Localization in lattice and continuum models of reinforced random walks

K e y w o r d s A g g r e g a t i o n , Lattice walks, Forward-backward parabolic, Coarsening process. 1. I N T R O D U C T I O N M o v e m e n t is a f u n d a m e n t a l process for a lmos t all biological o rganisms , rang ing f rom t h e single cell level to t h e p o p u l a t i o n level, and two m a j o r classes of models a re wide ly used to descr ibe movemen t . In s p a c e j u m p processes, m o v e m e n t is v ia a sequence of pos i t ion j u m p s a t r a n d o m t ime intervals , whi le in v e l o c i t y j u m p processes m o v e m e n t consis ts of s t ra igh t l ine m o t i o n p u n c t u a t e d by r a n d o m changes in ve loc i ty at r a n d o m t imes [1]. Space j u m p processes inc lude t h e famil iar K. J. Painter's research has been supported by SHEFC research developmental Grant 107. D. Horstmann was supported by the Deutschen Forschungsgemeinschaft (DFG) and H. G. Othmer's research has been supported by NIH-GM29123 and NSF-DMS9805494. 0893-9659/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by JtA.fS-TEX PII: S0893-9659(02)00208-2 376 K.J. PAINTER et al. lattice walks, and the simplest nearest-neighbor version in one space dimension leads to the master equation dP-'A~ = ~-1P~-1 2~p, + ~+1P,+1 (1) dt for the number density at site i. When the transition rates ~ are constant this leads to the diffusion equation for the continuum density p(x, t). A general theory of random walks with timeand space-independent waiting times and jump kernels leads to a renewal equation, and from this both the above master equation description and the continuum (PDE) limits can be derived by special choices of the kernels [1]. However, in many biological applications the transition rates depend on external fields or the number density, and the external fields may be altered by signals produced by the walkers. Here the theory is not as well developed and analysis proceeds from more specialized descriptions, because a general formulation that leads to either discrete or continuum limits is not available as yet. One tractable generalization of walks with constant transition probabilities begins with a reinforced random walk, in which a walker on a one-dimensional lattice modifies the transition probability for succeeding passages [2]. One example of this arises in the motion of gliding bacteria such as Myxococcus xanthus, which glides along slime trails it produces [3]. We suppose that the transit ion rates depend on the density of a control substance w tha t evolves according to dw~ dt = "~(P' W) , (2) where P = (Pz, . . ,Pu) and W = (wz , . . . ,wN). Continuum limits of such walks were studied in [4], where it is shown that a variety of asymptotic states are possible, ranging from blowup in finite time to collapse to a uniform distribution. Here we assume tha t ~ = Td(wi), i.e., the sensing is strictly local, which corresponds to reinforcement at the lattice sites rather than the intervals between sites. Let h be the lattice spacing and assume there is a sealing ~ ( W ) = ATe(W/h) such that limh-~0,x-~oo Ah 2 = D; then the diffusion limit of (1) is Op 0 2 0-7 = D-o~x2 ( ~ ( w ) p ) . (3) Whether this limit is legitimate for solutions tha t are not smooth was not addressed in [4]. For 7-d(U) = uK~/ (K~ + u 2) and jr = (p _ w)/e, numerical simulations indicate tha t there are only two possible asymptot ic states, blowup in finite time (or more precisely, essentially complete localization at one lattice point, since mass conservation implies tha t no finite-dimensional approximation to (3) can blow up in finite time), or convergence to a spatially-uniform solution. An example of the former is shown in Figure la. I t can be proven tha t the solution exists for a finite time for e > 0, and in Figure lb we show how the computed numerical blowup time depends on e. To understand why there appear to be at most two attractors for the evolution, we consider here the singular limit e = 0, and show tha t for any finite-dimensional approximation to (3) there are at most two stable steady states. .•200 ool io! 0.5 1 lOs [ ** g,o' t . . .~ u ,L** 1 0 4 1 0 ° 1 0 4 Figure 1. Left: The solution of (3) for e = 1, Kd = 1, and v = 1 on a lattice of 201 points. Right: The numerically-computed blowup time as a function of e. Reinforced R a n d o m Walks 377 2. A N A L Y S I S O F T H E S I N G U L A R M O D E L For the choice of Td(p) used here, (1) becomes dpi 2 2 • Kd P~-I ~ K~ p~ K~ P~+I ~ Z / z ' ~ u 2 _2 K 2 _k ~ 2 dt Kd + lJi1 d "~,i + u K~ + p~+ 1 (4) By scaling p by K d and t by u we can assume tha t K d 1 and u = 1. Let A d be the lattice Laplacian with zero Neumann boundary data; then (4) reads d P d---t = Ad(Td(P)P) ' (5) where (Td(P)P) i = Td(pi)pi. The flows defined by (5) and by (3) (with homogeneous Neumann data) both conserve the total mass. We first focus on (5), and let H be the simplex given by the intersection of the plane ~ Pi = P0 with thenonnega t ive cone of R N. Steady states of (5) are such tha t Td(PS)P 8 is proport ional to the eigenfunction of Ad belonging to the zero eigenvalue, and therefore, Td(p~)p~ = p for i = 1 , . . . , N where # is a constant. Since Td(p)p is monotone increasing for p E [0, 1) and decreasing for p E (1, oo), there are exactly two possible values for each p~, p*, and p. 1/p*, and (Td(p)py < 0 for one of these solutions. All admissible s teady states can be characterized by the fractions a) 1 and w2 = 1 wl of sites at p* and 1/p*. Admissible pairs ( p , o ) l ) , p E [0, oo), Wl : i / N , i = 1 , . . . , N satisfy w l p + W 2 _p0 M d (6) p N where M4 is the average number per site. The solutions are p+ = Md, which exists for all M~, and at most 2 (N 1) others given by P~I Md :t: ~ / M ' ~ 4WlW2 : 2021 (7") Since p ~ = 1/p~2, the admissible nonuniform solutions are obtained by pairing the branches as