Stochastic Models of Chemotaxis Processes
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STOCHASTIC MODELS OF CHEMOTAXIS PROCESSES Ya. I. Belopolskaya∗
UDC 519.2
Probabilistic representations of weak solutions to the Cauchy problem are constructed for systems of nonlinear parabolic equations arising in chemotaxis. These equations include as a special case the Keller–Segel model. Bibliography: 12 titles.
1. Introduction Chemotaxis is the movement of cells or organisms in response to chemical stimulus. The chemotaxis plays an important role in a number of biological processes. The most popular chemotaxis model describing the evolution of concentration ρ of cells and the chemical concentration c is given by nonlinear parabolic equations system ∂ρ = 12 ∇ · (α1 (ρ, c)∇ρ − α2 (ρ, c)ρ∇c) + α3 (ρ, c), ∂t (1) ∂c = 12 ∇ · (σ 2 ∇c) + α4 (ρ, c)ρ − α5 (ρ, c)c. ∂t We are looking for a probabilistic representation of generalized solution to the Cauchy problem for this system with initial data ρ(0, x) = ρ0 (x), c(0, x) = c0 (x). Here, t ∈ [0, T ], x ∈ Rd , d ≥ 3, and x · y =
d k=1
xk yk is the inner product in Rd .
We assume that the coefficients in (1) possess the following properties: • the chemotaxis sensitivity χ = α2 (ρ, c)ρ is proportional to the cell concentration ρ; attraction and repulsion correspond to positive and negative χ, respectively; • the cell diffusion coefficient α1 (ρ, c) is a real positive function; • α2 (ρ, c) is a real function responsible for the mean increase or decrease of the cell concentration; • α4 (ρ, c) is a positive function describing the chemical influence (this function is proportional to the concentration ρ, when chemical exposure comes from cell vital functions). Finally, α5 (ρ, c) is a positive function describing the rate of chemical consumption. The aim of the present paper is to construct nonlinear Markov processes associated with (1), interpret them as a system of direct Kolmogorov equations, and derive probabilistic representations of a generalized (weak) and a mild solution of (1) in some functional classes or in a class of measures. Parabolic systems of form (1) were intensively studied by many authors in the framework of PDE theory; the corresponding results and more detail can be found in [1–3]. A specific feature of these systems is a triangular shape of the matrix of higher derivatives. In addition, treating the second equation as a nonuniform linear parabolic equation, we can compute both the solution c(t, x) and its gradient ∇c(t, x) in terms of ρ(t, x) and substitute the expression for ∇c(t, x) into the first equation. This reduces solution of (1) to solution of an equation of the McKean-Vlasov type. Probabilistic approaches to investigation of nonlinear PDE systems with cross diffusion systems developed intensively within the last decade are based mainly on treating the system under consideration as the limit of the PDE systems describing the behavior of multiparticle ∗
St.Petersburg State University of Architecture and Civil Engineering, St.Petersburg, Russia; St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia, e-mail: yana@yb1
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