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Question Are there any relations/applications of integrable system theory (take it as broadly as one can: ODE, PDE, quantum, box-ball,...) to mathematical biology (in particular pharmacokinetics, pharmacodynamics ) ? )

Background on biology

Nowdays some commercial pharma-companies builds mathematical models how the drugs behave in body. That is called pharmacokinetics, pharmacodynamics (quoting Wikipedia: "Pharmacokinetics may be simply defined as what the body does to the drug, as opposed to pharmacodynamics which may be defined as what the drug does to the body.")

There are some use of ODE and PDE in these models as well as emprical and statistical ingredients.

The narrowed context of my question - I wonder what is mathatical nature of these ODE and PDE ? Are they somehow related to integrable systems ?

Well, in general I would be happy to hear about any other applications of integrable system theory to biology/pharmacology.

Probably it is well-known that some ODE like Lotka-Volterra equation (also known as predator–prey equations) are used in biology to model the "the dynamics of biological systems in which two species interact, one a predator and one its prey." However these equations seems not to be integrable, am I wrong ?

Background on "integrable systems (=solitonic)" dynamical systems

Roughly speaking dynamical system is when point is moving in some set. The character of movement may be chaotic either "integrable", or something in the middle. Chaotic - means point is like a generator of random numbers - behaviour is easyly predictable statistically, but not individually. Integrable is opposite - point has quite regular behaviour and conservation laws confine its movement to certain subsets.

There are integrable ODE (Toda, Calogero, ...) PDE (Korteweg-de Vries, Sine-Gordon, ...) see list in Wikipedia. One can consider quantum and discrete analogs. Recent surprising discover that certain cellular automata like box-ball system are certain limits of discrite soliton systems.

Integrable system theory nowdays consumes lots of advanced mathematics. Integrablity is related to certain hidden symmetry of the system and that is why Lie, quantum groups arise in the study of integrable systems. Typically Liouville tori are Jacobians of some algebraic curves and that it is one of the ways how algebraic geometry appears.

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1 Answer 1

There are some integrable Lotka--Volterra systems, see e.g. the paper O.I. Bogoyavlenskij, Integrable Lotka--Volterra systems, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 6, pp. 543–556.

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Can someone send me this paper ? I do not have access to Springer:( Thank you very much for yours answer ! –  Alexander Chervov Apr 23 '12 at 5:57

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