- Email: [email protected]
The red queen's walk
Physica A 190 (1992) 218-237 North-Holland
The Red Queen's walk Harald Freund and Peter Grassberger Physics Department, University of Wuppertal, W-5600 Wuppertal 1, Germany Received 1 August 1992
We study a new type of walk with memory which might serve as a toy model for the behavior one must adopt to avoid exhaustion of resources and attraction of parasites and predators. The walk takes place on a regular square lattice with periodic boundary conditions. Although the walk is completely deterministic, it mimics a "true" self-avoiding walk, i.e. a random walk with weak autocorrelation. This shows that the Red Queen effect can lead to aperiodic behavior. In addition to the case of single walkers in a fiat landscape we also study the cases of hilly landscapes and of several walkers performing simultaneous walks.
I. Introduction " N o w , here, y o u see, it t a k e s all t h e r u n n i n g you can d o , to k e e p in t h e s a m e p l a c e . If y o u w a n t to get s o m e w h e r e else, y o u m u s t run at least twice as fast as that." These words of the Red Queen towards Alice have the reputation of h a v i n g " p r o b a b l y b e e n q u o t e d m o r e o f t e n [ . . . ] t h a n a n y o t h e r p a s s a g e in t h e Alice b o o k s " [1]. In p a r t i c u l a r t h e y a r e the basis for w h a t biologists call s o m e t i m e s t h e " R e d Q u e e n e f f e c t " [2,3]: a species which d o e s n o t c h a n g e a t t r a c t s p a r a s i t e s a n d p r e d a t o r s , it e x h a u s t s its r e s o u r c e s , a n d if it is itself a p r e d a t o r , its p r e y d e v e l o p s efficient d e f e n s e s t r a t e g i e s . O n e w a y to a v o i d all t h e s e d r a w b a c k s is to c o n s t a n t l y c h a n g e . This p e r m a n e n t c h a n g e m i g h t s i m p l y consist in a w a n d e r i n g o f i n d i v i d u a l s o r o f flocks in real s p a c e , b u t m o r e i n t e r e s t i n g l y it m a y also d e s c r i b e a w a n d e r i n g in t h e a b s t r a c t s p a c e o f p r o p e r t i e s a n d strat e g i e s in which e a c h s p e c i e s is o n e p o i n t , a n d e a c h q u a s i s p e c i e s a c l o u d of points. O f c o u r s e , a s i m i l a r w a n d e r i n g is e n f o r c e d also by e x o g e n i c c h a n g e s of t h e e n v i r o n m e n t , c a u s e d e.g. b y climatic c h a n g e s , v o l c a n i c activities, m e t e o r i t e i m p a c t s , etc. It is c l e a r t h a t t h e s e e x o g e n i c v a r i a t i o n s a r e a m a j o r d r i v i n g f o r c e 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
H. Freund, P. Grassberger / The Red Queen's walk
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in evolution. The most dramatic and most popularized evolutionary event, caused most likely by one of these agents, is the extinction of dinosaurs. What is much less obvious is whether there would be any evolution possible if there were no such exogenic changes at all. Early theories of the succession of species like that of Cuvier assumed that such external influences were the only driving forces. Today this radical view is no longer held by most evolutionary theorists, but it seems that the opinion is still split on how important endogenic causes are as compared to exogenic ones. While arguments for climatic influences are collected in ref. [4], Maynard Smith stated in ref. [5] that "I see no reasons why extinctions [ . . . ] should not take place in the absence of any change in the physical environment, although [ . . . ] . " The obvious candidate for endogenic evolution would be the Red Q u e e n effect. Being no experts on evolution ourselves, we do not want to enter the discussion on its importance. But even as a secondary effect, it should have a non-negligible impact, and is worth further studies. Thus it would be of great interest if we had a general mathematical theory which would allow us to estimate this influence. In the lack of such a theory, it would be of interest to have at least a mathematical model which demonstrates explicitly that the Red Q u e e n effect can lead to evolution in the absence of any randomness from the environment. Since we are not aware of any mathematical notion of "evolution" in the biological sense (i.e., increase of some sort of complexity), we cannot have such a model either. What we want to present in the rest of the paper is a class of toy models which show at least that the Red Queen e f f e c t - t a k e n rather l i t e r a l l y ! - i s able to produce nonstationary and n o n p e r i o d i c behaviour. The models are formally strictly deterministic, but the outcome is essentially a r a n d o m walk. This is similar in spirit to the emergence of chaotic behavior in formally deterministic smooth dynamical systems [6], though the details are very different. In the next section we will discuss the simplest version of the model. Variants will be presented in section 3, and a discussion will be given in section 4. Before we go on, we should say one more word about more conventional models. The usual way to formulate prey-predator relationships is via L o t k a Volterra equations. It was indeed shown that Lotka-Volterra type systems with more than two partners are able to produce chaotic motion [7]. Again this falls short of genuine evolutionary activity, but it is the best one can hope for from differential systems with few degrees of freedom, and chaotic behavior seems a necessary prerequisite to evolution. Our models have a phase space structure very different from Lotka-Volterra models, though they try to describe a basically similar mechanism. They could a priori lead to genuine evolution, which was our original motivation for studying them.
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H. Freund, P. Grassberger / The Red Queen's walk
2. A single queen in a s w a m p
We consider a walker on a finite square lattice of size L x L. B o u n d a r y conditions are periodic, i.e. a walker leaving the lattice at one side re-appears on the other side. At each site i of the lattice is attached a nonnegative real n u m b e r ( " d e p t h " ) h i. Originally, at time t = 0, we assume the landscape is fiat, h i = 0, and the walker starts at a randomly chosen site (e.g. at the origin), facing a randomly chosen direction. T h e depth h i measures the degree to which a walker in the past has spoiled the site i. Thus, while the walker stands on i, she sinks as if she would stand in a swamp, and the depth of the ground increases by a fixed amount per time unit. We can take this to be one unit of depth, i.e.
hi=hi + l.
(1)
E m p t y sites relax during one time unit with some law h~ = f ( h i ) .
(2)
T h e detailed form of the function
f(h)>-O
'
f(h)
f(h) is not important. All we need is that
forh>0
'
d f I>0 dh '
f(0)=0
(3)
and
f(h) < 1
for 0 < h ~< 2 .
(4)
Eq. (3) implies in particular that a site not visited by the walker will relax monotonically towards h = 0. Later we will c o m m e n t on what happens when eq. (4) is dropped. The dynamics of the model is fixed by the rule that the walker always takes a step to that site in her neighborhood which has the smallest depth, i.e. she wants to avoid the scars in the environment she has created herself in the past. This is done in a shortsighted way, i.e. no long-term strategy is used, and the decision is based only on the depths of the present site and its four neighbors. Equations (3), (4) imply that the walker never will stand still or m a k e a 180 degree turn, i.e. the R e d Q u e e n indeed always runs as envisaged by Lewis Carrol. For standing still to be an optimal move, we would need 1 q- h i < f ( h j )
(5)
for some pair {hi, h/}. This cannot be. Similarly, a 180 degree turn would need f(1 + hi)
H. Freund, P. Grassberger / The Red Queen's walk
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If there are two or more sites with the same smallest depth, the walker adopts a deterministic rule which site to prefer. If we restrict ourselves to isotropic (rotationally symmetric) rules, we have 3 possible rules together with their left-right exchanged partners. They are defined by the preferences: straight, left, right (rule " S L R " ) ; left, straight, right (LSR); and left, right, straight (LRS). As we shall see, the choice of a specific rule influences the behavior on small lattices and for short times, but it is irrelevant for the long time behavior on large lattices. T h e r e are a n u m b e r of easy theorems which follow immediately from the above definitions. First of all, the walk must ultimately cover the entire lattice, i.e. there cannot be any site which is never visited. Indeed, every site is visited infinitely often. For the proof, r e m e m b e r that the lattice is finite, so some sites must be visited infinitely often. Take such a site, and assume it has a neighbor j which is visited only a finite number of times. Thus, after some finite time, the walker always prefers to take another direction from this site, although the depth hj is tending towards zero, and is lower than the depths at the sites visited last time. Obviously this cannot be. This argument can indeed be much sharpened. Due to eqs. (3), (4), the ordering of the depths of the neighbors of a site i is exactly the ordering of the times of their last visits. It also determines the ordering in which the neighbors will be visited again when the queen leaves site i again, unless a neighbor has been visited in the mean time by a walk which entered site i or did not pass through site i. This means that the continuous variable h i can be replaced by a discrete "spin"-like variable which stores information about the ranking of its four neighbors. This can be done by a variable with 4! = 24 possible values, s i = 1 . . . . . 24 #1. In terms of these variables the dynamics is somewhat complicated since not only the spins on the visited sites have to change, but also the spins on their neighbors. The important result is that the spin dynamics (and thus also the walks) are independent of the detailed form of the function f(h). Thus phase space is indeed finite for this model, with N ( L ) = 24 L2 possible states, and the walk ultimately has to be periodic. For small lattices, we were indeed able to find the periodicities and the periodic walks themselves. As already said, they depended strongly on the detailed implementation, i.e. on the preference between equally high sites. The periodicities depended also very strongly on the values of L. In general no monotonic behavior was found. In table I we give estimates of the transient times on small lattices for the three rules. They were obtained by running 2 lattices simultaneously, both starting with the same fiat landscape but one running twice as fast as the other. The ~ We t h a n k A r k a d y Pikovsky for pointing this out to us. Similar models with spins being c h a n g e d by a walker have been considered by C o h e n et al. [8,9] and Bunimovich et al. [10].
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H . F r e u n d , P. Grassberger / The R e d Q u e e n ' s w a l k
Table I Estimates of transient times on small lattices for the three rules. Rule L
SLR
LRS
LSR
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
8 18 48 286 72 98 192 3608 200 242 432 1413720 392 450 768 335203407
8 18 48 150 108 816 256 1411 26580 5166 54458 338275 2622708 5289144 41426880 245621434
8 18 48 75 144 1666 2432 5146 18100 11193 456624 2503976 207600 1072802 48245600 103122840
4n
3L 2
-
-
4n+l 4n + 2 4n + 3
2L 2 2L 2
-
-
times s h o w n in table I are those at which both depth distributions coincide. It is easily seen that they are bounded by the transient time T 0 from below, and by T O + Tp from above, where Tp is the period of the periodic orbit. For rule SLR, T Oincreases only quadratically with L if L ~ 4n + 1. Otherwise, T 0 seems to increase roughly as e aL2. The constant a is definitely less than ln24, suggesting that most of the phase space is not accessible for a walk starting from a flat surface. Let us finally mention that these periodic walks in general did not visit all sites equally often (though they visited of course all sites at least once during one period), as one might naively have guessed. Thus the walk does not lead to equipartition in general. For larger values of L (and for L = 4n + 1, in case of rule SLR), transient times are so long that only the transient behavior is of practical interest. In that case, the walk starts off very regularly, and gets more and more "random" as N increases. This is illustrated in fig. 1. There, in each panel are shown 5000 steps of a walk on a lattice of size 121 x 121 for rule SLR. While there are still rather regular patches at N = 105, they have disappeared at N = 4 x 105. A walk
H. Freund, P. Grassberger / The Red Queen's walk
223
~j
(a)
1
(b)
(cl Fig. 1. A Q u e e n ' s walk on a lattice of size 121 x 121 with rule SLR. In panel (a) are shown the 5000 steps from N = 35 001 to 40 000; in panel (b) those from 220 001 to 225 000; in panel (c) those from 430 001 to 435 000.
at v e r y long time on a lattice of size 361 x 361 is shown in fig. 2. C o m p a r i n g lattices o f sizes up to L = 1000, we f o u n d that it takes typically a time T r ~- L 25 for the walk to r a n d o m i z e , for all three rules. A f t e r this, the walk has the same statistical b e h a v i o u r for all three rules. This is illustrated in fig. 3, w h e r e we show the n u m b e r of straight steps, of left turns, a n d of right turns on lattices with L = 1001, as a function of time. While the b e h a v i o r d e p e n d s on the rule for small times, it is the same for all 3 rules for N > 108. It is also the same for walks starting not on a fiat landscape, but on a
224
H. Freund, P. Grassberger / The Red Queen's walk
Fig. 2. 20 000 steps of a walk on a 361 × 361 lattice after a very long transient time.
r a n d o m landscape with h i set randomly between 0 and 1 initially (for the latter, all three rules produce identical walks). We thus conjecture that the process b e c o m e s indeed ergodic and rule independent in the limit of infinite lattices. This is also supported by all results reported below. A very conspicuous result seen in fig. 3 is that the n u m b e r of left and right turns is always the same, up to very small fluctuations. Thus while the rules b r e a k chiral invariance explicitly, chiral symmetry is restored statistically. Finally, let us discuss the statistics of the walks for N >> T r. In fig. 4 we show the distribution P ( l ) of events in which the walker makes at least l straight steps in succession. Except for the first two points we see a perfect exponential curve, indicative of short term m e m o r y . The latter would suggest a truly r a n d o m walk behavior asymptotically, Rn = (Ix,+~ 2
_ _
m
xtl2> ~ n
.
(6)
For m o r e direct tests of this, we show R n2/ n in fig. 5, and the autocorrelation c. =
with Ax, = xt+ 1
-
-
(7)
X,, in fig. 6. Both show roughly the behavior expected from
H. Freund, P. Grassberger / The Red Queen's walk
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H. Freund, P. Grassberger / The Red Queen's walk . . . .
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random walks, but not precisely. For n ~<200, the autocorrelation rather suggests a power law with exponent a ~ - 0 . 9 superimposed with very strong periodic oscillations and with strong deviations for small n. This would suggest that R~ ~ n 2 - a = n El. From fig. 5 we see that this is indeed the behavior at n ~ 100, but for very large n the curve continues to bend down, indicating that R 2 -- n asymptotically. If we try a logarithmic correction, R 2 ~ n(ln n) 0 ,
(8)
then we find /3 ~ 0 . 3 3 with rather large uncertainty (at least -+0.05, due to clearly visible deviations from eq. (8)). We see thus that asymptotically the walk is just a (deterministic!) random walk, eventually with logarithmic corrections. The latter indicate that we might sit exactly at the upper critical dimension for this problem, i.e. that the upper critical dimension is d = 2. Essentially, the last observation tells us that the tendency to avoid stepping in the latest footsteps is sufficient to restore chiral symmetry, but is not enough to give a nontrivial long-time behavior. This is in contrast to "self-avoiding walks" (SAW's) [11] where the walker is killed if she attempts to step into her own footsteps. It is similar to "true self-avoiding walks" (TSAWs) [12]. In the
H. Freund, P. Grassberger / The Red Queen's walk
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latter, the walker tries to avoid stepping into her own steps, but is not killed if this is impossible. Both of the latter are random processes. For SAWs, one has R 2 - - n 2 v with v > ½, in contrast to the present model, while for TSAWs (in 2 dimensions) R 2 - n up to logarithmic corrections. That the self-avoidance is not very strong in the present model is also seen from the distribution of recurrence times. The recurrence time t is defined as the number of steps between successive visits of the same site. In 2D random walks it is known that P(t) ~ 1/t, while P(t) is of course 0 for SAW's. In fig. 7 we show P(t) for the Q u e e n ' s walk. A p a r t from deviations at small t, we see indeed a decay ~ l / t . Thus the tendency to avoid the own track is sufficient to randomize the walk, but is not enough to actually avoid self-crossings.
3. Extensions and generalizations
3.1. Dynamic randomness All walks in the last section were strictly deterministic. We considered also probabilistic versions where the above rules in each step are obeyed with probability 1 - p, while a random step is taken with probability p. The result is
H. Freund, P. Grassberger / The Red Queen's walk
229
as expected: for all three rules, the walks became random on large lattices and at times much larger than L 2. Moreover, its statistical properties changed continuously for small p and were in the limit p--~ 0 those of the nonrandom models. For rule SLR the latter applies of course only to lattices with L = 4n + 1, and shows that the nonrandom behavior on square lattices of other sizes is atypical and unstable. Stability of the pseudorandom behavior against truly random noise was also observed in the modified models studied below.
3.2. Infinite lattices U p to now we have considered the case of a finite lattice with fiat initial conditions. For an infinite and originally flat lattice, the problem would of course be trivial. But we can also consider an infinite lattice with disordered initial conditions. If the walker would not modify the landscape, the walk would be r a n d o m and thus recurrent in 2 dimensions. The fact that the traces left by the steps also produce a random walk suggest that for the actual outcome we have R ~ - N. This was indeed verified in simulations. In fig. 8 we show R~/N as obtained
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Fig. 8. Log-log plot of random mean square end-to-end displacements for short walks on random landscapes. The lattice is sufficiently large that no wrap-around happens.
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H. Freund, P. Grassberger / The Red Queen's walk
by 3 × 10 6 walks, each starting on a newly randomized landscape. We see that the walk is even more inflated than the walks in the previous section (see fig. 5), and it seems that again R ~ - - N ( l n N ) ~ asymptotically, this time with /3 ~ 0.4 (again with rather larger uncertainty).
3.3. Slow relaxation As we have pointed out, our previous results did not depend on the specific function f(h), provided they satisfied eq. (4). This guaranteed that the walk is governed by the times of last visits, and no accumulation of depth over successive visits has to be taken into account. This would no longer be true if relaxation is sufficiently slow. Actually, we found that eq. (4) can be replaced by considerably less stringent conditions without changing the result at all. We have worked out more generous sufficient conditions such that the walks are exactly the same as in the previous section. But they seem far from the necessary conditions for that, and we have not found a general method to obtain the necessary conditions. Thus we do not want to give any details. If relaxation is still sufficiently fast, then the walk will occasionally differ from that obtained in the last section, but the long time statistics and the qualitative behavior will be unchanged. This is no longer true for very slow relaxation, and in the limit of no relaxation at all. In the rest of this subsection we shall only discuss the latter. In the latter case, the walk is determined only by the number of previous visits, not by their times: it continues towards that neighbor which has been visited least often before. Also 180 degree turns and standstills are now possible unless they are explicitly forbidden, and chirality is not restored statistically. In the following we shall assume that backsteps and standstills are forbidden, in order to avoid proliferation of the number of rules. For the rules SLR and L S R we found that the walk becomes ordered and periodic on all quadratic lattices with flat initial conditions. It becomes disordered with very long and slow transients on lattices with small disorder (e.g., a single site with h i ~ 0). For rule LRS, in contrast, the walk becomes quickly random even on initially flat landscapes, and the landscape becomes a random surface. In fig. 9 we show R~ in the stationary state, and in fig. 10 we show the variance of the height distribution. For the latter, we have taken an average over times which were integer multiples of the lattice size L2 in order to avoid effects of partially filled layers. We see that again the mean displacement is somewhat larger than in a purely random walk but still asymptotically --~v~-~up to logarithmic corrections (/3 =0.61). Naively, one might take this as an
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H. Freund, P. Grassberger / The Red Queen's walk
indication that the landscape in the long time limit should become a random surface, but this is definitely not the case. As seen from fig. 10, the variance of the height distribution increases only like (log L ) 11 instead of --L. Qualitatively similar results were obtained with rules where backsteps are allowed. Notice that our models without relaxation are essentially deterministic versions of the "true self-avoiding walk" problem [12]. Also for the latter it is conjectured that R2n- n and 8h-~ const, up to logarithmic corrections [13]. 3.4. Hilly landscapes Up to now, the landscape relaxed always towards the flat landscape h i = 0. Instead of that, we now consider the case where each site i has a different function f~(h) with a fixed point h*. Actually we take f
(h) = f ( h - h * ) + h*
(9)
with f ( h ) as before. Relaxation now leads to a hilly landscape h i = h*. The walker still wants to reach the highest point. This model might be indeed biologically more interesting than the model of section 2, as it describes the Red Q u e e n effect in a situation where not all states are a priori equally good. With fixed depths, it would be just greedy optimization where the walker goes towards the nearest local maximum and stays there. The 'swampiness' of the landscape prevents the latter, in particular if relaxation is slow. Put differently, the hilltops of the landscape represent niches which can be occupied by an immobile species only for a finite time, but which can be re-visited (and will be good niches) later. If the niche is wide and deep enough, the walker will not be expelled from it completely, but will be able to survive in some of its parts until the other parts have relaxed. Thus we expect transitions between localized walks to delocalized ones, as we make the landscape flatter a n d / o r narrower. In the delocalized regime, typical walks will be intermittent since the walker approaches a local maximum, stays in its neighborhood until it is too much spoiled, and then moves on to some other local maximum. Both in the localized and delocalized regimes walks will not be periodic, unless the niches are very small and deep. Details depend not only on the properties of the surface {h*), but also on the speed of relaxation. In fig. 11 we show the x coordinate of a walk in a double well landscape h *(x, y) = a sin(2"rrx/L ) s i n ( 2 ~ y / L ) with a = 0.05, L = 100, and with
(10)
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H. Freund, P. Grassberger / The Red Queen's walk 100
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h f(h)-
1+ h "
(11)
We see indeed the expected intermittency (the delocalization threshold is at a = 0.085). On a surface with niches (or hilltops) of varying heights, this could lead to anomalous diffusion due to a wide distribution of trapping times. Finally, we tried to decide whether the transition between localization and delocalization in a single niche is sharp. If it resembles the thermally activated escape from a potential well, the escape rate would never be exactly zero, but would decrease exponentially with the depth of the well. On the other hand, if it is similar to a crisis [14] in a smooth dynamical system, then the transition would be sharp. Even very long runs (5 × 108 steps) were not able to decide this. If there is a sharp transition, then the escape rate rises very slowly, with a power (a - a c r ) ~' with y > 1. 3.5. Several walkers
As a last modification, we let several walkers walk simultaneously on a large but finite and initially flat landscape.
H. Freund, P. Grassberger / The Red Queen's walk
234
There are two possibilities to proceed. In the first, all walkers decide simultaneously where to go and perform the actual walking step after that, again simultaneously. This can easily be implemented on a massively parallel computer like the Connection Machine. It leads to a coalescence of walks, because all walkers will continue with identical paths as soon as they have chosen once the same site. Another possibility is to make the decisions and steps not simultaneously but one after the other. So if one walker has chosen a site, the next walker already sees the depression and avoids this site. All results stated below were obtained with this rule. Since walkers try to avoid the scars made in the swamp by previous steps, they feel effectively a repulsion which leads to negative correlations. In spite of the smallness of the effect of the (self-)repulsion on R, (fig. 5), this repulsive correlation is surprisingly strong. In fig. 12 we show the normalized two-point density correlation C(r) as a function of the Manhattan distance, for 8 different densities of walkers. Both the strength and the range of the correlation decrease with the number N of walkers, and are largest for N = 2. One might
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H. Freund, P. Grassberger / The Red Queen's walk
235
have guessed that the range of the correlation is given by the average nearest-neighbor distance, but this is not so. As fig. 12 shows, the increase with N is much slower than 1/X/-N, and is clearly visible (for L = 201) only for N > 10. Except for very high densities, there is a region of r where C(r) is linear in log r, with density independent slope. An amusing effect is the local maximum of the correlation at Manhattan distance 2 which is clearly visible in fig. 12. It is caused by pairs with ~x = -+ 1 and ~y = _1. Watching the evolution on-screen suggests that such diagonally placed pairs have a tendency to stay together for several time steps. In section 2 we had observed that chiral symmetry was restored statistically although it was broken explicitly by each rule. With several walkers, this needs not be true any more. But we found that it was again restored statistically. In addition, back steps are now chosen with a probability proportional to the third power of the walker's density.
4. Discussion
The walks studied in this paper have some similarities to random walks with m e m o r y studied during the last years in the physics literature. Examples are self-avoiding walks [11], 'true' self-avoiding walks [12], and Laplacian walks [15] (for a review, see ref. [13]). In contrast to our models, all these models are however stochastic processes, i.e. the walks are r a n d o m walks. In the present paper, we were mainly interested to see whether a weak form of self-avoidance can lead to random or even to structurally complex behavior in formally deterministic systems. We found that this is indeed the case. Several variants of the model lead to walks whose end-to-end distance scales as x/~. In all cases there were strong corrections to this which were compatible with being logarithmic. This suggests that d = 2 is the upper critical dimension. In this respect, our walks are like true self-avoiding walks. Our models are also similar to the flipping mirror models of refs. [8-10]. These models are deterministic like the present ones, but they take place in a landscape with already disordered itlitial conditions. The randomness of the trajectories in these models is a straightforward consequence of the randomness in the initial conditions. Also, the modification of the landscape does not lead to self-repulsion in these models. While we have interpreted our walks as metaphors for wanderings in an abstract space of phenotypic features, we can also interpret it more realistically but less excitingly as real walks of a foraging animal. Thus our results show that shortsighted search for the least foraged spots in an otherwise uniform field can lead to erratic wanderings, even if a deterministic strategy is used. This should
236
H. Freund, P. Grassberger / The Red Queen's walk
not be considered as a drawback of the shortsightedness of the walk (alternatively, the animal could also look further ahead, and could go to the optimal place in a larger neighborhood). Indeed, with more sophisticated strategies one runs the risk of encountering less small-scale erratic wanderings, but more large-scale disasters. For examples of chaotic behavior in computer networks which becomes worse with more sophisticated strategies, see ref. [16]. As we have already said in the introduction, the very fact that a system not perturbed by the outside can show aperiodic and unpredictable behaviour, is not new anymore. That this applies also to biological systems should be obvious, but it seems that its consequences are not always fully appreciated. Our very modeling ansatz seems new, but otherwise both the possibility of spontaneous chaotic behavior and its importance for evolution have been pointed out repeatedly. The best known models which can lead to chaotic behavior are n-species Lotka-Volterra systems with n 1>3 [7], or 2-species L o t k a - V o l t e r r a systems with spatial degrees of freedom [17]. Of some relevance in this respect are also the findings of Lindgren [18] that the iterated prisoner's dilemma leads to ever improving (and thus changing) strategies, and the instabilities in stochastic co-evolving systems found in ref. [19]. In connection with the need for chaotic behavior in ecologies, we might quote the concept of homeochaos introduced in ref. [20]. W h e n we started this work, we hoped to find not only random but also complex behavior. By the latter we mean behavior with structural complexity, in contrast to algorithmic complexity as defined by Kolmogorov, Chaitin and others [21,22]. In systems where only some sort of average behavior is interesting, the latter is not a very useful complexity measure as it cannot distinguish between randomness and structural complexity [23]. The best known class of models which can lead to such complexity are cellular automata [24]. Although our models can be understood as cellular automata with very sparse action, we did not find any really complex behavior. We should however point out that such behavior can be hidden rather subtly [25], and we might have just missed it. Also, the logarithmic effects might suggest that we are at the border to complex behavior. Finally, in the limit of infinitely slow relaxation, our model leads to a new (class of) surface growth model(s). Previously, models for surface growth [26] included unbiased growth at all surface sites (Eden growth), preferred growth at exposed tips (diffusion limited aggregation), and growth at temporally active sites which inherit their activity from their neighbors (epidemic models). All these processes lead to surfaces whose height fluctuations increase like some power of the surface area. In contrast, our models lead to height fluctuations which increase only logarithmically. Whether this has applications in solid state physics remains to be seen.
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237
Acknowledgements W e t h a n k A r k a d y P i k o v s k y f o r m a n y u s e f u l d i s c u s s i o n s a n d for his i n t e r e s t i n t h i s w o r k . T h a n k s go also to R o b e r t W o l t e r for h e l p w i t h g r a p h i c a l o u t p u t , a n d to D r s . J. D u a r t e a n d H . N a k a n i s h i for i n f o r m a t i o n o n T S A W s . T h i s w o r k w a s s u p p o r t e d b y t h e V o l k s w a g e n - S t i f t u n g , c o n t r a c t A Z 1/66 995.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
M. Gardner/L. Carrol, The Annotated Alice (Clarkson N. Potter, New York, 1960) p. 210. L. Van Valen, Evol. Theor. 1 (1973) 1. J. Maynard Smith, The Evolution of Sex (Cambridge Univ. Press, Cambridge, 1978). R. Pearson, Climate and Evolution (Academic Press, London, 1978). J. Maynard Smith, The causes of extinction, in: Evolution and Extinction, W.G. Chaloner et al., eds. (The Royal Society, London, 1989). P. Berg6, Y. Pomeau and Ch. Vidal, Order within Chaos (Wiley, New York, 1986). J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems (Cambridge Univ. Press, Cambridge, 1988). Th. W. Ruijgrok and E.G.D. Cohen, Phys. Lett. A 133 (1988) 415. X.P. Kong and E.G.D. Cohen, J. Stat. Phys. 62 (1991) 737. L. Bunimovich and S.E. Troubetzkoy, J. Stat. Phys. 67 (1992) 289. P.G. deGennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1979). D. Amit, G. Parisi and L. Peliti, Phys. Rev. B 27 (1983) 1635. L. Peliti, Random walks with memory, in: Fractals in Physics, L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986). C. Grebogi, E. Ott and J.A. Yorke, Physica D 7 (1983) 181. J.W. Lyklema and C. Evertsz, The Laplacian random walk, in: Fractals in Physics, L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986). B.A. Huberman and T. Hogg, in: Ecology of Computation (North-Holland, Amsterdam, 1988) p. 77. J. Zhuo, G. Murthy and S. Redner, preprint (1991). K. Lindgren, Evolutionary phenomena in simple dynamics, in: Artificial Life II, C.G. Langton et al. eds. (Addison-Wesley, Redwood City, 1992). H. Freund and R. Wolter, to be published. K. Kaneko and T. Ikegami, Physica D 56 (1992) 406. A.N. Kolmogorov, Problems Inform. Theor. 1 (1959) 2. G.J. Chaitin, Towards a mathematical definition of 'life', in: The Maximum Entropy Principle, R.D. Levine and M. Tribus, eds. (MIT Press, Cambridge, MA, 1979). P. Grassberger, Randomness, information, and complexity, in: Proc. 5th Mexican Summer School on Stat. Phys., F. Ramos-Gomez, ed. (World Scientific, Singapore, 1991). S. Wolfram, ed., Theory and Applications of Cellular Automata (World Scientific, Singapore, 1986). P. Grassberger, J. Stat. Phys. 45 (1986) 27. T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989).
The Red Queen's walk Harald Freund and Peter Grassberger Physics Department, University of Wuppertal, W-5600 Wuppertal 1, Germany Received 1 August 1992
We study a new type of walk with memory which might serve as a toy model for the behavior one must adopt to avoid exhaustion of resources and attraction of parasites and predators. The walk takes place on a regular square lattice with periodic boundary conditions. Although the walk is completely deterministic, it mimics a "true" self-avoiding walk, i.e. a random walk with weak autocorrelation. This shows that the Red Queen effect can lead to aperiodic behavior. In addition to the case of single walkers in a fiat landscape we also study the cases of hilly landscapes and of several walkers performing simultaneous walks.
I. Introduction " N o w , here, y o u see, it t a k e s all t h e r u n n i n g you can d o , to k e e p in t h e s a m e p l a c e . If y o u w a n t to get s o m e w h e r e else, y o u m u s t run at least twice as fast as that." These words of the Red Queen towards Alice have the reputation of h a v i n g " p r o b a b l y b e e n q u o t e d m o r e o f t e n [ . . . ] t h a n a n y o t h e r p a s s a g e in t h e Alice b o o k s " [1]. In p a r t i c u l a r t h e y a r e the basis for w h a t biologists call s o m e t i m e s t h e " R e d Q u e e n e f f e c t " [2,3]: a species which d o e s n o t c h a n g e a t t r a c t s p a r a s i t e s a n d p r e d a t o r s , it e x h a u s t s its r e s o u r c e s , a n d if it is itself a p r e d a t o r , its p r e y d e v e l o p s efficient d e f e n s e s t r a t e g i e s . O n e w a y to a v o i d all t h e s e d r a w b a c k s is to c o n s t a n t l y c h a n g e . This p e r m a n e n t c h a n g e m i g h t s i m p l y consist in a w a n d e r i n g o f i n d i v i d u a l s o r o f flocks in real s p a c e , b u t m o r e i n t e r e s t i n g l y it m a y also d e s c r i b e a w a n d e r i n g in t h e a b s t r a c t s p a c e o f p r o p e r t i e s a n d strat e g i e s in which e a c h s p e c i e s is o n e p o i n t , a n d e a c h q u a s i s p e c i e s a c l o u d of points. O f c o u r s e , a s i m i l a r w a n d e r i n g is e n f o r c e d also by e x o g e n i c c h a n g e s of t h e e n v i r o n m e n t , c a u s e d e.g. b y climatic c h a n g e s , v o l c a n i c activities, m e t e o r i t e i m p a c t s , etc. It is c l e a r t h a t t h e s e e x o g e n i c v a r i a t i o n s a r e a m a j o r d r i v i n g f o r c e 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
H. Freund, P. Grassberger / The Red Queen's walk
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in evolution. The most dramatic and most popularized evolutionary event, caused most likely by one of these agents, is the extinction of dinosaurs. What is much less obvious is whether there would be any evolution possible if there were no such exogenic changes at all. Early theories of the succession of species like that of Cuvier assumed that such external influences were the only driving forces. Today this radical view is no longer held by most evolutionary theorists, but it seems that the opinion is still split on how important endogenic causes are as compared to exogenic ones. While arguments for climatic influences are collected in ref. [4], Maynard Smith stated in ref. [5] that "I see no reasons why extinctions [ . . . ] should not take place in the absence of any change in the physical environment, although [ . . . ] . " The obvious candidate for endogenic evolution would be the Red Q u e e n effect. Being no experts on evolution ourselves, we do not want to enter the discussion on its importance. But even as a secondary effect, it should have a non-negligible impact, and is worth further studies. Thus it would be of great interest if we had a general mathematical theory which would allow us to estimate this influence. In the lack of such a theory, it would be of interest to have at least a mathematical model which demonstrates explicitly that the Red Q u e e n effect can lead to evolution in the absence of any randomness from the environment. Since we are not aware of any mathematical notion of "evolution" in the biological sense (i.e., increase of some sort of complexity), we cannot have such a model either. What we want to present in the rest of the paper is a class of toy models which show at least that the Red Queen e f f e c t - t a k e n rather l i t e r a l l y ! - i s able to produce nonstationary and n o n p e r i o d i c behaviour. The models are formally strictly deterministic, but the outcome is essentially a r a n d o m walk. This is similar in spirit to the emergence of chaotic behavior in formally deterministic smooth dynamical systems [6], though the details are very different. In the next section we will discuss the simplest version of the model. Variants will be presented in section 3, and a discussion will be given in section 4. Before we go on, we should say one more word about more conventional models. The usual way to formulate prey-predator relationships is via L o t k a Volterra equations. It was indeed shown that Lotka-Volterra type systems with more than two partners are able to produce chaotic motion [7]. Again this falls short of genuine evolutionary activity, but it is the best one can hope for from differential systems with few degrees of freedom, and chaotic behavior seems a necessary prerequisite to evolution. Our models have a phase space structure very different from Lotka-Volterra models, though they try to describe a basically similar mechanism. They could a priori lead to genuine evolution, which was our original motivation for studying them.
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2. A single queen in a s w a m p
We consider a walker on a finite square lattice of size L x L. B o u n d a r y conditions are periodic, i.e. a walker leaving the lattice at one side re-appears on the other side. At each site i of the lattice is attached a nonnegative real n u m b e r ( " d e p t h " ) h i. Originally, at time t = 0, we assume the landscape is fiat, h i = 0, and the walker starts at a randomly chosen site (e.g. at the origin), facing a randomly chosen direction. T h e depth h i measures the degree to which a walker in the past has spoiled the site i. Thus, while the walker stands on i, she sinks as if she would stand in a swamp, and the depth of the ground increases by a fixed amount per time unit. We can take this to be one unit of depth, i.e.
hi=hi + l.
(1)
E m p t y sites relax during one time unit with some law h~ = f ( h i ) .
(2)
T h e detailed form of the function
f(h)>-O
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f(h)
f(h) is not important. All we need is that
forh>0
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f(0)=0
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and
f(h) < 1
for 0 < h ~< 2 .
(4)
Eq. (3) implies in particular that a site not visited by the walker will relax monotonically towards h = 0. Later we will c o m m e n t on what happens when eq. (4) is dropped. The dynamics of the model is fixed by the rule that the walker always takes a step to that site in her neighborhood which has the smallest depth, i.e. she wants to avoid the scars in the environment she has created herself in the past. This is done in a shortsighted way, i.e. no long-term strategy is used, and the decision is based only on the depths of the present site and its four neighbors. Equations (3), (4) imply that the walker never will stand still or m a k e a 180 degree turn, i.e. the R e d Q u e e n indeed always runs as envisaged by Lewis Carrol. For standing still to be an optimal move, we would need 1 q- h i < f ( h j )
(5)
for some pair {hi, h/}. This cannot be. Similarly, a 180 degree turn would need f(1 + hi)
H. Freund, P. Grassberger / The Red Queen's walk
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If there are two or more sites with the same smallest depth, the walker adopts a deterministic rule which site to prefer. If we restrict ourselves to isotropic (rotationally symmetric) rules, we have 3 possible rules together with their left-right exchanged partners. They are defined by the preferences: straight, left, right (rule " S L R " ) ; left, straight, right (LSR); and left, right, straight (LRS). As we shall see, the choice of a specific rule influences the behavior on small lattices and for short times, but it is irrelevant for the long time behavior on large lattices. T h e r e are a n u m b e r of easy theorems which follow immediately from the above definitions. First of all, the walk must ultimately cover the entire lattice, i.e. there cannot be any site which is never visited. Indeed, every site is visited infinitely often. For the proof, r e m e m b e r that the lattice is finite, so some sites must be visited infinitely often. Take such a site, and assume it has a neighbor j which is visited only a finite number of times. Thus, after some finite time, the walker always prefers to take another direction from this site, although the depth hj is tending towards zero, and is lower than the depths at the sites visited last time. Obviously this cannot be. This argument can indeed be much sharpened. Due to eqs. (3), (4), the ordering of the depths of the neighbors of a site i is exactly the ordering of the times of their last visits. It also determines the ordering in which the neighbors will be visited again when the queen leaves site i again, unless a neighbor has been visited in the mean time by a walk which entered site i or did not pass through site i. This means that the continuous variable h i can be replaced by a discrete "spin"-like variable which stores information about the ranking of its four neighbors. This can be done by a variable with 4! = 24 possible values, s i = 1 . . . . . 24 #1. In terms of these variables the dynamics is somewhat complicated since not only the spins on the visited sites have to change, but also the spins on their neighbors. The important result is that the spin dynamics (and thus also the walks) are independent of the detailed form of the function f(h). Thus phase space is indeed finite for this model, with N ( L ) = 24 L2 possible states, and the walk ultimately has to be periodic. For small lattices, we were indeed able to find the periodicities and the periodic walks themselves. As already said, they depended strongly on the detailed implementation, i.e. on the preference between equally high sites. The periodicities depended also very strongly on the values of L. In general no monotonic behavior was found. In table I we give estimates of the transient times on small lattices for the three rules. They were obtained by running 2 lattices simultaneously, both starting with the same fiat landscape but one running twice as fast as the other. The ~ We t h a n k A r k a d y Pikovsky for pointing this out to us. Similar models with spins being c h a n g e d by a walker have been considered by C o h e n et al. [8,9] and Bunimovich et al. [10].
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H . F r e u n d , P. Grassberger / The R e d Q u e e n ' s w a l k
Table I Estimates of transient times on small lattices for the three rules. Rule L
SLR
LRS
LSR
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
8 18 48 286 72 98 192 3608 200 242 432 1413720 392 450 768 335203407
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8 18 48 75 144 1666 2432 5146 18100 11193 456624 2503976 207600 1072802 48245600 103122840
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times s h o w n in table I are those at which both depth distributions coincide. It is easily seen that they are bounded by the transient time T 0 from below, and by T O + Tp from above, where Tp is the period of the periodic orbit. For rule SLR, T Oincreases only quadratically with L if L ~ 4n + 1. Otherwise, T 0 seems to increase roughly as e aL2. The constant a is definitely less than ln24, suggesting that most of the phase space is not accessible for a walk starting from a flat surface. Let us finally mention that these periodic walks in general did not visit all sites equally often (though they visited of course all sites at least once during one period), as one might naively have guessed. Thus the walk does not lead to equipartition in general. For larger values of L (and for L = 4n + 1, in case of rule SLR), transient times are so long that only the transient behavior is of practical interest. In that case, the walk starts off very regularly, and gets more and more "random" as N increases. This is illustrated in fig. 1. There, in each panel are shown 5000 steps of a walk on a lattice of size 121 x 121 for rule SLR. While there are still rather regular patches at N = 105, they have disappeared at N = 4 x 105. A walk
H. Freund, P. Grassberger / The Red Queen's walk
223
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at v e r y long time on a lattice of size 361 x 361 is shown in fig. 2. C o m p a r i n g lattices o f sizes up to L = 1000, we f o u n d that it takes typically a time T r ~- L 25 for the walk to r a n d o m i z e , for all three rules. A f t e r this, the walk has the same statistical b e h a v i o u r for all three rules. This is illustrated in fig. 3, w h e r e we show the n u m b e r of straight steps, of left turns, a n d of right turns on lattices with L = 1001, as a function of time. While the b e h a v i o r d e p e n d s on the rule for small times, it is the same for all 3 rules for N > 108. It is also the same for walks starting not on a fiat landscape, but on a
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H. Freund, P. Grassberger / The Red Queen's walk
Fig. 2. 20 000 steps of a walk on a 361 × 361 lattice after a very long transient time.
r a n d o m landscape with h i set randomly between 0 and 1 initially (for the latter, all three rules produce identical walks). We thus conjecture that the process b e c o m e s indeed ergodic and rule independent in the limit of infinite lattices. This is also supported by all results reported below. A very conspicuous result seen in fig. 3 is that the n u m b e r of left and right turns is always the same, up to very small fluctuations. Thus while the rules b r e a k chiral invariance explicitly, chiral symmetry is restored statistically. Finally, let us discuss the statistics of the walks for N >> T r. In fig. 4 we show the distribution P ( l ) of events in which the walker makes at least l straight steps in succession. Except for the first two points we see a perfect exponential curve, indicative of short term m e m o r y . The latter would suggest a truly r a n d o m walk behavior asymptotically, Rn = (Ix,+~ 2
_ _
m
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For m o r e direct tests of this, we show R n2/ n in fig. 5, and the autocorrelation c. =
with Ax, = xt+ 1
-
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X,, in fig. 6. Both show roughly the behavior expected from
H. Freund, P. Grassberger / The Red Queen's walk
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H. Freund, P. Grassberger / The Red Queen's walk . . . .
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random walks, but not precisely. For n ~<200, the autocorrelation rather suggests a power law with exponent a ~ - 0 . 9 superimposed with very strong periodic oscillations and with strong deviations for small n. This would suggest that R~ ~ n 2 - a = n El. From fig. 5 we see that this is indeed the behavior at n ~ 100, but for very large n the curve continues to bend down, indicating that R 2 -- n asymptotically. If we try a logarithmic correction, R 2 ~ n(ln n) 0 ,
(8)
then we find /3 ~ 0 . 3 3 with rather large uncertainty (at least -+0.05, due to clearly visible deviations from eq. (8)). We see thus that asymptotically the walk is just a (deterministic!) random walk, eventually with logarithmic corrections. The latter indicate that we might sit exactly at the upper critical dimension for this problem, i.e. that the upper critical dimension is d = 2. Essentially, the last observation tells us that the tendency to avoid stepping in the latest footsteps is sufficient to restore chiral symmetry, but is not enough to give a nontrivial long-time behavior. This is in contrast to "self-avoiding walks" (SAW's) [11] where the walker is killed if she attempts to step into her own footsteps. It is similar to "true self-avoiding walks" (TSAWs) [12]. In the
H. Freund, P. Grassberger / The Red Queen's walk
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latter, the walker tries to avoid stepping into her own steps, but is not killed if this is impossible. Both of the latter are random processes. For SAWs, one has R 2 - - n 2 v with v > ½, in contrast to the present model, while for TSAWs (in 2 dimensions) R 2 - n up to logarithmic corrections. That the self-avoidance is not very strong in the present model is also seen from the distribution of recurrence times. The recurrence time t is defined as the number of steps between successive visits of the same site. In 2D random walks it is known that P(t) ~ 1/t, while P(t) is of course 0 for SAW's. In fig. 7 we show P(t) for the Q u e e n ' s walk. A p a r t from deviations at small t, we see indeed a decay ~ l / t . Thus the tendency to avoid the own track is sufficient to randomize the walk, but is not enough to actually avoid self-crossings.
3. Extensions and generalizations
3.1. Dynamic randomness All walks in the last section were strictly deterministic. We considered also probabilistic versions where the above rules in each step are obeyed with probability 1 - p, while a random step is taken with probability p. The result is
H. Freund, P. Grassberger / The Red Queen's walk
229
as expected: for all three rules, the walks became random on large lattices and at times much larger than L 2. Moreover, its statistical properties changed continuously for small p and were in the limit p--~ 0 those of the nonrandom models. For rule SLR the latter applies of course only to lattices with L = 4n + 1, and shows that the nonrandom behavior on square lattices of other sizes is atypical and unstable. Stability of the pseudorandom behavior against truly random noise was also observed in the modified models studied below.
3.2. Infinite lattices U p to now we have considered the case of a finite lattice with fiat initial conditions. For an infinite and originally flat lattice, the problem would of course be trivial. But we can also consider an infinite lattice with disordered initial conditions. If the walker would not modify the landscape, the walk would be r a n d o m and thus recurrent in 2 dimensions. The fact that the traces left by the steps also produce a random walk suggest that for the actual outcome we have R ~ - N. This was indeed verified in simulations. In fig. 8 we show R~/N as obtained
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by 3 × 10 6 walks, each starting on a newly randomized landscape. We see that the walk is even more inflated than the walks in the previous section (see fig. 5), and it seems that again R ~ - - N ( l n N ) ~ asymptotically, this time with /3 ~ 0.4 (again with rather larger uncertainty).
3.3. Slow relaxation As we have pointed out, our previous results did not depend on the specific function f(h), provided they satisfied eq. (4). This guaranteed that the walk is governed by the times of last visits, and no accumulation of depth over successive visits has to be taken into account. This would no longer be true if relaxation is sufficiently slow. Actually, we found that eq. (4) can be replaced by considerably less stringent conditions without changing the result at all. We have worked out more generous sufficient conditions such that the walks are exactly the same as in the previous section. But they seem far from the necessary conditions for that, and we have not found a general method to obtain the necessary conditions. Thus we do not want to give any details. If relaxation is still sufficiently fast, then the walk will occasionally differ from that obtained in the last section, but the long time statistics and the qualitative behavior will be unchanged. This is no longer true for very slow relaxation, and in the limit of no relaxation at all. In the rest of this subsection we shall only discuss the latter. In the latter case, the walk is determined only by the number of previous visits, not by their times: it continues towards that neighbor which has been visited least often before. Also 180 degree turns and standstills are now possible unless they are explicitly forbidden, and chirality is not restored statistically. In the following we shall assume that backsteps and standstills are forbidden, in order to avoid proliferation of the number of rules. For the rules SLR and L S R we found that the walk becomes ordered and periodic on all quadratic lattices with flat initial conditions. It becomes disordered with very long and slow transients on lattices with small disorder (e.g., a single site with h i ~ 0). For rule LRS, in contrast, the walk becomes quickly random even on initially flat landscapes, and the landscape becomes a random surface. In fig. 9 we show R~ in the stationary state, and in fig. 10 we show the variance of the height distribution. For the latter, we have taken an average over times which were integer multiples of the lattice size L2 in order to avoid effects of partially filled layers. We see that again the mean displacement is somewhat larger than in a purely random walk but still asymptotically --~v~-~up to logarithmic corrections (/3 =0.61). Naively, one might take this as an
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H. Freund, P. Grassberger / The Red Queen's walk
indication that the landscape in the long time limit should become a random surface, but this is definitely not the case. As seen from fig. 10, the variance of the height distribution increases only like (log L ) 11 instead of --L. Qualitatively similar results were obtained with rules where backsteps are allowed. Notice that our models without relaxation are essentially deterministic versions of the "true self-avoiding walk" problem [12]. Also for the latter it is conjectured that R2n- n and 8h-~ const, up to logarithmic corrections [13]. 3.4. Hilly landscapes Up to now, the landscape relaxed always towards the flat landscape h i = 0. Instead of that, we now consider the case where each site i has a different function f~(h) with a fixed point h*. Actually we take f
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We see indeed the expected intermittency (the delocalization threshold is at a = 0.085). On a surface with niches (or hilltops) of varying heights, this could lead to anomalous diffusion due to a wide distribution of trapping times. Finally, we tried to decide whether the transition between localization and delocalization in a single niche is sharp. If it resembles the thermally activated escape from a potential well, the escape rate would never be exactly zero, but would decrease exponentially with the depth of the well. On the other hand, if it is similar to a crisis [14] in a smooth dynamical system, then the transition would be sharp. Even very long runs (5 × 108 steps) were not able to decide this. If there is a sharp transition, then the escape rate rises very slowly, with a power (a - a c r ) ~' with y > 1. 3.5. Several walkers
As a last modification, we let several walkers walk simultaneously on a large but finite and initially flat landscape.
H. Freund, P. Grassberger / The Red Queen's walk
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There are two possibilities to proceed. In the first, all walkers decide simultaneously where to go and perform the actual walking step after that, again simultaneously. This can easily be implemented on a massively parallel computer like the Connection Machine. It leads to a coalescence of walks, because all walkers will continue with identical paths as soon as they have chosen once the same site. Another possibility is to make the decisions and steps not simultaneously but one after the other. So if one walker has chosen a site, the next walker already sees the depression and avoids this site. All results stated below were obtained with this rule. Since walkers try to avoid the scars made in the swamp by previous steps, they feel effectively a repulsion which leads to negative correlations. In spite of the smallness of the effect of the (self-)repulsion on R, (fig. 5), this repulsive correlation is surprisingly strong. In fig. 12 we show the normalized two-point density correlation C(r) as a function of the Manhattan distance, for 8 different densities of walkers. Both the strength and the range of the correlation decrease with the number N of walkers, and are largest for N = 2. One might
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H. Freund, P. Grassberger / The Red Queen's walk
235
have guessed that the range of the correlation is given by the average nearest-neighbor distance, but this is not so. As fig. 12 shows, the increase with N is much slower than 1/X/-N, and is clearly visible (for L = 201) only for N > 10. Except for very high densities, there is a region of r where C(r) is linear in log r, with density independent slope. An amusing effect is the local maximum of the correlation at Manhattan distance 2 which is clearly visible in fig. 12. It is caused by pairs with ~x = -+ 1 and ~y = _1. Watching the evolution on-screen suggests that such diagonally placed pairs have a tendency to stay together for several time steps. In section 2 we had observed that chiral symmetry was restored statistically although it was broken explicitly by each rule. With several walkers, this needs not be true any more. But we found that it was again restored statistically. In addition, back steps are now chosen with a probability proportional to the third power of the walker's density.
4. Discussion
The walks studied in this paper have some similarities to random walks with m e m o r y studied during the last years in the physics literature. Examples are self-avoiding walks [11], 'true' self-avoiding walks [12], and Laplacian walks [15] (for a review, see ref. [13]). In contrast to our models, all these models are however stochastic processes, i.e. the walks are r a n d o m walks. In the present paper, we were mainly interested to see whether a weak form of self-avoidance can lead to random or even to structurally complex behavior in formally deterministic systems. We found that this is indeed the case. Several variants of the model lead to walks whose end-to-end distance scales as x/~. In all cases there were strong corrections to this which were compatible with being logarithmic. This suggests that d = 2 is the upper critical dimension. In this respect, our walks are like true self-avoiding walks. Our models are also similar to the flipping mirror models of refs. [8-10]. These models are deterministic like the present ones, but they take place in a landscape with already disordered itlitial conditions. The randomness of the trajectories in these models is a straightforward consequence of the randomness in the initial conditions. Also, the modification of the landscape does not lead to self-repulsion in these models. While we have interpreted our walks as metaphors for wanderings in an abstract space of phenotypic features, we can also interpret it more realistically but less excitingly as real walks of a foraging animal. Thus our results show that shortsighted search for the least foraged spots in an otherwise uniform field can lead to erratic wanderings, even if a deterministic strategy is used. This should
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not be considered as a drawback of the shortsightedness of the walk (alternatively, the animal could also look further ahead, and could go to the optimal place in a larger neighborhood). Indeed, with more sophisticated strategies one runs the risk of encountering less small-scale erratic wanderings, but more large-scale disasters. For examples of chaotic behavior in computer networks which becomes worse with more sophisticated strategies, see ref. [16]. As we have already said in the introduction, the very fact that a system not perturbed by the outside can show aperiodic and unpredictable behaviour, is not new anymore. That this applies also to biological systems should be obvious, but it seems that its consequences are not always fully appreciated. Our very modeling ansatz seems new, but otherwise both the possibility of spontaneous chaotic behavior and its importance for evolution have been pointed out repeatedly. The best known models which can lead to chaotic behavior are n-species Lotka-Volterra systems with n 1>3 [7], or 2-species L o t k a - V o l t e r r a systems with spatial degrees of freedom [17]. Of some relevance in this respect are also the findings of Lindgren [18] that the iterated prisoner's dilemma leads to ever improving (and thus changing) strategies, and the instabilities in stochastic co-evolving systems found in ref. [19]. In connection with the need for chaotic behavior in ecologies, we might quote the concept of homeochaos introduced in ref. [20]. W h e n we started this work, we hoped to find not only random but also complex behavior. By the latter we mean behavior with structural complexity, in contrast to algorithmic complexity as defined by Kolmogorov, Chaitin and others [21,22]. In systems where only some sort of average behavior is interesting, the latter is not a very useful complexity measure as it cannot distinguish between randomness and structural complexity [23]. The best known class of models which can lead to such complexity are cellular automata [24]. Although our models can be understood as cellular automata with very sparse action, we did not find any really complex behavior. We should however point out that such behavior can be hidden rather subtly [25], and we might have just missed it. Also, the logarithmic effects might suggest that we are at the border to complex behavior. Finally, in the limit of infinitely slow relaxation, our model leads to a new (class of) surface growth model(s). Previously, models for surface growth [26] included unbiased growth at all surface sites (Eden growth), preferred growth at exposed tips (diffusion limited aggregation), and growth at temporally active sites which inherit their activity from their neighbors (epidemic models). All these processes lead to surfaces whose height fluctuations increase like some power of the surface area. In contrast, our models lead to height fluctuations which increase only logarithmically. Whether this has applications in solid state physics remains to be seen.
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Acknowledgements W e t h a n k A r k a d y P i k o v s k y f o r m a n y u s e f u l d i s c u s s i o n s a n d for his i n t e r e s t i n t h i s w o r k . T h a n k s go also to R o b e r t W o l t e r for h e l p w i t h g r a p h i c a l o u t p u t , a n d to D r s . J. D u a r t e a n d H . N a k a n i s h i for i n f o r m a t i o n o n T S A W s . T h i s w o r k w a s s u p p o r t e d b y t h e V o l k s w a g e n - S t i f t u n g , c o n t r a c t A Z 1/66 995.
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