Spreadability of transport systems

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Spreadability of transport systems

A. EL JAI ^a & K. KASSARA ^a

a IMP/CNRS—University of Perpignan , 52, Avenue de Villeneuve, Perpignan Cedex, 66860, France

Published online: 16 May 2007.

To cite this article: A. EL JAI & K. KASSARA (1996) Spreadability of transport systems, International Journal of Systems Science, 27:7, 681-688, DOI: 10.1080/00207729608929265

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lnternational Journal of Systems Science, 1996, volume 27, number 7, pages 681-688

Spreadability of transport systems

A. EL JAlt and K. KASSARA t

The spreadability concept of distributed parameter systems was essentially motivated by environment systems and various environment systems are governed by transport equations. The aim of this paper is to study the spreadabilit.y of systems modelled by transport equations. We shall be concerned by linear autonomous systems as well as nonlinear systems. For those systems we give characterisation results for spreadabilit y.

f1J'z(x, t) -ee- (x,t,z(x, t))EK

where K c Q x I x IR is a set of state constraints. In our case we consider

where 8:Q -> IR is a desired target state and we should say that the system S is 8-spreadable or null-spreadable in the case where 8= O.

As stressed in EI Jai and Kassara (1994) the main motivation is related to vegetation dynamics where, for instance, z denotes the spacial biomass density. In this case the expansion of vegetation may be described by a sequence of subdomainsWI such that

w,= {xEQlz(x, r) ;:::zm;n}

The case where z= 0 may hold for the case of desertifica- tion considering the biomass density as equal to zero.

Various other examples may concern the pollution problem, cancerous cells development, fire dynamics, etc.

whereI = [0; t_{f ]} is the time interval; letf1J' be a property acting on the range of z and consider the family of su bsets

W, = {x^EQIf1J'z(x,t)}

The notation f1J'z(x, r) means that, for x Ew, and teI, z(x, t)satisfies the propertyf1J'.Then we remind ourselves of the following definition.

Definition I: The system S is said to be f1J'-spreadable during the time interval I if the family (w,),.t is non-

decreasing. D

As regards the property f1J', several situations may be considered (El Jai and Kassara 1994). A general and more realistic case consists in considering

f1J'z(x, t) ^{¢ >} z(x, t)=^8(x) (1) (tEI),

z(t) =z(-,t):Q -> IR S whose state is denoted by

Received 9 June 1995. Revised version received I December 1995.

Accepted 18 December 1995.

t IMP/CNRS-University of Perpignan, 52, Avenue de Villeneuve, 66860 Perpignan Cedex, France. E-mail: eljai@univ-perpJr.

I. Introduction

In order to describe the phenomenon of expansion which arises in environmental processes such as vegetation dynamics or pollution, we have introduced (EI Jai and Kassara 1994) the concepts of spreadability and spray control for distributed parameter systems. See also the references in EI Jai and Kassara (1994) for the motivation and the mathematical statements.

Various ecological modelling studies show that space models describing natural phenomena are transport models (see Betrami 1987, Berger and Tricot 1988, Nahout and Mahrer 1989, Diaz and Lyons 1993, Brufau et al. 1994, Friedman and Littman 1994, Kurnetov et al.

1994, Brufau 1995, etc.). So we focus the present paper on the spreadability of such models when they concern vegetation dynamics. In vegetation dynamics modelling, the 'seed effect' hypothesis is linked to seed distribution as generated by wind or animals. This hypothesis leads to a transport term in the model. For this hypothesis consideration, see Nahout and Mahrer (1989), Brufau et al. (1994), Brufau (1995) and El Jai et al. (1994,1996).

For vegetation dynamics modelling in general one can refer to Horn (1975a), Levain (1976), Shugart (1984), Budyko (1986), Huston et al. (1988), Nahout and Mahrer (1989), Smith et al. (1992), Monserud et al. (1993), Solomon and Shugart (1993) and EI Jai et al. (1996). Let us recall that spreadability consists in a sequence of increasing subdomains where the state of a distributed system satisfies a given spacial property. More precisely, for a given open-bounded set Q in the Euclidean space IR" with ^II = 1,2 or 3, and a distributed parameter system

Downloaded by [Carnegie Mellon University] at 08:34 25 January 2015

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682 A. El Jai andK. Kassara

(4)

(5) in ^Q x I

in Q z(O) =^Zo

OZ-+v·Vz=az

ot

Consider a distributed system governed by the following transport equation:

where v^E~n is a constant velocity field and a⁼ rr(x, I) is the coefficientof generation. The initial condition is given by

(2) inQ x I

Firstly we try to investigate some models that describe the motivating processes. It is well known that the modelization of these processes is based on the mass conservation principle (or the eontinuity equation) (Beltrami 1987) and leads to a class of transport systems (Banks and Kunish 1989, Dautray and Lions 1987) governed by the following partial differential equation:

oz-+v·Vz=¢(z)+q

or

where ^Zo^EU(Q) with p ;:::: I. As regards the boundary conditions, various situations for the transport models (4) have been studied by Dautray and Lions (1987). They generally involve the velocity field vand do not concern the whole boundary oQ while the problem is well posed. Amongst them we select the following absorbing condition (see Fig. I):

where v is the outward normal vector to oQ.

F, ={xEoQlv·v(x) < O}

where, z= z(x, ^{I) E}~ is the density of matter, q = q(x, I) E~ is the sink/source term and v= v(x, t)E ~n is the velocity field. The operator ¢(z) may contain a generation term and/or an eventual diffusion, and may be nonlinear (Levain 1976). Naturally (2) is completed with initial and boundary conditions.

In environmental science, numerous processes are governed by partial differential equations of the form (2).

One can cite, for instance, population dynamics (Levain 1976, Beltrami 1987), for which ¢(z) =az whereais the coefficient of generation. Interacting species models in a continuous environment (Levain 1976) are governed by (2) with

with

Z(/)/r_v =0 (tE1) (6)

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On the other hand, the class of systems (2) may be considered in water pollution or air-quality modelling.

In this case they involve a diffusion term as follows (see Diaz and Lions 1993, Friedman and Littman 1994):

¢(z) =-V dVz

¢(z)=pz(l - az) (3)

2.2. Solution

The study of existence, uniqueness and smoothness of the solutions of the system «4); (5); (6» is detailed by Dautray and Lions (1987). The assumption ZoEU(Q) leads to a solution in U(Q). Let A be the transport operator defined by

Considering the semi-group approach, it is well known that the densely defined and unbounded operator A generates a reo-semi-group (S(/»,>O on U(Q) given by the following formula:

whered holds for the diffusion coefficient.

The main contribution of this paper concerns the geometric assumption which characterizes the spreadability of the homogeneous system (2). This is developed in §3 and is illustrated by an example. Noticing the nonlinear aspect of the environmental processes (Diaz and Lyons 1993) which motivate the concept of spreadability we give, in §4,a class of nonlinear null-spreadable systems. The results obtained will be applied to the model of interacting species «2), (3» where the Shauder fixed- point theorem is considered for the proof of the solution.

A~= -v·V~

with domain

~(A)=g ^EU(Q)lv'V~^EU(Q) and ~/rv=O}

{^~(X ^{- VI)} ^if^I.(X)^> ^t

(S(/)~)(X)= 0

ift.(x) :::; I,

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(9)

2. The homogeneous transport equation 2.1. Mtuhematical statement

This section concerns mathematical preliminaries for the transport equation. In what follows we shall denote by {f= g} the set of pointsxEQ such that f(x) = g(x) where bothf and y mapQ into ~. IfB c Q and aE~n, 1/+^Bwill stand for the set ofa+ysuch thatyEB,and B' for the complementary set ~n\B. For I s ps co,

U(Q) denotes the space of. maps f: Q-> IR such that

Sillf I"dx < co, Finally, the gradient operator with respect to space variable will be designated by V.

for all xEQ and where

r.(x)=sup{t^EIlx - VSEQ; "Is< t},

Figure 1. Absorbing part of the boundary.

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Spreadabi/ity of transport systems 683

ift.(x)> t

Figure2. Segment s';

(x EQ) fT_x= (x - vI)n Q

3.1. Null-spreadability

Consider the case where the initial zone satisfies

be the portion of the segment [x - vtl; x] contained inside the domain Q. We can now show the following result.

Wo={zo =O} #-0

and let (see Fig. 2) Remark I: In the case where the domain Q is convex

then, by (10), we have

t.(x) > t -ee- X - vt^EQ

It follows that the semi-group (S(t)) may be easily expressed by replacing in (9) t,,(x) > t by x - vtEQ. 0

Now let us assume the term^(J to be in LP(Q x I) and let y =y(x, t) be defined by

y(x,t)=

t

^{x• ,)}

^,+ ^J:

rr(x - us, H )

"j

stands for the time that a particle, initially located at x, requires to hit the boundary aQ, under the velocity field v.

(13) Proof: Let us denote, for tE1,

W, ={z(t) =O}

where z(t) =z(·,r)stands for the solution of the system

«4);(5);(6». It follows from (12) and Remark I that Theorem I: The system «4);(5); (6)) is null-spreadable from Wo during the time interval I if and onlyif

1

Q

y(t)/lv = 0 y(O)=0

(II) where z is the solution of the system «4);(5);(6». By differentiating (11) we obtain the following system:

ay-+v'V'y=O Qx1

at

whose solution may be expressed by means of the semi-group operator(S(t» as follows:

y(t) = S(t)zo

Hence so that

Now, assume that (13) holds and let tEl and x^EW,;

then we have two exclusive situations.

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W,= {vt +(Q\wo)"} n Q

W, =[(vt +_{w o)}u (vt +QC)] n Q or equivalently

{

XEvt +Wo

X Ew, -ee- or

x Evt+QC

{

X - vtEWo x EW, ee- or

x - vt^EQc

ift.(x):<;; t ift.(x)> t From (9) we obtain

) {

zo(X - vt) ift.(x)> t y(x,^t =

o ift.(x):<;;t Consequently, it follows from (11) that z(x, t) =

t"(x - "')

^exp

u:

^{.(x -}

^{us, ,- sj}

^d

^j

XEVS+^Q' ^c w,.

Indeed, if it is not the case, we obtain by the convexity of^Q

x - vs= ;(x - vs)+

(I -

^;)x^E^Q,

and this implies (12)

Obviously formula (12) may be used in order to express the solution of the inhomogeneous system «4); (5); (6»

where a sink/source term qis added to (4).

3. Spreadability of linear transport systems

In this section we shall derive the conditions which make the system «4); (5); (6» spreadable. Thus let v,(J and Zo be as in the previous section and assume the domainQ

to be convex in order to apply Remark I.

(I) x - vtEQC which implies that x-vsEQC (s ~ t)

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684 A. £1 lai and K. Kassara (2) x - vtEWo: in this case let x' =^{x -} vt. By (13) we

have ?7,' ^cwo' ^{Now since}

x - vs=x' - v(s - t)E,rx'u Q',

thenxEi»,and so the family (W,),elis non-decreasing,

IEx] - I, IE, The initial zone Wo is limited by a parabola, Figure 5 illustrates the distributed state z(t) of the system and the associate increasing zones W, =

{z(t) =O} over discrete times. This is a very significant illustration of null-spreadability of the transport equation.

Conversely, let us assume the system((4); (5); (6»)to be null-spreadable from Woduring the time interval I and let x^E^Woo

Since the family (W,),el is non-decreasing we have

XEW, (tEI)

3.3, The general case: il-spreadability

Here we will analyse the more general case of 0- spreadability defined in (1).So let 0: Q -+ IR be a given state and consider the following set:

It follows from (14) that ( 15)

Remark 2:

I. It is convenient to mention that null-spreadability condition (13) depends neither on thc generation coefficient ⁽¹ nor, on the values of the initial state Zo

(thai is to say on the zonewo),

2.Clearly the condition (13) may be seen as a geo- metrical property involving both the velocity field v and the initial zone wo' Indeed the condition (13) stipulates that segments ,rx ^{(see Fig,} 2) remain in the zone Wo whenever x belongs to Woo This fact is

illustrated in Fig, 3. 0

and therefore ?7,^CWo' This ends the proof.

Proof: The solutionzof the system((4); (5);(6» is given by

Theorem 2 (A sufficient condition): Suppose that rhe qeneration coefficient⁽¹is null. Then the system((4); (5);(6»

is O-spreadoble from ^Wo and during the time interval I if

we haverliefollowing relation:

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(18) (17) (s~ t)

(x, t)ES ee- (x, s)EY' For all xEQ let

Y' =^{(x, t)EQ x II(x, x - vr)Egr}

Then we have the result,

) {

zo(X - vt) if x - vtEQ z(x,t =

o ^if ^{x -} vtrfQ

o

(I EI) x - VIEWou Q'

Similarly it is convenient to give some situations leading to non-null-spreadable systems (Fig, 4),

3,2, Simulation of a spreadable transport model

In this subsection we give numerical simulation results obtained by Matlab, They concern the homogeneous transport equation over a square domain Q =] - I,

Furthermore consider the zones

W, =^{z(t) =O} (tEI) Therefore, according to (15) and (16) we obtain

{

(x , x - vt)Egr

XEW, -ee- or

O(x)=⁰ ^and ^{x -} vtrf^Q

v

----+ ^v----+

Figure 3. Case of null-spreadability.

v ----+

Figure 4. Case of non-null-spreadability.

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686 A. EI Jai and K. Kassara

n

and let

) {

zo(X - vt) exp {g(x, t)}

y(x,t =

o

if x - vtEQ ifx - vtEQ' (23)

J

J Q e(t)jrv=⁰

e(O) =0

ay +v''Vy =yi/J(f)Nf Q x J at

y(t)jr_v=0 yeO) =^Zo

Letting e =y - z,it follows that

ae- +v''Ve = ei/J(f)Nf Q x J

at

Now, comparing (12) with ^(J = i/J(Z)Nf with (23), this implies that y is exactly the solution of the following partial differential equation:

where the boundary condition on r^v is due to (ii).

Conseq uently, e= 0and hence y= f, therefore (19)

Q x J

- +az ^V''Vz =^IL(Z)Nz

at

WII n

Figure 7. The set'!l.

4.1. A class of nonlinear spreadable systems

Let us consider the system described by the following partial differential equation:

where N: !?iJ(N) c U(Q) --+U(Q) is an operator which may be nonlinear and II maps IR intoIR and is such that

Moreover, we assume that equation (19) is completed by both a boundary condition on aQ x J and an initial condition z(O)=zo, providing a unique solution f(t) E

!?iJ(N), tEJ,and let

In the case of nonlinear O-spreadability, the difficulty comes from the fact that the solution has generally no explicit form.

Noticing that in (23) the exponential term never vanishes, one can deduce by application of theorem I that the family w, is non-decreasing and furthermore we have

(22). D

(t^E J)

w, ⁼ ^{f(t) =O} =^{yet) ⁼ O}

(21) (20)

(tEJ) (yEIR) IL(Y) = yi/J(y)

w, =^{f(t) =O}

4.2. Application to an interacting species model

Let us consider the system described by the following nonlinear partial differential equation

Then we have the following result:

Theorem 3: Further assumption (20) suppose that (i) The couple (wo,v) satisfies the null-spreadability

condition (13).

az- +^v''Vz⁼ ^{pz(1 -} ^(Xz)

at (24)

w, ={vt +^(Q\w_{o)' }}ⁿ ^Q (22) is locally integrable on the interval Jfor almost everyX EQ. Then the system (19)isnull-spreadable and furthermore we have

Proof: Given x^EQ and t EJsuch thatx - vtEQ,then by the assumption (iii) one can define

g(x, r)=

I

^{i/J(f(x -} vs, t - s»Nz(x - vs, t - s) ds t --+i/J(f(x,t»Nf(x, t)

(26) (zEJ) (25)

z(O)= Zo EU(Q) z(t)jfv = 0

First of all, we have to derive sufficient conditions under which the system «24); (25); (26» has a solution. That is the aim of the following result.

where z describes the density of a species submitted to interactions in a continuous environmentQwith another species. The coefficients p and(X are dependent upon the space variable x and the state of the other species. The velocity field is assumed to be constant. Now let us suppose that we have

so that there is no species on the absorbing subset I', (7). Moreover, assume the initial condition to be as follows:

(t^EJ) f(t)jf_v =0

(ii)

(iii) TIle map

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Spreadability of transport systems 687

(f<::0) where

where

(<jJEU(Q))

(tEI) (yEIR) /ley)= y

z(t)/f_v= 0 N<jJ =p(1 - rx<jJ) and

5. Conclusion

This paper shows that the transport equation in models describing processes which are subject to space expansion can be connected with the spreadability concept. In the case of the homogeneous transport equation, the conditions of spreadability have been derived from the properties of the semi-group generated by the transport operator.

These conditions concern both the velocity field and the initial zone. The results proven in the nonlinear case are of no less interest because they concern the processes that are more realistic in environmental modelling. Amongst the open studied questions, one may cite the following.

(I) H ow can the general case of the inhomogeneous transport equation be studied? This situation is very interesting since it is directly related to spray Now by Proposition I the system «24); (25); (26)) admits a unique solution z^{such that}

which is closed convex and bounded in the reflexive Banach space U(Q x I),so that e maps % into itself.

Moreover, it is easy to show by essentially the hypotheses (i) and (ii) that the operatore maps weakly convergent sequences into strongly convergent sequences so that it is weakly continuous.

Summarizing, the weakly continuous operatoremaps the closed convex and bounded subset % into itself, and therefore it admits (by the Shauder fixed-point theorem) a fixed point zwhich solves the system «24), (25), (26)).

D

Consequently, we are exactly in the circumstances of Theorem 3 which implies the null-spreadability of the

system «24); (25); (26)). D

Clearly the above result may be interpreted as follows.

In the case where the species is assumed to be extinct upon both absorbing set rv and initial zone wo, then it would be progressively extinguished over the whole domainQ. Nevertheless it is useful to mention that this fact is not affected by the interactions caused by the other species; this is due to the expression of the nonlinearity in the describing equation (24).

Proposition I: Under the hypotheses of the theorem (4),

ifthe couple (wo, v) satisfies the spreadability condition (13) then the nonlinear system «24); (25); (26)) is nul/- spreadable.

Proof: We have to apply Theorem 3. The system

«24); (25); (26)) may be written in the form (19) by considering

(31) (29) (27)

if x - vtE QC

ifx - vtEQ with (J =^rxp and

k(x,r) =

toe,

^~^''1^exp

u:

^p(x^~^us)

^d'}

in the case where x - vtEQ. Equivalently, it is easy to see that

{

k(X,t)G(x,t,z) if x - vtEQ

z(x, t)=

o if x - vtE QC

Consequently, this leads us to a fixed-point formulation as follows:

This allows us to examine the set z =ez

where the nonlinear operator EI is formally given by (ef)(x, t) = {k(X,t)G(x,t, f) ~f^{x -} ^vt^E^Q ⁽³²⁾

o ^If ^{x -} vtE QC

g(x, t) =exp

{t

p(x - vs)(1 - rx(x - vs) x z(x - vs,t - s)) dS} (28)

G(x,t, z) =^exp

{t -

fJ(x - vs)z(x - vs,t - s)dS}

(30)

%=UEU(QxI)IO~f~k}

Theorem 4: Under the fol/owing assumptions (i) ^ZoEU(Q); ^Zo <::0

(ii) po:EU(Q); o«<::0

where I < p< 00 and q=p/( p - I), then the system

«24); (25); (26)) has a solution satisfying

z<::0; ZEU(Q)

Proof: Let us consider the system «24); (25); (26)) where the term p(1 - rxz) is regarded as generation term, and apply formula (12). It follows that ifz is a solution of the system «24); (25); (26)) then it obeys the following equation for almost everywhere (x, t)EQ x I:

_ {zo(X - vt)g(x,r) ifx - vtEQ z(x, t) -

o ^{if x - vt}^{E QC}

for all f ^EU(Q x I). By the assumptions (i) and (ii) we can note that

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688 Spreadability of transport systems

control theory (EI Jai and Kassara 1994, 1996) and concerns the search of controls making a general transport distributed system spreadable.

(2) What happens when the absorbing boundary condi- tion is not homogeneous (i.e. z(t)/rv =⁹ where

(JEU(r_v))?

Acknowledgments

We thank B. Noumare and M. C. Simon for their valuable assistance in drawing the figures.

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