meta-dynamical adaptive systems and their applications to a fractal algorithm and a biological model

$: Meta-dynamical adaptive systems and their applications to a fractal algorithm and a biological model$
Physica D 207 (2005) 79–90

Meta-dynamical adaptive systems and their applications toa fractal algorithm and a biological model

Emmanuel Moulaya,b,∗, Marc Baguelinc,d,1

a Laboratoire Paul Painlevé, UFR de Mathématiques Pures et Appliquées, Université de Sciences et Technologies de Lille, Franceb LAGIS (UMR-CNRS 8146), Ecole centrale de Lille, Cité Scientifique, B.P. 48, 59651 Villeneuve d’Ascq Cedex, France

c Department of Zoology, University of Cambridge, Downing Street, Cambridge CB2 3EJ, UKd Animal Health Trust, Lanwades Park, Kentford, Newmarket CB8 7UU, UK

Received 5 October 2004; received in revised form 6 May 2005; accepted 20 May 2005Available online 13 June 2005

Communicated by S. Kai

Abstract

In this article, one defines two models of adaptive systems: themeta-dynamical adaptive systemusing the notion of Kalmandynamical systems and theadaptive differential equationsusing the notion ofvariable dimension spaces. This concept of variabledimension spaces relates the notion of spaces to the notion of dimensions. First, a computational model of the Douady’s Rabbitfractal is obtained by using the meta-dynamical adaptive system concept. Then, we focus on a defense-attack biological modeldescribed by our two formalisms.©

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2005 Elsevier B.V. All rights reserved.

SC:93A05; 37F50; 92D15; 92D25

eywords:Dynamical systems; Adaptive systems; Biological systems; Fractal algorithm

. Introduction

In the two last decades, there has been much inter-st in the study and formalization of complex adaptiveystems (see[1,2]). Many different approaches haveeen proposed: artificial chemistries, evolutionary for-alism, cellular automata. However, from a more the-

∗ Corresponding author. Tel.: +33 3 20 33 54 50.E-mail addresses:[email protected] (E. Moulay);

[email protected] (M. Baguelin)1 Tel.: +44 1223 336676.

oretical viewpoint, few mathematical formalisms efor adaptive systems. Though we may cite the cha“Categorical System Theory” proposed by A.H. Loin [3] who discuss the relationship between natand formal systems, most of the studies are masimulation-based (see[2,4]). With the recent evolution of physics and biology, a general mathemaformalism for adaptive systems would be very useActually, behind their apparent heterogeneity, mosthe adaptive systems share one important featureare dynamical objects whose structures are somemodified by a top level automation-like rule. On

167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.physd.2005.05.013

80 E. Moulay, M. Baguelin / Physica D 207 (2005) 79–90

basis of this observation, we have built a two-level for-malism that helps us to design an algorithm for fractalsand a biological model of co-evolution in a bacterium-phage system.

The main goal of this article is to present a conceptof adaptive systems, general enough to be used in dif-ferent fields (mathematics, physics or biology) and toapply this framework to a fractal implementation anda biological system. Besides, a new concept of spacesis developed. Since the second part of the 19th cen-tury, various definitions of dimensions, in particularthose developed by Cantor and Peano, have appeared(see[5]). These new definitions lead to the concept ofa fractal model and with it, a global view of spaces(see[6]). In order to study adaptive systems, we willuse a sort of “adaptive” space calledvariable dimen-sion spacewhose dimension changes. This new kindof space brings the notion of space and the notion ofdimension together.

The paper is organized as follows. In Section2,a very general adaptive system concept calledmeta-dynamical adaptive systemis given by using an ex-tension of Kalman systems. Section3 is dedicated tothe Douady’s Rabbit fractal implemented as a meta-dynamical adaptive system. In Section4, the conceptof variable dimension space leads to a special modelof adaptive system: theadaptive differential equation.Finally, we use our concepts of adaptive systems to de-scribe a biological example of a defense-attack modelin Section5.

2

ap-t la ur-i nto them epto fK om-m canp tiono imei (toc ms).T bles

characterizing the system) and the transition func-tion. The transition function defines the trajectoryin the state set starting from an initial state. Onlya few axioms are required to characterize these ob-jects and allow them to form a “dynamical system”.Among the more important ones are direction of time,consistency, composition property and causality. Letus recall the fundamental definition of Kalman (see[9]).

A dynamical system respecting Kalman axiomsisdefined by the following axioms:

(1) There is a given time setT, a state setX, a set ofinput valuesU, a set of acceptable input functionsΩ = ω : T → U, a set of output valuesY, and aset of output functionsΓ = γ : T → Y,

(2) (Direction of time)T is an ordered subset of thereals,2

(3) The input spaceΩ satisfies the following condi-tions:(a) (Nontriviality)Ω is nonempty,(b) (Concatenation of inputs) An input segment

ω]t1,t2] is ω ∈ Ω restricted to ]t1, t2] ∩ T . Ifω,ω′ ∈ Ω and t1 < t2 < t3, there is anω′′ ∈Ω such thatω′′

]t1,t2] = ω]t1,t2] and ω′′]t2,t3] =

ω′]t2,t3] ,

(4) There is given a state-transition function

ϕ : T × T ×X×Ω → X.

f

ractalc

. Meta-dynamical adaptive system

It is possible to give a general definition of adive systems by using the concept ofmeta-dynamicadaptive systemdeveloped by one of the author d

ng his Ph.D.[7,8]. But, in order to be able to preseur formal approach, we will have a look atain ideas behind the formalization of the concf dynamical system by Kalman in[9]. The aim oalman’s approach is to show that some very con mathematical structures plus a few axiomsrovide a very general framework where the nof dynamical systems (of all kinds) is captured. T

s modelled as an ordered subset of the realsover both the continuous and discrete paradighe important objects are the state set (the varia

whose value is the statex(t) = ϕ(t, τ, xτ, ω) ∈ X

resulting at timet ∈ T from the initial statexτ =x(τ) ∈ X at initial timeτ ∈ T under the action othe inputω ∈ Ω. ϕ has the following properties:(a) (Direction of time)ϕ is defined for allt ≥ τ

but not necessarily for allt < τ.(b) (Consistency)ϕ(t, t, x, ω) = x for all t ∈ T , all

x ∈ X and allω ∈ Ω.(c) (Composition property) For anyt1 < t2 < t3,

we haveϕ(t3, t1, x, ω) = ϕ(t3, t2, ϕ(t2, t1, x, ω), ω)for all x ∈ X and allω ∈ Ω.

(d) (Causality) If ω,ω′ ∈ Ω and ω]τ,t] = ω′]τ,t]

thenϕ(t, τ, x, ω) = ϕ(t, τ, x, ω′).

2 Such a general definition could include exotic sets such as fantor sets, for instance, in practice, the sets used are part ofR orN.

E. Moulay, M. Baguelin / Physica D 207 (2005) 79–90 81

(5) There is given a readout mapη : T ×X → Y

which defines the outputy(t) = η(t, x(t)). The map]τ, t] → Y given by σ → η(σ, ϕ(σ, τ, x, ω)), σ ∈]τ, t] is an output segment, that is, the restrictionγ]τ,t] of someγ ∈ Γ to ]τ, t].

A dynamical system is referred asΣ = T,X,U,Ω,Y, Γ, ϕ, η.

Now, inspired by the definition of dynamical sys-tems of Kalman, we propose a formalization of ourmeta-dynamical adaptive system.

Definition 1. A meta-dynamical adaptive systemMis a composite mathematical concept defined by thefollowing axioms:

(1) The dynamical level: The suitably indexed set

Σi,j = T0, Xi, U,Ω, Y, Γ, ϕi,j

is a dynamical system respecting Kalman axiomsfor all (i, j) ∈ I × J whereϕi,jj∈J are transitionfunctions on state setXi:

ϕi,j : T0 × T0 ×Xi ×Ω → Xi.

(2) The meta-dynamical level: LetX = ⋃i∈I Xi be the

set of all the possible states of the system andD =ϕ be the set of all possible transition

time

level,t

Φ

w utf

If t ∈ T1 andΦ(t, x, ϕ) = (x, ϕ), Φ is said tobemuteat (t, x, ϕ) else (t, x, ϕ) is acommutationpoint.

(3) Evolution rule between dynamical and meta-dynamical levels: Let xt = ϕ(t, t1, x1, ω) with ω

an input function, there exists a “meta”-transitionfunction

Ψ : T1 × T1 ×X×D×Ω → X×D

such that:(a) (Dynamical phase) IfΦ is mute on (t, xt, ϕ) for

all t ∈ [t1, t2[ ∩ T1, thenΨ is defined betweent1 andt2 and(i) if t2 /∈ T1, then

Ψ (t2, t1, x1, ϕ, ω) = (xt2, ϕ),(ii) elset2 ∈ T1 and

Ψ (t2, t1, x1, ϕ, ω) = Φ(xt2, ϕ).(b) (Concatenation rule) If there existst2 ∈ ]t1, t3[

such thatΨ is defined betweent1 and t2 andbetweent2 andt3 thenΨ is defined betweent1andt3 and

Ψ (t3, t1, x1, ϕ, ω)

= Ψ (t3, t2, Ψ (t2, t1, x1, ϕ, ω), ω).

(c) (Stopping rule)Ψ is defined betweent1 andt2if, respecting previous axioms, there is only afinite number of commutation points in [t1, t2].

The meta-dynamical rule in point (2) can operatea notn ly ah eta-d itionf ther h then int( icalr witht

, ourd l ando t usr hsc sedo ents(

i,j (i,j)∈I×Jfunctions, then there exists a meta-dynamicalT1 and a meta-dynamical rule3

Φ : T1 ×X×D → X×D

such that:(a) (Temporal hierarchy)T1 ⊆ T0,(b) (Coherence states/transitions) IfΦ(t, x1,

ϕi1,j1) = (x2, ϕi2,j2) for t ∈ T1, then x1 ∈Xi1 ⇒ x2 ∈ Xi2.

3 It is possible to consider input values at the meta-dynamicalhenΦ is defined as follows

: T1 ×X×D×Ω1 → X×D

hereΩ1 = ω : T1 → U1 is the set of acceptable “meta”-inpunctions.

t instants for which the system is defined andecessarily at all of them (see point 2(a)); actualigher level is usually slower. Moreover, as the mynamical rule can change the state and the trans

unction, we have to consider that both match well:esulting state has to belong to the state set on whicew transition function operates (point 2(b)). In po3) we describe how dynamics and the meta-dynamule combine together to make the system changeime.

To take up notions mainly used in social scienceynamical rule can be seen as a heterarchical leveur meta-dynamical level as a hierarchical level. Leecall that aheterarchyis a network of elements whichare the same “horizontal” position level in ahierar-hy. Each level in a hierarchical system is compof a heterarchy which contains its constituent elemsee[10]).


Remark 2. In the second case of (3(a)), whent2 ∈ T1,we deliberately consider that (t2, Φ(xt2, ϕ)) is mute.We do not consider the case where the meta-dynamicswould “rebound” and have several commutations at thesame time. If the system has several commutations, itis always possible to consider this set of commutationas one, with the final state of the last commutation(provided that, we know from 3(c) that the numberof commutations is finite). Axiom 3(c) is to avoidZeno-style system with an infinite number of commu-tations in a finite amount of time (e.g. with a quantityx(t) = sin(2π/1 − t) with a commutation each timex(t) = 0 on [1− ε,1 + ε]).

Remark 3. The index setJ is associated with thechange of dynamics in the same state space. We recog-nize here the framework of hybrid systems (see[11]).

Remark 4. It is also important to differentiate betweenthe continuous or discrete dynamics and the continu-ous or discrete meta-dynamics because the confusionis easy. The first case is well known. The second oneshows the difference between a meta-dynamical timeT1 which is continuous (see Section3) and a meta-dynamical time which is discrete (see Section5).

Example 5. On Fig. 1, we can see a meta-dynamicalsystem in action. Att5 = tc, we pass from a two-dimensional state set to a three-dimensional one.Tt ont -n (a)).E e

meta-dynamics is mute ont6, t7 and t8. The junctionbetween the two of them is made by using the con-catenation rule. As there is only one commutation, thesystem is defined.

Now, it is possible to expand some classical notionsof dynamical system, as the notion of trajectory.

Definition 6. Let M be a meta-dynamical adaptivesystem with the meta-transition functionΨ , if for t ∈T1,

Ψ (t, t0, x0, ϕ0, ω) = Φ(xt, ϕt)

with xt ∈ Xt then (Xt, ϕt) is ameta-stateofM in t.The sequence (Xti , ϕti )0≤i≤n,0≤n≤+∞ of meta-state

ofM such that for allt ∈ [ti, ti+1[, (Xt, ϕt) = (Xti , ϕti )and (Xti , ϕti ) = (Xti+1, ϕti+1) is called meta-orbit ofM.

If n < +∞, (Xtn, ϕtn ) is said to be anabsorbentmeta-state.

The sequence (ti, Xti , ϕti )0≤i≤n,0≤n≤+∞ satisfyingthe same conditions is called atrajectoryofM.

One can extendDefinition 1to stochastic systems.

Definition 7. Let us consider the dynamical rule ofDefinition 1with the following meta-dynamical rule

pΦ : T ×X×D → (X×D → [0,1])

wr

p

axiom

he dynamics are continuous, soT0 = [t0, tend] andhe meta-dynamical time setT1 is discrete:T1 =t1, t2, t3, . . . , t8. Since the meta-dynamics is mute1, t2, t3 andt4, the evolution fromt0 to t5 is a purely dyamical phase, ended by a commutation (Axiom 3volution fromt5 to tend is also purely dynamical (th

Fig. 1. Illustration of some

1

herepΦ is a probability distribution onX×Dwhichepresents the probability

Φ(t, x, ϕ) · (xf , ϕf )

s of a meta-dynamical system.


that (x, ϕ) becomes (xf , ϕf ) at t. The stochastic evolu-tion rule between dynamical and meta-dynamical lev-els is given by

pΨ : T1 × T1 ×X×D×Ω → (X×D → [0,1])

wherepΨ is a probability distribution onX×Dwhichrepresents the probability

pΨ (t1, t2, x, ϕ, ω) · (xf , ϕf )

that (x, ϕ) becomes (xf , ϕf ) betweent1 and t2 withthe input functionω. The properties ofpΦ andpΨare the same asΦ andΨ given inDefinition 1. Such asystem is called a stochastic meta-dynamical adaptivesystem.

The interest ofDefinition 7 is its generality. It isfor example well adapted to model a large number ofcomplex adaptive systems, in particular the biologicalmodel we develop in Section5.

3. Algorithm for the Douady’s Rabbit fractal

The goal of this example is to give an algorithmbased onDefinition 1allowing to describe the Douady’sRabbit Fractal. In this example, the setI and thus thefamily Xii∈I of Definition 1 is not countable. TheR 2 hee -g LetK

if

Fs

O

a tx1ax

f

Now, let us recall the definition of the Julia set (see[12]). Let Rbe a non-constant rational function onS

2.The Fatou setof R is the maximal open subset ofS

2

on whichRnn∈N is equicontinuous whereRn = R . . . R. TheJulia setJR of R is the complement of theFatou set onS2. Thefilled in Julia setKR of a functionR is all the points which are not attracted to the super-attracting fixed point at infinity, that is

KR = z ∈ C : Rn(z) → ∞.

This closed set includes the Julia set as its boundary,JR = ∂KR. Theescape setIR of a functionR is all thepoints that “escape” to infinity, that is

IR = z ∈ C∞ : Rn(z) → ∞.

If Ris a polynomial of degree 2,JR is called a quadraticJulia set. The following result can be found in[13]: ifP is a polynomial of degreed ≥ 2, thenJP is closedand dense within itself.

Here, we are interested in quadratic Julia sets with

Pc(z) = z2 + c (1)

wherec ∈ C. For small values ofc, the Julia set isdistorted by varying degrees from the unit circle, inthese cases the Julia set has an infinite length. For largevalues ofc, the Julia set becomes an infinite set oftotally disconnected points, often said to be dust like( e areo cted– hev ted.I

An isa

b

b

( ab-b wer.W

iemann SphereS is mapped one-to-one onto txtended complex planeC∞ = C ∪ ∞ by stereoraphic projection. Let us recall some definitions.= R or C andB be the unit ball ofK. If f : K → K

s a function,x ∈ X is periodic of period n ∈ N ifn(x) = x and for allk ∈ 1, . . . , n− 1, f k(x) = x.or x periodic of periodn, the cycle O(x) is theet

(x) = x, f (x), . . . , f n−1(x)

nd its cardinal isn. Moreover, iff is differentiable a, x is stable, quasi stableor unstableif |(fn)′(x)| <, |(fn)′(x)| = 1 or |(fn)′(x)| > 1. If x is stable,x isttractiveif there exists an intervalVstrictly containingso that for allx ∈ V

n(x′) −→n→+∞ x.

in the sense of Cantor). In the quadratic case thernly these two possible – connected and disconnetypes. TheMandelbrot setis the space containing t

aluec for which the associated Julia set is connect is generated by the quadratic sequence

zn+1 = z2n + c

z0 = c(2)

complex point z = a+ ib ∈ C will be de-oted by (a; b). The Douady’s Rabbit fractal

Mandelbrot set with c = (−(3/2) + (1/2)(a+); (

√3/2)(a− b)) wherea = 3

√(25+ √

621)/54 and

= 3√

(25− √621)/54, soc (−0.12256; 0.744862)

see[14]). As the points get closer to the Douady’s Rit fractal, the speed of convergence becomes sloith the choice ofc, (c2 + c)2 + c = 0 so the origin is


an attractive cycle4 of period 3 ofPc

O(0) = 0, c, c2 + c.

The boundary points move chaotically. Thus, the ideais to change the points by using the speed of conver-gence as an adaptive value. The principle of our algo-rithm is to have a set of points evolved to the fractalboundary. For this, one gives a weight to each point.This weight varies with a dynamics which “rewards”the most adapted points (the points which are less at-tracted by limit values) and “penalizes” the least in-teresting ones. When the most efficient points reach acertain threshold, they are allowed to be multiplied intheir neighborhood. The least efficient points disappearwhen they reach a minimal threshold. This system ofthresholds which changes the dynamical structure isour meta-dynamics.

3.1. Dynamical level (DL)

With the formalism ofDefinition 1, we haveT0 = N.The index setI is the set of all finite sets of points ofC

(so i is a set of points ofC andI is not countable). Forall i ∈ I, Xi = N

card(i) is the set of the point weightsof i. At each given pointz ∈ C, we attach a selectivevalue

µn : C → R+

z → min|Pn(z)|, |Pn(z) − c|, |Pn(z)

3,1 n-i

pF

O

ab| lV

f

1/Pnc (z) →n→+∞ 0. One increases the weight of thepoints which are close to the boundary. In order to dothis, we organize a “competition” by comparing theirmutual slowness of convergence. So, the weights of thepointsz ∈ i are given by

ωz(t + 1) = ωz(t) +∑

q∈i,q =zδ(z, q) with δ(z, q)

=

1 if µt(z) > µt(q)

0 if µt(z) = µt(q)

−1 if µt(z) < µt(q)

ωz(0) = 0

wheret ∈ N. Using the notation of Section2, we havethe following transition function

ϕi(t + 1, t, ωz(t)z∈i) = ωz(t + 1)z∈i

for all t ∈ N. For each state space there is only oneassociated transition function, so the use of the indexsetJ is unnecessary.

3.2. Meta-dynamical level (ML)

With the formalism ofDefinition 1, we haveT1 =kN. When the weight of a point reach an upper thresh-oldM > 0, the point is allowed to give birth to a newpoint, randomly in its neighborhood. When the weightof a point reaches a lower thresholdm < 0, the point isremoved. This can be modelled by the meta-dynamicalrule

Φ

w

i

s edp -b pte

·

A mesm f

c c c

− c2 − c|, 1|Pnc (z)|

0, c, c2 + c is the attractive cycle of period/Pnc (z) is referring to the attraction to the infity seeing thatPnc (z) →n→+∞ ∞ is equivalent to

4 Let K = R or C, f : K→ K be function,x ∈ X is periodicoferiodn ∈ N if fn(x) = x and for allk ∈ 1, . . . , n− 1,f k(x) = x.or x periodic of periodn, thecycleO(x) is the set

(x) = x, f (x), . . . , f n−1(x)

nd its cardinal isn. Moreover, if f is differentiable atx, x is sta-le, quasi stableor unstable if |(fn)′(x)| < 1, |(fn)′(x)| = 1 or(fn)′(x)| > 1. If x is stable,x is attractiveif there exists an interva

strictly containingx so that for allx ∈ V

n(x′) −→n→+∞ x.

(t, ωz(t)z∈i(n), ϕi(n)) = (ωz′ (t)z′∈i(n+1), ϕi(n+1))

here

(n+ 1) = z+ 3ε : 3 is a random point inB, z ∈ i(n)

andωz(t) > M ∪ z ∈ i(n) : ωz(t) ≥ much thatε > 0 given. The weight of the reproducoint is shared between itselfωz(t) and the new neighouring pointωz+3ε(t). The other weights are kequal. Let us sum up the evolution rule:

· · → ϕi(n)(t) →DLϕi(n)(t + 1)→

DL· · ·

→DLϕi(n)(t + k) →

MLϕi(n+1)(t + k) → · · ·

s the time increases, the selective value becoore and more accurate, i.e.i(n) tends to a set o


Fig. 2. The Douady’s Rabbit fractal.

points belonging to the Julia spaceJPc or to the emptyset whenn → +∞. This is an interesting point be-cause the calculations are concentrated on the fractalboundary. In a lot of classical algorithms, the calcula-tion time is squandered for points in the interiorKPc

(see[15]). Indeed, this meta-dynamical adaptive sys-tem produces a cloud of points which gather roundthe Julia spaceJPc called the Douady’s Rabbit fractal(seeFig. 2).

Though we have developed this algorithm in the set-ting of Julia sets, the same framework can be used to ex-plore many complex frontiers, for example other fractalstructures where the boundary properties are largelyunknown. Indeed, this kind of algorithm is really in-teresting because the calculation is concentrated on aspecific region.

4. Adaptive differential equations

First, let us recall the classical notion of Hausdorfffractal dimension (see[16]). Let E be a subspace of ametric spaceM andρ a positive number, one definesRρ as the set of all coverings (Bi, ρi)i of E by ballsBiwith diameter 0< ρi < ρ. For each positive numberα,one denotes:

Hαρ (E) = inf

∑i

ραi : (Bi, ρi)i ∈ Rρ

.

Hα(E) = limρ→0Hαρ (E) is called theα-dimensional

Hausdorff measure ofE and belongs to [0,+∞]. Then,let

dim(E) = inf α > 0 :Hα(E) = 0,

it is theHausdorff dimensionof E. On the one hand, theHausdorff dimension is defined for all metric spaces.On the other hand, in the case of a classical space(non fractal), it is identical to its topological dimen-sion (for example the Hausdorff dimension ofR

n is n).In the case of a simple linear fractal, such as fractalswith internal homothetia obtained by an homothetic it-eration with constant ratio, the Hausdorff dimensionis equal to the homothetic dimension dimh(E) givenby:

dimh(E) = ln(n)

ln(k)= logk(n)

wheren is the number of subsets obtained during thehomothetic process of reduction with ratio 1/k (see[17]). For more information on dimension theory thereader may refer to[18] or [19].

The variable dimension space is defined as follows:

Definition 8. Let M be a metric space andΛ aparameter space. One defines two mapsd : Λ →[0,+∞], λ → d(λ) andF : Λ → 2MwhereF (λ) ver-i -e -s ei e.da -u arityoi cet ifi

thef

E -i oft

fies dimF (λ) = d(λ) and 2M is the family of nonmpty subsets ofM. F (Λ) is a set of variable dimenion spaces. IfF (Λ) is a totally ordered set for thnclusion, one calls (F, d) a variable dimension spac

is the dimension function andd(F (λ)) ≤ d(M) forll λ ∈ Λ. If Λ is a topological space thenF is a set valed function and we may take account of the regulf d. The variable dimension space (F, d) is continuous

f d is continuous. Moreover, ifΛ is an ordered spahen (F, d) is increasing (respectively decreasing)ds increasing (respectively decreasing).

To illustrate these definitions, we can considerollowing example:

xample 9. Let 1/4 ≤ λ ≤ 1/2, one defines four simlarities from the familyKc of the compact subsethe squarec = [0,1]2 with value inKc by


Fig. 3. Van Koch snowflake generator.

• s1,λ(x) = λx,

• s2,λ(x) = λ

1

2− λ −

√λ− 1

4√λ− 1

4

1

2− λ

x+

(λ

0

),

• s3,λ(x) = λ

1

2− λ

√λ− 1

4

−√λ− 1

4

1

2− λ

x

+

1

2√λ− 1

4

and

• s4,λ(x) = λx+(

1 − λ

0

).

Then, one defines the functionΩλ : Kc → Kc, x →s1,λ(x) ∪ s2,λ(x) ∪ s3,λ(x) ∪ s4,λ(x).Ωλ being contracting, one definesF (λ) as the fixed

point5 of this function for the Hausdorff distance. In[6], one may find the following result: the dimension ofF (λ) is d(λ) = ln 4/ln(1/λ). So, (F, d) is a continuousincreasing variable dimension space onΛ = [ 1

4,12],

called the Van Koch snowflake (seeFig. 3). A sim-ilar construction allows us to turn continuously an-dimension space into a (n+ 1)-dimension space.

An interesting application of the above concepta states er ap es”o dap-t

bsetso lso aBf e[

Definition 10. Let X be a Banach space of fi-nite dimension called the possible state space. Cons-ider

(1) a subdivisiontii∈N of R+,(2) an applicationd : R+ ×X → N, (t, y) → d(t, y)

such thatd(0, y) is given,(3) an applicationg : R+ × R

dim(X) → Rdim(X),

such that one has the systemxi(t) = fi(t, xi(t)), t ∈ [ti, ti+1[, xi(t)

∈ Rd(ti,yi), i ∈ N

xi(ti) = g(ti, yi)

(3)

wherey0 = x0(0),yi = limt→t−i

xi−1(t) for i ∈ N∗, xi(·) is

the right derivative ofxi(·) and

fii∈N : [ti, ti+1[ × Rd(ti,yi) → R

d(ti,yi)

is a family of applications. Such a system is called anadaptive differential equation. A trajectory of the sys-tem(3) is a family

x(t) = xi(t) : t ∈ [ti, ti+1[i∈N.

One may notice that dim(g(ti, yi)) = d(ti, yi). Thisc uc-c thei s-t enta

uc-t sys-t tionw+

rises when the variable dimension space is thepace of the solutions of a differential equation overiod of time. The union of the solutions then “movn a variable dimension space. We talk about an a

ive differential equation.

5 To prove this, one uses two classical results: the family of suf a Banach space endowed with the Hausdorff distance is aanach space and if the similarities are contracting inK2, then the

unctionΩλ is also contracting inKc with the Hausdorff distanc20].

oncept of adaptive differential equation is not a session of ordinary differential equations becausenitial condition of each systemi depends on the syemi− 1. The ordinary differential equations represcase whered is constant.The adaptive differential equation is a red

ive approach of the meta-dynamical adaptiveem to a system described by a differential equahose meta-dynamics is discrete and where dimX <

∞.


5. Application to a biological model

Here, we want to model the influence of thebacterium-phage interaction on the co-evolution of thepopulations of bacteria and phages. In the follow-ing we will need the definition of the Hamming dis-tance. Letn ∈ N, theHamming distanceis the functiondH : 0,1n → N defined by

dH (s1, s2) =nc∑k=0

|ski − skj |

where s1, s2 ∈ 0,1n. dH represents the number ofdiffering bits between the two binary stringss1 =s11s

21 . . . s

n1 ands2 = s12s

22 . . . s

n2.

The attack of a bacteria population by phages is as-sumed to be done by the lysis process: a phage hangson the surface of a bacterium cell, injects its DNA in itand then forces the bacterium to yield its own replicasinside the cell. When the cell is full, it bursts, releasinga huge quantity of copies of the infecting phage. Theefficiency of the attack (i.e. the probability of success ofthe infection), depends on the couple bacterium-phage.One of the other characteristics of bacterial and phagespopulations are their high variabilities. They frequentlymutate, creating new populations with new properties.Such a system has two dynamics to be taken into ac-count: the dynamics of the populations of bacteria andphages and the meta-dynamics of evolution geared bym t am oryo ellt

5

ua-t

w tbf -ta hep r theb

cal discussion). The populations are then characterizedby the different concentrations and the two lists of bitstrings. The setL of all the pairs of lists is taken as oursecond index (the one calledJ in Definition 1).

The model is described by a modified version ofMosekilde equations (see[21]). This set of ordinarydifferential equations describes the interactions of bac-terial populationsBi and phage populationsPj in achemostat.Bi andPj also symbolize the concentrationof these populations andSthe concentration of the nu-trient. The process of infection of bacteria by phagesis modelled by three infection stagesI1,j, I2,j andI3,j.One associates with the ecosystem(4) the state space

X1+nb+4np = R1+nb+4np+ . Over a period of time with-

out appearance or disappearance of any strain of bac-teria or phages, the dynamical evolution of the system(4) is modelled by the evolution of the concentrationsof S, the different bacteriaBi and phagesPj, and threeinfection stagesIk,j. With the formalism ofDefinition1, that means thatT0 = R and for a fixedl ∈ L thetransition functionϕ1+nb+4np,l is the integration of thefollowing set of differential equations

dBidt

= νSBi

κ + S− Bi

np∑j=1

αωijPj − ρBi

dI1,jdt

= Pj

nb∑i=1

αωijBi − 3I1,j

τ− ρI1,j

dI2,j = 3(I1,j − I2,j) − ρI2,j

w( a-t ion( la eriums

utations and extinctions. It is without any doubodel which is not in the scope of the classical thef dynamical systems, but our formalism applies w

o it.

.1. Dynamical level

It is made up of a set of ordinary differential eqions. Let us consider the following system

S, Bi0≤i≤nb, Pj, I1,j, I2,j, I3,j0≤j≤np, (4)

ith S the concentration of nutrient,Bi the differenacteria strains with 0≤ i ≤ nb andPj andIk,j the dif-

erent phages strains with 0≤ j ≤ np. To each populaionBi andPj is associated a binary stringsbi ∈ 0,1nnd spi ∈ 0,1n. These binary strings code for troperties of attack (for the phages) or defence (foacteria) facing the infection (see[7] for the biologi-

dt τ

dI3,jdt

= 3

τ(I2,j − I3,j) − ρI3,j

dPjdt

= 3βI3,j

τ− Pj

nb∑i=1

αBi +np∑j=1

3∑k=1

αIk,j

−ρPjdS

dt= ρ(σ − S) −

nb∑i=1

νγSBi

κ + S

(5)

ith 0 ≤ i ≤ nb, 0 ≤ j ≤ np, ρ the rate of dilutionρ = 0.0045 min−1), κ andν respectively the saturion term and the growth from the Monod equatκ = 10g ml−1, ν = 0.024 min−1), α the theoreticadsorption constant depending on phage and bactize (α = 10−9 ml min−1), τ a time constant (τ =


30 min),β the number of copies of phagej released dur-ing the burst of the infected bacterial cell (β = 100),σthe continuous supply of substrate (σ = 10g ml−1),γ the amount of nutrient consumed in each cellular di-vision (γ = 0.01 ng) and finallyωij the probability ofinfection ofBi by Pj which depends on the similaritybetween bit stringsbi (attached to bacterium populationBi) andspj (attached to phage populationPj) as follows

ωij =(

1 −dH (sbi , s

pj )

nc

)2

with nc the size of the binary string.Here, we are not interested in the identification of

the biological dynamical level which can be found in[7]. With the given size of a binary stringnc, there ex-ists a finite number of possible dynamical systems, heredifferential equations. Indeed, the number of possibledifferent populations of bacteria is equal to the numberof parts of the set of binary strings of sizenc. So, thereare 22

nc possible populations of bacteria. For the samereason, one deduces that the possible number of pop-ulations of phages is the same and thus the total num-ber of possible state spaces is 22nc × 22nc = 22nc+1.So, with the notation ofDefinition 1, it means thatcard(J) = 22nc+1. If we take for examplenc = 10, onehas 21025 3.6 × 10308 possible state spaces. Theo-retically, one can consider a 3.6 × 10308 dimensionalspace to embed the system(5). Nevertheless, when itcomes to numerical simulations, such a big system isimpossible to deal with. One thus understands the needof an adaptive system to describe the system(5).

5.2. Meta-dynamical level (meta-dynamicaladaptive system point of view)

This is the main difference with the model of Mosek-ilde given in[21, Chapter A]which is not evolutive.

Proposition 11. Consider a small interval of timeDt,the adaptive changes of the system(4) are given by thefollowing mechanisms

pΨ (t +Dt, t, δ, ϕ2+nb+4np,l1)

·(θ1, . . . , θk, ϕ1+kδ+nb+4np,l2) = e−λ(t) λ(t)k

k!(6)

and

Ψ (t +Dt, t, δ, ϕ2+nb+4np,l1,mδ) = ϕ2−εδ+nb+4np,l2

(7)

where(1) Formula (6) is the probability that the species

δ (a bacterium B or a phage P) gives birth tok ≥ 1 mutante strainθ1, . . . , θk on [t, t +Dt[ withλ(t) = δ(t)

δepe,pe the probability that a small group

of mutant species of sizeδe gradually replacingthe species of the parent populationδ. ϕi,j is thefunction defined by equations(5) for j = l1, l2 andkδ = k if δ is a bacterium andkδ = 4k if δ is aphage,

(2) Formula (7) gives the determinist rule of the ex-tinction of the speciesδ which depends on a giventhresholdmδ, εδ = 1 if δ is a bacterium andεδ = 4if δ is a phage.

pΨ is the stochastic transition function ofDefinition7which governs the meta-dynamical rule of the appear-ance.Ψ is the transition function ofDefinition 1whichgoverns the meta-dynamical rule of the extinction.

Proof. Consider the probabilitypBi (t, k) that the pop-ulationBi gives birth tokmutante strains on [t, t +Dt[and the probabilitype that a small group of mutantebacteria of sizeBe is gradually replacing the bacteriaof the parent populationBi. Such a reasoning gives abinomial probability for

p

w ya sonp

p

w llt nlyo u-t teria(p op-u n isa od-e ere

Bi (t, k) = Ckn(t)pke(1 − pe)

n(t)−k

ith n(t) = Bi(t)/Be. When n(t) is large, one mapproximate the binomial probability by the Poisrobability

Bi (t, k) e−λ(t) λ(t)k

k!(8)

ith λ(t) = (Bi(t)/Be)pe. Suppose that the birth of ahe populations to a Hamming distance of one (one bit is different) is equiprobable. The birth of a m

ant strain results in the change of a group of bacwhich is a part of a parent population) of sizeBe, in aopulation group with new characteristics. If this plation group already exists, the mutant populatiodded to it. The same mechanism governs the mlling of the phage mutation. For the extension, th


exists a thresholdmδ below which the extinction of thepopulation is certain. Every population under a giventhreshold (different for bacteriamB and phagesmP) isremoved from the system. Thus, the extinction is de-terminist.

We have defined the macroscopic birth of a mu-tant population as an event occurring on an interval[t, t +Dt[. There exists a time setT1 where the system(5) commutes. This commutation depends on the stateof the system(5)and on the coefficientpB etpP respec-tively coefficient of the Poisson law of the bacteriumand the phage. In order to simplify the model and tomake it computable, one may suppose that

T1 = t0 + iDt

even if in practice, the mutations have no reasons tobe periodically defined. This allows to define, with theprevious notations, the meta-dynamical rule

pΦ(t0 + iDt, δ, ϕ2+nb+4np,l1)

·(θ1, . . . , θk, ϕ1+kδ+nb+4np,l2) = e−λ(t) λ(t)k

k!

Φ(t0 + iDt, δ, ϕ2+nb+4np,1,mδ) = ϕ2−εδ+nb+4np,2.

Then, at each stepDt there are four possible commu-tations:

• birth of a new bacterial strain: a variableBnb+1 isadded, the dimension of the system(5) increases by

• are

• ari-

• on-sys-

m ac

tT paces( itht siono

5.3. Meta-dynamical level (adaptive differentialequation point of view)

We have described the biological model as astochastic meta-dynamical adaptive system by usingDefinition 7. This modelling corresponds to a stochas-tic view of the system(4)where the space of all stochas-tic realizations is infinite. There exists a family of pointstii∈N where the system(5) commutes. We have sup-posed that this commutation is given byti = t0 + iDt.Then, at each stepDt the system may commute. Tosee a possible evolution of the system(4), we choose astochastic realization at each pointti (only for the ap-pearance of a strain because the extinction of a strainis determinist)g(ti, yi) whereyi = lim

t→t−ixi−1(t) with

xi−1(t) the solution of the system(5) on [ti−1, ti[. Itdescribes the determinist evolution of the system(4).By extension, for us astochastic realizationis a func-tion

g : R+ × Rdim(X) → R

dim(X)

defined at least on(ti, yi)i∈N. For a given realizationg, the system(4)may be modelled by an adaptive differ-ential equation whose equations are given by the system(5). This modelling belongs to the variable dimensional

spaceRnb+4np+1+ wherenb + 4np + 1 depends ongand

follows the rule given onFig. 4. A detailed study of thisbiological model with implementation can be found in[22].

In this example, one sees that the framework of theadaptive differential equations corresponds to the caseof a transition function defined by a set of differential

Fig. 4. Evolution of the state space withnb andnp respectively thenumber of bacterial and phage strains before the transition.

one;birth of a new phagical strain: four variablesadded:Pnp+1, I1,np+1, I2,np+1 andI3,np+1, the di-mension of the system(5) increases by four;extinction of a bacterial strain: the concerned vable is removed, the dimension of the system(5)decreases by one;extinction of phagical strain: variables of the ccerned phage are removed, the dimension of thetem(5) decreases by four.

The different possible state spaces resulting froommutation are given onFig. 4.

If more than one event occurs at each moment ∈1, one composes the possible change of state sfor example, the extinction of a phage combined whe birth of a bacterial strain decreases the dimenf the system(5) by 4− 1 = 3).


equations and a meta-dynamical time reduced to a dis-crete subset ofR.

6. Conclusion

We started our research with a biological systemwhose dynamics changes in different dimensions.Seeing that there was no mathematical framework todescribe such a system, we have developed a math-ematical tool following Kalman’s dynamical systemcalled meta-dynamical adaptive system which wasappropriate to give a constructive algorithm for somefractals. It was also adapted to describe and analyzeour biological system. However, we have found ourtool too general and we decided to develop a specialtool for differential equations. The new system called“adaptive differential equation” is not a successionof differential equations because the initial conditionof each system depends on the previous system andgives the new dimension of the following system. Thismodel allows to describe a system of changing differ-ential equations, in particular the stochastic realizationof a stochastic meta-dynamical adaptive system. Thelast tool we use is the “variable dimension space”.This new kind of space links the notion of space anddimension in a changing dynamics. We think that ourwork will contribute to understand the huge numberof complex systems where the espace of explorationis too big to be investigate with classical means.

A

h sys-t NeRt llys raceB

R

dge

[2] M. Gell-Mann, The Quark and the Jaguar: Adventures in theSimple and the Complex, W. H. Freeman and Company, NewYork, 1994.

[3] R. Rosen, Theoretical Biology and Complexity: Three Essayson the Natural Philosophy of Complex Systems, AcademicPress, Orlando, 1985.

[4] J.H. Holland, Adaptation in Natural and Artificial Systems: AnIntroductory Analysis with Applications to Biology, Control,and Artificial Intelligence, MIT Press, 1992.

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[6] B. Mandelbrot, Les Objets Fractals, Edition du Seuil, Paris,1980.

[7] M. Baguelin, Modelisation de systemes complexes parmetadynamiques: Applicationa la modelisation de populationsen coevolution, Ph.D. thesis, Universite des sciences et tech-nologie de Lille, 2003.

[8] M. Baguelin, J. Lefevre, J.P. Richard, A formalism for modelswith a metadynamically varying structure, in: Proceedings ofthe IEEE European Control Conference, Cambridge, UK, 2003.

[9] R.E. Kalman, P.L. Falb, M.A. Arbib, Topics in MathematicalSystem Theory, International Series in Pure and Applied Math-ematics, McGraw-Hill, 1969.

[10] E. Jen, Stable or robust? What’s the difference? Complexity 8(3) (2003) 12–18.

[11] B.P. Zeigler, H. Praehofer, T.G. Kim, Theory of Modelling andSimulation: Integrating Discrete Event and Continuous Com-plex Dynamic Systems, second ed., Academic Press, 2000.

[12] D.S. Alexander, A History of Complex Dynamics: FromSchruder to Fatou and Julia, Braunschweig, 1994.

[13] H. Brolin, Invariant sets under iteration of rational functions,Arkiv for Matematik 6 (1965) 103–144.

[14] A. Douady, J.H. Hubbard, Etude dynamique des polynomescomplexes, (premiere et deuxieme partie), PublicationsMathematiques d’Orsay 84–02, 85–04.

[ ges,

[ Har-

[ (v):ond.

[ .–

[ ni-

[ rlag,

[ s in

[ h toings

tadt,

cknowledgements

The authors would like to thank Jacques LeFevre toave given a start to our study of evolutionary eco

ems and Jean-Pierre Richard at the head of the Syeam for his help in producing this article. Partiaupported by CNRS Maths-STIC and the Horseetting Levy Board.

eferences

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meta-dynamical adaptive systems and their applications to a fractal algorithm and a biological model

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