Transport, reaction, and delay in mathematical biology, and the inverse problem for traveling fronts

K. P. Hadeler

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 57-77.

Résumé

Reaction-diffusion equations are the standard models for reacting and moving particles in ecology, cell biology, and other fields of Biology. In many situations, a more detailed description of the movements of particles or individuals is required. Then reaction-transport systems, reaction Cattaneo systems, and Kramers–Langevin approaches can be used. The typical limit solutions in unbounded domains are traveling fronts which pose new mathematical problems in the case of transport equations. The inverse problem for traveling fronts is a novel problem, which is investigated here in great detail. Additional features are delays, which typically lead to oscillations, and quiescent phases, which can be shown to stabilize against the onset of oscillations. In particular, neutral delay equations can be rigorously derived from first-order hyperbolic equations with appropriate boundary conditions modelling age structure. Multi-species systems lead to study various phenomena such as Turing instability, interaction of diffusion and delay, and cross diffusion.

Bibliographie

[1] Bocharov G., Hadeler K. P., “Structured population models, conservation laws, and delay equations”, J. Differential Equations, 168 (2000), 212–237 | DOI | MR | Zbl

[2] Brailovsky I., Sivashinsky G. I., “On deflagration-to-detonation transition”, Comb. Sci. and Tech., 130 (1997), 201–230 | DOI

[3] Brezis H., Kamin S., Sivashinsky G., “Initiation of subsonic detonation”, Asymptot. Anal., 24 (2000), 73–90 | MR | Zbl

[4] Dkhil F., Hadeler K. P., “Travelling fronts in pressure-driven combustion”, SIAM J. Appl. Math., 66 (2006), 1473–1481 | DOI | MR | Zbl

[5] Hadeler K. P., “Travelling fronts and free boundary value problems”, Numerical treatment of free boundary value problems (Workshop, Oberwolfach, 1980), 1982, 90–107 | Zbl

[6] Hadeler K. P., “Reaction telegraph equations and random walk systems”, Stochastic and spatial structures of dynamical systems, North-Holland, Amsterdam, 1996, 133–161 | MR | Zbl

[7] Hadeler K. P., “Nonlinear propagation in reaction transport systems”, Fields Inst. Commun., 21 (1999), 251–257 | MR | Zbl

[8] Hadeler K. P., “Reaction transport systems in biological modelling”, Lecture Notes in Math., 1714, 1999, 95–150 | MR | Zbl

[9] Hadeler K. P., “The role of migration and contact distribution in epidemic spread”, Frontiers Appl. Math., 28 (2003), 199–210 | MR

[10] Hadeler K. P., “Quiescent phases and stability” (to appear)

[11] Hadeler K. P., Coexistence of two species on a common gradient, Gotovitsya k pechati

[12] Hadeler K. P., Bocharov G., “Delays in population models and where to put them, in particular in the neutral case”, Canad. Appl. Math. Quart., 11 (2003), 159–173 | MR | Zbl

[13] Hadeler K. P., Bocharov G., Quasilinear neutral delay equations, Gotovitsya k pechati

[14] Hadeler K. P., Hillen T., “Coupled dynamics and quiescent phases”, Proc. of the Conference on Deterministic and Stochastic Modelling in Biomedicine, Economy, and Industry (Milano, 2005) (to appear)

[15] Hadeler K. P., Hillen T., Lutscher F., “The Langevin or Kramers approach to biological modeling”, Math. Models Methods Appl. Sci., 14 (2004), 1561–1583 | DOI | MR | Zbl

[16] Hadeler K. P., Lewis M. A., “Spatial dynamics of the diffusive logistic equation with a sedentary compartment”, Canad. Appl. Math. Quart., 10 (2002), 473–499 | MR | Zbl

[17] Hadeler K. P., Rothe F., “Travelling fronts in nonlinear diffusion equations”, J. Math. Biol., 2 (1975), 251–263 | DOI | MR | Zbl

[18] Hale J. K., Lunel S. M. V., Introduction to functional differential equations, Springer, New York, 1993 | MR

[19] Hess P., “Periodic-parabolic boundary value problems and positivity”, Pitman Research Notes in Math., 247 (1996)

[20] Hillen T., Hadeler K. P., “Hyperbolic systems and transport equations in Mathematical Biology”, Analysis and numerics for conservation laws, Springer, Berlin, 2005, 257–279 | MR | Zbl

[21] Hofbauer J., So J. W.-H., “Diagonal dominance and harmless off-diagonal delays”, Proc. Amer. Math. Soc., 128 (2000), 2675–2682 | DOI | MR | Zbl

[22] Mimura M., Kawasaki K., “Spatial segregation in competitive interaction-diffusion equations”, J. Math. Biol., 9 (1980), 49–64 | DOI | MR | Zbl

[23] Sánchez-Garduño F., Maini P. K., “Travelling wave phenomena in some degenerate reaction-diffusion equations”, J. Differential Equations, 117 (1995), 281–319 | DOI | MR | Zbl

[24] Schumacher K., “Travelling-front solutions for integro-differential equations. I”, J. Reine Angew. Math., 316 (1980), 54–70 | DOI | MR | Zbl

[25] Schwetlick H., “Travelling fronts for multidimensional nonlinear transport equations”, Ann. Inst. H. Poincaré Anal. Non Lineaire, 17 (2000), 523–550 | DOI | MR | Zbl

[26] Tang T., Othmer H. G., “A $G$ protein-based model of adaptation in Dictyostelium discoideum”, Math. Biosci., 120 (1994), 25–76 | DOI | Zbl

[27] Weinberger H. F., Lewis M. A., Li B., “Analysis of linear determinacy for spread in cooperative models”, J. Math. Biol., 45 (2002), 183–218 | DOI | MR | Zbl