaplain, Anderson) or tumor growth and endothelial cell movement (Owen, Byrne, Preziosi's group, Bellomo's group), or chemotaxis in the immune response (is there literature?).

5. Underlying energy prinicple: For most of the modified models (i)-(iv) there is no Lyapunov-function known, or no energy functional. Although, numerical solutions behave as if there was an energy which the solutions try to minimize (Painter, Hillen).

6. Continuation after blow-up: Some groups work on the continuation of solutions after blow-up. I don't see the point. If blow-up has happend the model has proven itself as un-useful for further description. Why continue after blow-up? Can anyone explain to me why this is important? 

thillen(at)ualberta(dot)ca