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Saddle point behaviour for a reaction-diffusion system: Application to a class of epidemic models
Mathematics and Computers in Simulation XXIV (1982) 540-547 North-Holland Publishing Company
540
SADDLE POINT BEHAVIOUR FOR A REACTION-DIFFUSION SYSTEM: APPLICATION TO A C L A S S OF E P I D E M I C M O D E L S V. CAPASSO and L. M A D D A L E N A Istituto vii Analisi Matematica, Palazzo Ateneo, 70121 Bari, Italy
i. Introduction In [5] the following system of two ordina ry differential equations was proposed to m o d e T the cholera epidemic which spread in the European Mediterranean regions in 1973 dz I d---~ - = -allzl+al2z2 (1.1) dz 2 d---~ = -a22z2+g(zl ) supplemented by suitable initial conditions. Here z I denotes the average concentration bacteria and z 2 the infective human population in a urban community. Hence the term -allZ 1 describes the natural growth of the bacterial population, while the term, al2z 2 is the contrib~ tion of the infective humans to the growth rate of bacteria. In the second equation, the term -a22z 2 describes the natural damping of the infective population due to the finite mean duration time of the infectiousness of humans. The last term g(z I) is the infection rate of humans under the assumption that the total susceptible human population is constant during the evolution of the epidemic. This kind of mechanism seems to be appropriate to interpret other epidemics with oro-fae cal transmission such as typhoid fever, infectious hepatitis, polyomelitis etc. with suitable modifications. Otherrelevant epidemic phenomena such as schi stosomiasis(see [i] and its references) can be mo deled as in (i.i) with a different meaning of the terms. In [15],~ a similar system of ODE was studied, modeling a cellular control process. To let the model be more realistic, in [2, 3] it was assumed that the bacteria diffuse randomly in the habitat due to the particular transmission mechanism in the regions where usual ly these kinds of epidemics spread. With this in mind, system (i.I) was modified as follows 3u 1 3---~ - = dlAUl-allul+al2u2
(1.2) 3u 2 -~¥- = d2Au2-a22u2+g(ul )
0378-4754/82/0000-0000/$02.75
with suitable boundary and initial conditions. Now u I and u 2 respectively denote the spa tial density of the bacterial population and of the infective human population in the habitat (usually a bounded domain in sn, n=1,2,3). The diffusion coefficients dl,d 2 are assumed to be greater or equal to zero. As for as the infecti ve human population is concerned the random dif f u s i o n may actually be neglected with respect to that of bacteria, but (for completeness) it has been in general allowed. The case d2=O was discussed with some detail in [3]. The case in which the parameters depend periodically on time (to take into account seasonal fluctuations) was studied in [4]. The exi stence and the asymptotic stability of periodi~ solutions was shown in this case for systems (I.I) and (1.2). In [2,3] assumptions of monotonicity and concavity were made on the function g(.) so that at must only two equilibrium points are present for system (i.i). Correspondingly the same happens for system (1.2), if we assume for this PDE system general boundary conditions of the third type. Threshold parameters were given which di scriminate between the extinction of the epide 7 mic and the asymptotic tendency to a nontrivial endemic level. Here we consider the same system (1.2) in the case in which the function g(-) has an inflection point being convex before it and conca ve after it. This case was considered in [I] for sch~tosomiasis, and [15] for the ODE system. The general problem of the behaviour of the solution near a saddle point for reaction-diffusion systems has been discussed by various authors (see e.g. [6, IO]). Here we study the gl! bal behaviour of our system in the positive cone. From an epidemiological point of view the case in which g is initially convex and then con cave is very interesting since it seems quite un realistic that the initial derivative of g(.) i~ greater than zero. Anyhow for large values of z, g should be sublinear and often tending to a con stant level [I, 5]. The origin is always locally asymptotically stable for system (i.i) and other two nontrivial equilibrium states may appear the "largest" of which being itself locally asymptotically sta ble, whenever it exists. In this case the smaller of the two points is a saddle point whose stable manifold divides the positive quadrant in two regions, one of attraction for the origin, the other of attraction for the largest equili-
© 1982 I M A C S / N o r t h - H o l l a n d
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system brium point. In this paper we study the influence of the presence of the saddle point for the behaviour of the PDE system (1.2) with homogeneous term and with general third type boundary condi tions.
2. Notations
and preliminary
results.
The aim of this paper is to study the following system of semilinear parabolic equations ~ t Ul(X;t)=dlAUl(X;t)-allUl(X;t)+al2u2(x;t) ~ t u2(x;t)=d2Au2(x;t)-a22u2(x;t)+g(ul(x;t))
541
L~I=I~11+I~21 its norm. In order to study system (2.1) we need to introduce the Banach space X:=C(~,R 2) (of the functions u=(ul,u2)' continuous in ~)-endowed with the norm IiuIl= su 2 Iuf(x)l+suplu2(x)I. x is x~ x~ an ordered Banach space with the pointwise partial order induced by the above said order ~ on R2; i.e. we set u~v in X iff u(x)~v(x) in ~. X+ ~ill denote the positive cone of X, X+={u~Xlu~O }. We shall say that ueX is minorized by veX+ if v~u; u~X+ is majorize~ by vEX+ if u~v; in both cases we say that u and v are comparable. If u~v we may introduce the order interval
[u,v]:={w~Xlu~w~v}. , (x;t)e~×(O,+=)
(2.1)
subject to boundary conditions
Let A be defined as the operator DA=diag (dl,d2)g with domain
of the form D(A):={u=(ul,u2)'cXIDAu~X,~i~ui+
Bi(x)~
(2.2)
ui(x;t)+~i(x)ui(x;t)=O (2.1b) , (x;t)eS~x(O,+~),
i=1,2
,
Here ~ denotes an open bounded subset of sn(n=l,2,3) whose boundary an is sufficiently regular; 3/3v denotes the outward normal derivative on 3~; ai(- ) and Bi(. ) are sufficiently smooth nonnegative ~i(x)+Bi(x)>O, x ~ ,
functions i=1,2.
System (2.1), initial conditions
(2.1b)
ui(x;O ) = u °i(x)
, in g
by
(2.1o)
not both identically zero. In (2.1) the diffusion coefficients may be assumed to be nonnegative real constants, but we shall discuss the case di>O , i=1,2; the analysis for the signifialong the same
lines [3]. We shall assume that all , a22 , a12 are p! sitive real constants and that the function g:$+->R+ satisfies the following assumptions: (i) if O
J
"
" K+
" [fl(~)/
(a) f (0)=0 (b) f is a quas~monotone nondecreasing function on $, i.e. fi(6) is nondecreasing in Cj with i~j (i,j=l,2). (c) for any ~e~ a ~oC~ exists, 0<<~o s.t. ~ ¢ o and f(~o)<
u(O) =u cX+
(iii) g is twice continuously differentiable, and a ~ $ + - { O } exists such that g"(z)>O
for
O
for
z*
(iv) li~ sup g(z) < alia22 z÷+~ z a12 As usual we shall make use of the following notations. Let g=(gl,g2)' and n=(ql,n2)' b E means that gig~i , i=1,2;
{>~
stands for ~an and ~#~; ~>>n stands for {i>ni , i=1,2. This establishes
a partial ordering
R~ We shall denote by ~:=$+x$+ ne of R2. If ~=(gi,~2)'e$2
2
i.e. F[u](x)=f(u(x) , ueX, xc~. Thanks to the assumptions made on g we can state the following properties for the function f:
o
g"(z)
~allg#al2g2/
~=(~i,~2)~=f(~)=If2(~)]:=~(~l)_a22~a]e $ (2.3)
d d--{ u(t) = A u(t)+F[u(t)]
then O
(ii) g(O)=O
long to R 2 ; ~ n
and let F denote the Nemitskii (substitution) operator generated by the nonlinear function
such that
is supplemented
cant case d2=O can be discussed
+~iuil3~=O,i=l,2}
on
, t>O
(2.4) (2.4o)
Under the above assumptions it canbe shown [7, 16] that a unique solution exists for problem (2.4), (2.4o) for any u°cX+. A bounded invariant "rectangle" can be defined for any solution u(t) of such system, thanks to (c), and this implies that any solution can be extended to the whole [0,+~). This solution is in fact a classical solution of system (2.1), (2.1b), (2.1o) if f is sufficiently smooth (we may assu me that fcC2($~)). For any u°eX+, we shall den~ te by U(t)u ° the unique solution of problem (2.4), (2.4o). It can be shown [3, 7] that operator U(t):X+÷X+, tE[O,+~) satisfies the following:
the positive co-
we shall denote by
U(O) = I
(2.5a)
542
1I. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system
U(t)U (s)=U(t+s)
,
U(t)O : O
s, te__R+
,
O
O
O
O
O
O
O
O
teR+
u ,v EX+,u .
O
O
(2.5b)
~*=(~,~)'~
(2.5c)
O is unstable while ly stable in 5-{O}.
(2.5d)
u ,v cX+,u .U(t)u <
O
,t>O (2.5e)
for any t>.O, the map u°cX ÷ U(t)u°EX is continuous, uniformly in t£[tl,t2]cR+
(2.5f)
for any u°~X, the map tE[O,+~) -~U(t)u ° is continuous.
(2.5g)
of the system f(~)=O.
Proof. - We have already remarked that, due to (2.5c) and (2.5e), the evolution operator V(t) of system (3.1) is positive; this can be dSrectl shown by observing that if we denote by B 2 the Jacobian matrix Jf(O+)
B2:=
-all
a12 I
g' (O) We may observe that the strict monotonic i ty properties (2.5d), (2.5e) with (2.5c) imply the positivity of the operator U(t), tcR+. Here we shall reduce as in [3] the problem of the existence and stability of the stationary solutions of system (2.4) to the corresponding problem for suitable ODE systems. 3. The ODE system. We start with the analysis of the ODE system associated with system (2.4) d
d-T {(t) = f(~(t))
,
t>O.
(3.1)
We shall denote by V(t), teR+, the evolution operator of (3.1) in K. It is obvious that properties analogous to (2.5a) - (2.5g) are valid for V(fl), too. The following theorems hold for system
(3.1). Proposition
Proof.
(3.6)
-a22 /
then c 6>0 exists s.t. for any
(3.7) ~e~,i~i<~:f(~)~B2~
.
Since B 2 is a quasimonotone trix, the positivity son theorems. Hence, V(t)¢°~
,
increasing ma-
of V(t) follows via compar ! if ~°¢K-{O} we have that
t>O
We can then state that in correspondence of a ~ E K - { O } and a time t>O a suitable rectangle Re, exists such that V(t) $ o
R
It can be shown that R can be chosen in such a way that ~*e~, and V~3R
3.1 - If
g(z) < alla22 - , for any z>O (3.2) z a12 then the trivial solution is globally asimptot! cally stable in the positive cone ~ for system (3.1).
In this case
¢* is globally asymptotical-
: f(~).~
(3.8)
This implies that R is an invariant rectangle containing only one equilibrium point ~*. By means of the contracting rectangles technique [14] it can be shown that ~* is then global ly asymptotically stable in 5-{0}. We study now the "intermediate" case respect to the two just analyzed.
- Let B 1 denote the matrix
-all
B1 =
a12)
a21 -
u
(3.3)
-a22
and the two nontrivial
g(z~
where a21=s p
z
. Then
=(~i,~2)
ZE~+
f(~)~B 1 < , for any <~$ .
(3.4)
Due to (3.2), B 1 has negative eigenvalues,
g'(O)al2 -
-
n =(nl,n2)' and
Let 0<<~*<<~*; we have that ~* is a saddle [i0, p. 106], in fact the Jacobian matrix
Jf(n*) has one negative
and one positive
eigen-
If we denote by M+ the stable manifold
3.2 - If e:=
point
solutions
of the system f(~)=O.
value.
and the theorem follows. Proposition
Proposition 3.3 - If @
>
I
(3.5)
alla22 then system (3.1) admits two equilibrium solutions in the positive cone K. They are given by the origin O=(O,O)'andthe only nontrivial solution
of
n*, it divides the cone K in two attraction domains for 0 and for $* respectively (we shall denote them by dom(0) and dom(~*) respectively). 0 and ~* are globally asymptotically stable in their own domain of attraction. Proof.
- Observe first that the Jacobian
of f is
543
I4 Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system contained respectively We would like to the above arguments is tonicity of f and upon of v(t).
given by -all Jf(g) =
a12
g,(61 )
6 =
-a22
~K 62
Under the hypotheses of the proposition, it is easy to show that Jf(O +) and Jr(6*) have both negative eigenvalues; hence O and ~* are locally asymptotically stable for system (3.1). On the other hand Jf(n*) has one positive and one negative eigenvalues; hence D* is a saddle point. Due to standard results [iO; p.lO7] it can be shown that in K these exist two curves M+ and M respectively known as stable manifold stable manifold of n* (see Fig. i).
and u~
in dom(O) and in dom(~*). stress that the proof of bas:,~! u p o n the quasimono the implied monotonicity
Remark 3.4 - We may remark here that the parame ter @ cannot be considered by itself alone a threshold parameter. In particular if @>I the nontrivial endemic level corresponding to the e quilibrium point 6" is globally asymptotically-stable. But if @
R, Rz
~
~
fl=O
2 =0
We distinguish two main cases respect to the boundary conditions (2.1b); homogeneous Neu mann boundary conditions (~i=~2=O) and general third type boundary conditions (which includes the case of homogeneous Dirichlet boundary conditions). 4.1. Homogeneous
•
Neumann boundary conditions.
'
Observe first that, due to the fact that f is quasimonotone, in this case the following lemma holds (see e.g. [12]). Lemma 4.1 - If initially u°£X+ is such that 4R
o'
~6 Fig. 1
M+ is positively
invariant
any 6(M+; M+ is negatively lim V(t)<=n*for t÷-~
and lim V(t)~=n * for t-~+~ invariant and
any 6eM_.
" , The lines $I = nl and ~2=n2 divide the po-
sitive cone K into four (closed)
rectanglesRi,
i=1,2,3,4 as shown in Fig. i. By an analysis of the vector field f it is not difficult to show that M_cR2u-R 4 and in particular it is contained in the (invariant) region between the two isoelines fl=O and f2=O. On the other hand it can be shown, by the contracting rectangles technique (see e.g. [2, 14]), that R2cdom(6* ) and R4cdom(O); it is o
o
~°~u°(x)~q °
, x~
(4.1)
oo with ¢ ,n E~, then
.
clear then that M+cRIUR3U{6
}.
Now if ~°eRi-M + then V(t)6°~M+,t>.O and lim V(t)$°#~ * [I0, p.107]. This implies that t-~+~ [17, p.280] the trajectory of 6 ° must leave R 1
V(t)6°~U(t)u°zV(t)~ °, te$+
(4.2)
where V(t) is the evolution operator of (3.1) and U(t) is the evolution operator of (2.4). It is clear then that Theorems 3.1 and 3.2 hold also for the PDE problem. Under the conditions of Prop.3.3 the only constant equilibrium solutions of system (2.4) are given by those of system (3.1), O,n*, 6*. Following [6, II] it can be shown that two sets M'+ and M' exist, both of them contained in X+, respectiv[ly called stable manifold and unstable manifold of n* such that if u°(Mi then U(t)u ° ÷ ~ * ( a s que backward
t÷+~), while if u°eM~,
then a un !
extension [(t;u°),te(-~,O]
for the solution
U(t)u °
exists
such that
o u(t;u )+n * (as t+-~). The structure of these manifolds is not clear in general but in this case we can establish a structural correspondence be tween M+, M_ in Theorem 3.3 and M$, M i. In fact
the following
theorem holds.
and enter R 2 if t ° is "above" M+; hence 6°e dom(~*). If ~o is "below" M it must enter R4; hence ~o£ dom(O). In an analogous way it can be shown that M+ divides
R 3 in two regions
Theorem 4.2 - Let the assumptions of Proposition 3.3• be satisfied. Then M+cM+, '" moreover , if o o u°cX+ zs such that two elements ~ ,N eM+ exist
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system
544 for which ~°~u°(x)~n °
, x~
(4.3)
which the range of the values of u ° in ~ "crosses" the manifold M+. The numerical results are displayed in Figs. 2-4.
then
0.5
Hu(t)u°-n*ll= o
lim
t'
t,O
_
.
t++~
0.4
F u r t h e r m o r e M_cM~; and i f B i s t h e n e g a t ! ve eigenvahe
of Jr(n*),
t h e n two n u m b e r s p,y>O
exist such that for any u°eB (n*), if to>O is P o such that for any t¢[O,to] , U(t)u ~B (n*), then P dist (U(t)u°,M~)
,
0.3 t
0.2
t
:
O
.
~
tE[O,to].
Proof. - The first part of the theorem follows from Lemma 4.1 if one takes into account Proposition 3.3. The second part follows from Theorem 3.3 in [6] and from the pro£erties of the spectrum o(A+Jf(n*)) [9].
~
5 t =2.5
o,
t-- 5
O~
t= 7.5
0
12
dl=dz-- 0.1
14
~6
a,z: 4
al 1. azz. 1
J8
I10
Ah:o.5 0=0
Remark 4.3 - If we denote by i
Fig. 2
*
R 2 := {ueX+I N ~u}
I
and
/t = 15
' t=]O
R~ := {u~X+Jn~1*} then, thanks to Theorem 4.2 we can state that
(R uR ) = Further properties of the domains of attraction dom (0) and dom (¢*) for the parabolic system (2.4) are given by the following theorem, which is a direct consequence of Lemma 4.1. Theorem 4.4 - Under the assumptions of Theorem o 4:2, if u c X + is majorized by a ~ d o m ( 0 ) , t h e n
t=3
0.8
t=l 0.6 0.4 t=O
0.2 0
lim IIU(t)u~I = O t++~
!
0 all:d2:0.01
I
2
4
a,f a22=1
a~2' 4
I
6
At :0.5
8
10
(~=0
o
If u EX+ is minorized by a ~°c dom (~*) Fig. 3
then
lim [IU(t)u°-~*ll= 0 . 3.5. It is important to make the following remark.
,~
Remark 4.5 - With homogeneous Neumann boundary conditions the results already obtained for the ODE system are reproduced for system (2.4) as far as the equilibrium solutions are concerned; the same is true for the asymptotic behaviour of solutions whose initial condition is eompara ble with elements of dom (O) or of dom (~*) (see Theorem 4.4). If this is not the case for u°~X+, it is rather difficult to give an analytlc answer about the qualitative behaviour of the solution U(t)u o. Anyhow numerical simulations have been carried out to show the behaviour of the solution in the particularly interesting case in
3. 2.5 2 1.5. 10.5_ 0
0
dl:az= 0.01
i
16
'4
a11. a22 ~ 1 a12s
4
Fig. 4
li
At =0.5 0=0
~10
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system The figures show the different behaviour of the solutions of system (2.4) for the Neumann (Figs.l,2),and Dirichlet (Fig.3) boundary conditions.In all cases the range of the values of the initial condition u°(x)#0 belongs in part to dom(O) and in part to dom(~*). Fig.l shows a ten dency toward O, while Fig.2 shows a tendency toward ~*. Fig.3 shows the eventual tendency of the solution toward a nontrivial nonhomogeneous equi librium state; initially the trivial solution "attracts" the lower parts of u°(x).
545
Let now ~M and ~M denote the first eigenvalue and the corresponding eigenfunction, spectively, of problem (4.5) when ~-~M = max max ~. (x) x~3~ ic{l,2} I ~---Bm= min xe~
re-
(4.10)
min B. (x) it{l,2} I
Theorem 4.7 - If g' (O) a12
(+)
@M :=
4.2. Boundary conditions of the third type.
> 1
(4.11)
(all+d IIM ) (a22+d21M)
When the coefficients ~.(.) in (2.1b) are 1 not both identically zero, the only spatially homogeneous equilibrium solution of system (2.4) is the origin. We give here conditions for the asymptotic stability of this point; i.e. conditions for the eventual extinction of the epidemic. We denote by Xm the first eigenvalue asso
Moreover two nontrivial equilibrium solutions
ciated with the boundary value problem
O<<~_~<~+ exist such that for any u°~X+-{O}:
A~ + I~ = 0 ~ ~-f~+
m
that for any u°~X+, u°#O: lim sup IIU(t)u~I >~ k t->+~
in on ~
= min min ~.(x) x~3g iE{l,2} l
(4.12)
lim dist(U(t)u °,[~_,~+])=O t->~
(4.5)
a ~ = O
.
(4.13)
Here dist denotes the distance associated with the sup norm.
when ~
then the trivial solution is an unstable equili brium solution; in particular a k>O exists such
(4.6)
B~BM= max max B.(x) x~3~ i~{1,2} I Theorem 4.6 - If
Proof. - Equation (4.11) implies in particular that (3.5) holds. Then, as stated in Proposition 3.2, system (3.1) admits two equilibrium solutions in the po sitive cone _Kc_R2. They are given by the origin-O=(O,O)' and-the only nontrivial solution =(~I,~2)'EK of the system f(¢)=O. Due to the
g(z) < (all+dlXm)(a22+d21m) , for any z>O (4.7) z a12 then the trivial solution is globally asymptot! cally stable in X+ for system (2.4). Proof. - If B I is defined as in (3.3) and if Tm(t) , t~O, denotes the evolution operator of the linear system d u(t) = AmU (t) +BlU(t) d-~
' t>O
(4.8)
where A m defined as in (2.2) with 6i(x)=B M and ~i(x)=~m
assumptions made on g, under condition (3.5) it is possible to find another function g:R+~R+ such that (i), (ii), (iii) hold for it; g is strictly concave in R+; in addition to this we may impose that
a)
g'(O)
b)
g(~l ) ~ g(~l )
for any ~I
c)
~(~i ) = g(~l )
for any ~ i ~ I
= g'(O)
Let
(i=1,2; xc3~), then f(O:=(-all~l+al2~21,
U(t)u°~Tm(t)u°
, for any u°eX+, and t>O.
~ ~[~i I c ~
(4.14)
(4.9) ~g (~I)-a22 ~2 /
SiNce, under the assumptions made, the origin is globally asymptotically stable for system (4.8), the theorem follows from (4.9) (see [13; Cor. 3]). (+) For homogeneous Dirichlet boundary conditions the space X has to be suitable modified; this does not affect the statements of the following theorems [4].
\~2/
and let U(t) denote the evolution operator of the following system d u(t) = Au(t)+F[u(t) ] , t>O d--t-
(4.15)
where F is the Nemitskii operator generated by Due to b) and c), by comparison theorems it is clear that
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system
546
U(t) ~ U(t)
, t~O
(4.16)
If we now proceed as in [3], it can be shown that for system (4.15) the origin is unstable and (4.12) holds for U(t), and hence for U(t). Moreover a unique nontrivial equilibrium solution 4>>O exists for system (4.15), which is globally asymptotically stable in X+-{O}. Furthermore for any choice of uOeX+-{O} a subsolution ! and a supersolution ~ * stem
of sy-
A~ + FIe] = 0
(4.17)
exists such that u o ~<¢Mn. In correspondence of ~, an c>O exists such that f(¢)~
for any ~EK
s.t. ~,
(4.23)
Due to (4.23) we have that g U(t)u ° .< TM(t)¢Mn
(4.24)
where T~(t) is the evolution operator of system (4.22). Since TM(t)~M~ monotonically
tends to zero
the theorem follows. can be found such that ¢~U(~)u°$~ for a suitable ~>O.
Observe now that since fsf in K is also a subsolution of
then
A¢ + FIe] = O Moreover
(4.18)
since $ ~ * ,
and f(~)=f(~)
for
~ ¢ * than ~ is also a supersolution of (4.18). Due to monotonicity arguments [3] we can then state that a ¢_>>O, and a ¢+~¢_ exist such that (4.13) holds. Theorem 4.8 - If @M
<
(all+dl%M)(a22+d2XM)
z
(4.19)
a12
is not satisfied for any z>O, then the trivial solution is locally asymptotically stable. Moreover if n** denotes the saddle point of the following ODE system d d-~ then,
= -%MD¢ + f(~)
for
any u°eX+,
,
t>O
(4.20)
o ** u <¢M ~ , we h a v e
Proof. - Consider the matrix
=
-all a12 ) g'(O) +Ct~ -a22 !
(4.21)
Since @MO such that the matrix
(-XMD+Bc) has both negative eigenvalues,
and then the trivial solution is globally asymptotically stable for the linear system d
d--t v(t) = AMV(t)
+ B v(t)
, t>O
(4.22)
where A M is defined as in (2.2) with 6i(x)=B m and ~i(x)=~M
(i=1,2; x e ~ ) .
Let u°~X+, u°<¢Mn
lim IIu(t)u°-¢+II = o t~++~
(4.25)
References [1] R.M. Anderson, The dynamics and control of direct life cycle helminth parasites, in Vito Volterra Symposium on Math. Models in Biology (C. Barigozzi, ed.) Lect. Notes in Biomath., 39, (1980), 278-322. [2] V. Capasso and L. Maddalena, A non linear diffusion system modelling the spread of oro faecal diseases, in Nonlinear Phenomena in Mathematical Sciences (V. Lakshmikantham, ed.), Ac. Press., New York, 1981. [3] V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biolosy, 13, (1981), 173-184. [4] V. Capasso and L. Maddalena, Periodic Solutions for a reaction-diffusion system model llng the spread of a class of epidemics, SIAM J. Appl. Math., to appear.
lim llUCt)u°II = O t~+~
B
Remark 4.9 - It can be easily shown that under the assumptions of Theorem 4.8 an equilibrium solution ¢+EX+ exists such that if u°EX+ u°>.¢+ then
. Then an ne~, ~
[5] V. Capasso and S.L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidem. et Sant~ Publ., 27, (1979). 121-132. Errata, Ibidem 28, (1980), 330. [6] N. Chafee, Behaviour of solutions leaving the neighborhood of a saddle point for a non linear evolution equation, J. Math. Anal. Appl., 58, (1977), 312-325. [7] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969. [8] L. Galeone, Positivity and asymptotic behaviour of the numerical solution of weakly coupled systems of semilinear parabolic dif ferential equations, Proc. IOth. I M A C S W o r n Congress, Montreal, 1982. [9] K.P. Hadeler, Diffusion
equations
in biolo-
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system gy, in Mathematics of Biology, C.I.M.E. L~ guori Editore, 1979, Napoli. [IO] J.H. Hale, Ordinary Differential Equations, Wiley-lnterscience, New York, 1969. [Ii] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. [12] V. Lakshmikantham, Comparison results for reaction-diffusion equations in a Banach space, Conf. Sem. Mat. Univ. Bari, 161, (1979). [13] R.H. Martin, Asymptotic stability and critical points for nonlinear quasimonotone p~ rabolic systems, J. Diff. Equations 30, (1978), 391-423. [14] J. Rauch and J.A. Smoller, Qualitative theo ry of the Fitzhugh-Nagumo equations, Advan ~ ces in Math. 27 (1978), 12-44. [15] J.F. Selgrade, Mathematical analysis of a cellular control process with positive feed back, SlAM J. Appl. Math., 36, (1979), 219-229. [16] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 229-310. [17] H.H. Wilson, Ordinary Differential Equations, Addison-Wesley Publ. Comp., London, 1971.
Ackowled~ement. It is a great pleasure to acknowledge the useful contribution of Prof.L.Galeone whose numerical work on this subject lead to Figs. 2-4.(See [8]7.
547
540
SADDLE POINT BEHAVIOUR FOR A REACTION-DIFFUSION SYSTEM: APPLICATION TO A C L A S S OF E P I D E M I C M O D E L S V. CAPASSO and L. M A D D A L E N A Istituto vii Analisi Matematica, Palazzo Ateneo, 70121 Bari, Italy
i. Introduction In [5] the following system of two ordina ry differential equations was proposed to m o d e T the cholera epidemic which spread in the European Mediterranean regions in 1973 dz I d---~ - = -allzl+al2z2 (1.1) dz 2 d---~ = -a22z2+g(zl ) supplemented by suitable initial conditions. Here z I denotes the average concentration bacteria and z 2 the infective human population in a urban community. Hence the term -allZ 1 describes the natural growth of the bacterial population, while the term, al2z 2 is the contrib~ tion of the infective humans to the growth rate of bacteria. In the second equation, the term -a22z 2 describes the natural damping of the infective population due to the finite mean duration time of the infectiousness of humans. The last term g(z I) is the infection rate of humans under the assumption that the total susceptible human population is constant during the evolution of the epidemic. This kind of mechanism seems to be appropriate to interpret other epidemics with oro-fae cal transmission such as typhoid fever, infectious hepatitis, polyomelitis etc. with suitable modifications. Otherrelevant epidemic phenomena such as schi stosomiasis(see [i] and its references) can be mo deled as in (i.i) with a different meaning of the terms. In [15],~ a similar system of ODE was studied, modeling a cellular control process. To let the model be more realistic, in [2, 3] it was assumed that the bacteria diffuse randomly in the habitat due to the particular transmission mechanism in the regions where usual ly these kinds of epidemics spread. With this in mind, system (i.I) was modified as follows 3u 1 3---~ - = dlAUl-allul+al2u2
(1.2) 3u 2 -~¥- = d2Au2-a22u2+g(ul )
0378-4754/82/0000-0000/$02.75
with suitable boundary and initial conditions. Now u I and u 2 respectively denote the spa tial density of the bacterial population and of the infective human population in the habitat (usually a bounded domain in sn, n=1,2,3). The diffusion coefficients dl,d 2 are assumed to be greater or equal to zero. As for as the infecti ve human population is concerned the random dif f u s i o n may actually be neglected with respect to that of bacteria, but (for completeness) it has been in general allowed. The case d2=O was discussed with some detail in [3]. The case in which the parameters depend periodically on time (to take into account seasonal fluctuations) was studied in [4]. The exi stence and the asymptotic stability of periodi~ solutions was shown in this case for systems (I.I) and (1.2). In [2,3] assumptions of monotonicity and concavity were made on the function g(.) so that at must only two equilibrium points are present for system (i.i). Correspondingly the same happens for system (1.2), if we assume for this PDE system general boundary conditions of the third type. Threshold parameters were given which di scriminate between the extinction of the epide 7 mic and the asymptotic tendency to a nontrivial endemic level. Here we consider the same system (1.2) in the case in which the function g(-) has an inflection point being convex before it and conca ve after it. This case was considered in [I] for sch~tosomiasis, and [15] for the ODE system. The general problem of the behaviour of the solution near a saddle point for reaction-diffusion systems has been discussed by various authors (see e.g. [6, IO]). Here we study the gl! bal behaviour of our system in the positive cone. From an epidemiological point of view the case in which g is initially convex and then con cave is very interesting since it seems quite un realistic that the initial derivative of g(.) i~ greater than zero. Anyhow for large values of z, g should be sublinear and often tending to a con stant level [I, 5]. The origin is always locally asymptotically stable for system (i.i) and other two nontrivial equilibrium states may appear the "largest" of which being itself locally asymptotically sta ble, whenever it exists. In this case the smaller of the two points is a saddle point whose stable manifold divides the positive quadrant in two regions, one of attraction for the origin, the other of attraction for the largest equili-
© 1982 I M A C S / N o r t h - H o l l a n d
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system brium point. In this paper we study the influence of the presence of the saddle point for the behaviour of the PDE system (1.2) with homogeneous term and with general third type boundary condi tions.
2. Notations
and preliminary
results.
The aim of this paper is to study the following system of semilinear parabolic equations ~ t Ul(X;t)=dlAUl(X;t)-allUl(X;t)+al2u2(x;t) ~ t u2(x;t)=d2Au2(x;t)-a22u2(x;t)+g(ul(x;t))
541
L~I=I~11+I~21 its norm. In order to study system (2.1) we need to introduce the Banach space X:=C(~,R 2) (of the functions u=(ul,u2)' continuous in ~)-endowed with the norm IiuIl= su 2 Iuf(x)l+suplu2(x)I. x is x~ x~ an ordered Banach space with the pointwise partial order induced by the above said order ~ on R2; i.e. we set u~v in X iff u(x)~v(x) in ~. X+ ~ill denote the positive cone of X, X+={u~Xlu~O }. We shall say that ueX is minorized by veX+ if v~u; u~X+ is majorize~ by vEX+ if u~v; in both cases we say that u and v are comparable. If u~v we may introduce the order interval
[u,v]:={w~Xlu~w~v}. , (x;t)e~×(O,+=)
(2.1)
subject to boundary conditions
Let A be defined as the operator DA=diag (dl,d2)g with domain
of the form D(A):={u=(ul,u2)'cXIDAu~X,~i~ui+
Bi(x)~
(2.2)
ui(x;t)+~i(x)ui(x;t)=O (2.1b) , (x;t)eS~x(O,+~),
i=1,2
,
Here ~ denotes an open bounded subset of sn(n=l,2,3) whose boundary an is sufficiently regular; 3/3v denotes the outward normal derivative on 3~; ai(- ) and Bi(. ) are sufficiently smooth nonnegative ~i(x)+Bi(x)>O, x ~ ,
functions i=1,2.
System (2.1), initial conditions
(2.1b)
ui(x;O ) = u °i(x)
, in g
by
(2.1o)
not both identically zero. In (2.1) the diffusion coefficients may be assumed to be nonnegative real constants, but we shall discuss the case di>O , i=1,2; the analysis for the signifialong the same
lines [3]. We shall assume that all , a22 , a12 are p! sitive real constants and that the function g:$+->R+ satisfies the following assumptions: (i) if O
J
"
" K+
" [fl(~)/
(a) f (0)=0 (b) f is a quas~monotone nondecreasing function on $, i.e. fi(6) is nondecreasing in Cj with i~j (i,j=l,2). (c) for any ~e~ a ~oC~ exists, 0<<~o s.t. ~ ¢ o and f(~o)<
u(O) =u cX+
(iii) g is twice continuously differentiable, and a ~ $ + - { O } exists such that g"(z)>O
for
O
for
z*
(iv) li~ sup g(z) < alia22 z÷+~ z a12 As usual we shall make use of the following notations. Let g=(gl,g2)' and n=(ql,n2)' b E means that gig~i , i=1,2;
{>~
stands for ~an and ~#~; ~>>n stands for {i>ni , i=1,2. This establishes
a partial ordering
R~ We shall denote by ~:=$+x$+ ne of R2. If ~=(gi,~2)'e$2
2
i.e. F[u](x)=f(u(x) , ueX, xc~. Thanks to the assumptions made on g we can state the following properties for the function f:
o
g"(z)
~allg#al2g2/
~=(~i,~2)~=f(~)=If2(~)]:=~(~l)_a22~a]e $ (2.3)
d d--{ u(t) = A u(t)+F[u(t)]
then O
(ii) g(O)=O
long to R 2 ; ~ n
and let F denote the Nemitskii (substitution) operator generated by the nonlinear function
such that
is supplemented
cant case d2=O can be discussed
+~iuil3~=O,i=l,2}
on
, t>O
(2.4) (2.4o)
Under the above assumptions it canbe shown [7, 16] that a unique solution exists for problem (2.4), (2.4o) for any u°cX+. A bounded invariant "rectangle" can be defined for any solution u(t) of such system, thanks to (c), and this implies that any solution can be extended to the whole [0,+~). This solution is in fact a classical solution of system (2.1), (2.1b), (2.1o) if f is sufficiently smooth (we may assu me that fcC2($~)). For any u°eX+, we shall den~ te by U(t)u ° the unique solution of problem (2.4), (2.4o). It can be shown [3, 7] that operator U(t):X+÷X+, tE[O,+~) satisfies the following:
the positive co-
we shall denote by
U(O) = I
(2.5a)
542
1I. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system
U(t)U (s)=U(t+s)
,
U(t)O : O
s, te__R+
,
O
O
O
O
O
O
O
O
teR+
u ,v EX+,u .
O
O
(2.5b)
~*=(~,~)'~
(2.5c)
O is unstable while ly stable in 5-{O}.
(2.5d)
u ,v cX+,u .
O
,t>O (2.5e)
for any t>.O, the map u°cX ÷ U(t)u°EX is continuous, uniformly in t£[tl,t2]cR+
(2.5f)
for any u°~X, the map tE[O,+~) -~U(t)u ° is continuous.
(2.5g)
of the system f(~)=O.
Proof. - We have already remarked that, due to (2.5c) and (2.5e), the evolution operator V(t) of system (3.1) is positive; this can be dSrectl shown by observing that if we denote by B 2 the Jacobian matrix Jf(O+)
B2:=
-all
a12 I
g' (O) We may observe that the strict monotonic i ty properties (2.5d), (2.5e) with (2.5c) imply the positivity of the operator U(t), tcR+. Here we shall reduce as in [3] the problem of the existence and stability of the stationary solutions of system (2.4) to the corresponding problem for suitable ODE systems. 3. The ODE system. We start with the analysis of the ODE system associated with system (2.4) d
d-T {(t) = f(~(t))
,
t>O.
(3.1)
We shall denote by V(t), teR+, the evolution operator of (3.1) in K. It is obvious that properties analogous to (2.5a) - (2.5g) are valid for V(fl), too. The following theorems hold for system
(3.1). Proposition
Proof.
(3.6)
-a22 /
then c 6>0 exists s.t. for any
(3.7) ~e~,i~i<~:f(~)~B2~
.
Since B 2 is a quasimonotone trix, the positivity son theorems. Hence, V(t)¢°~
,
increasing ma-
of V(t) follows via compar ! if ~°¢K-{O} we have that
t>O
We can then state that in correspondence of a ~ E K - { O } and a time t>O a suitable rectangle Re, exists such that V(t) $ o
R
It can be shown that R can be chosen in such a way that ~*e~, and V~3R
3.1 - If
g(z) < alla22 - , for any z>O (3.2) z a12 then the trivial solution is globally asimptot! cally stable in the positive cone ~ for system (3.1).
In this case
¢* is globally asymptotical-
: f(~).~
(3.8)
This implies that R is an invariant rectangle containing only one equilibrium point ~*. By means of the contracting rectangles technique [14] it can be shown that ~* is then global ly asymptotically stable in 5-{0}. We study now the "intermediate" case respect to the two just analyzed.
- Let B 1 denote the matrix
-all
B1 =
a12)
a21 -
u
(3.3)
-a22
and the two nontrivial
g(z~
where a21=s p
z
. Then
=(~i,~2)
ZE~+
f(~)~B 1 < , for any <~$ .
(3.4)
Due to (3.2), B 1 has negative eigenvalues,
g'(O)al2 -
-
n =(nl,n2)' and
Let 0<<~*<<~*; we have that ~* is a saddle [i0, p. 106], in fact the Jacobian matrix
Jf(n*) has one negative
and one positive
eigen-
If we denote by M+ the stable manifold
3.2 - If e:=
point
solutions
of the system f(~)=O.
value.
and the theorem follows. Proposition
Proposition 3.3 - If @
>
I
(3.5)
alla22 then system (3.1) admits two equilibrium solutions in the positive cone K. They are given by the origin O=(O,O)'andthe only nontrivial solution
of
n*, it divides the cone K in two attraction domains for 0 and for $* respectively (we shall denote them by dom(0) and dom(~*) respectively). 0 and ~* are globally asymptotically stable in their own domain of attraction. Proof.
- Observe first that the Jacobian
of f is
543
I4 Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system contained respectively We would like to the above arguments is tonicity of f and upon of v(t).
given by -all Jf(g) =
a12
g,(61 )
6 =
-a22
~K 62
Under the hypotheses of the proposition, it is easy to show that Jf(O +) and Jr(6*) have both negative eigenvalues; hence O and ~* are locally asymptotically stable for system (3.1). On the other hand Jf(n*) has one positive and one negative eigenvalues; hence D* is a saddle point. Due to standard results [iO; p.lO7] it can be shown that in K these exist two curves M+ and M respectively known as stable manifold stable manifold of n* (see Fig. i).
and u~
in dom(O) and in dom(~*). stress that the proof of bas:,~! u p o n the quasimono the implied monotonicity
Remark 3.4 - We may remark here that the parame ter @ cannot be considered by itself alone a threshold parameter. In particular if @>I the nontrivial endemic level corresponding to the e quilibrium point 6" is globally asymptotically-stable. But if @
R, Rz
~
~
fl=O
2 =0
We distinguish two main cases respect to the boundary conditions (2.1b); homogeneous Neu mann boundary conditions (~i=~2=O) and general third type boundary conditions (which includes the case of homogeneous Dirichlet boundary conditions). 4.1. Homogeneous
•
Neumann boundary conditions.
'
Observe first that, due to the fact that f is quasimonotone, in this case the following lemma holds (see e.g. [12]). Lemma 4.1 - If initially u°£X+ is such that 4R
o'
~6 Fig. 1
M+ is positively
invariant
any 6(M+; M+ is negatively lim V(t)<=n*for t÷-~
and lim V(t)~=n * for t-~+~ invariant and
any 6eM_.
" , The lines $I = nl and ~2=n2 divide the po-
sitive cone K into four (closed)
rectanglesRi,
i=1,2,3,4 as shown in Fig. i. By an analysis of the vector field f it is not difficult to show that M_cR2u-R 4 and in particular it is contained in the (invariant) region between the two isoelines fl=O and f2=O. On the other hand it can be shown, by the contracting rectangles technique (see e.g. [2, 14]), that R2cdom(6* ) and R4cdom(O); it is o
o
~°~u°(x)~q °
, x~
(4.1)
oo with ¢ ,n E~, then
.
clear then that M+cRIUR3U{6
}.
Now if ~°eRi-M + then V(t)6°~M+,t>.O and lim V(t)$°#~ * [I0, p.107]. This implies that t-~+~ [17, p.280] the trajectory of 6 ° must leave R 1
V(t)6°~U(t)u°zV(t)~ °, te$+
(4.2)
where V(t) is the evolution operator of (3.1) and U(t) is the evolution operator of (2.4). It is clear then that Theorems 3.1 and 3.2 hold also for the PDE problem. Under the conditions of Prop.3.3 the only constant equilibrium solutions of system (2.4) are given by those of system (3.1), O,n*, 6*. Following [6, II] it can be shown that two sets M'+ and M' exist, both of them contained in X+, respectiv[ly called stable manifold and unstable manifold of n* such that if u°(Mi then U(t)u ° ÷ ~ * ( a s que backward
t÷+~), while if u°eM~,
then a un !
extension [(t;u°),te(-~,O]
for the solution
U(t)u °
exists
such that
o u(t;u )+n * (as t+-~). The structure of these manifolds is not clear in general but in this case we can establish a structural correspondence be tween M+, M_ in Theorem 3.3 and M$, M i. In fact
the following
theorem holds.
and enter R 2 if t ° is "above" M+; hence 6°e dom(~*). If ~o is "below" M it must enter R4; hence ~o£ dom(O). In an analogous way it can be shown that M+ divides
R 3 in two regions
Theorem 4.2 - Let the assumptions of Proposition 3.3• be satisfied. Then M+cM+, '" moreover , if o o u°cX+ zs such that two elements ~ ,N eM+ exist
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system
544 for which ~°~u°(x)~n °
, x~
(4.3)
which the range of the values of u ° in ~ "crosses" the manifold M+. The numerical results are displayed in Figs. 2-4.
then
0.5
Hu(t)u°-n*ll= o
lim
t'
t,O
_
.
t++~
0.4
F u r t h e r m o r e M_cM~; and i f B i s t h e n e g a t ! ve eigenvahe
of Jr(n*),
t h e n two n u m b e r s p,y>O
exist such that for any u°eB (n*), if to>O is P o such that for any t¢[O,to] , U(t)u ~B (n*), then P dist (U(t)u°,M~)
,
0.3 t
0.2
t
:
O
.
~
tE[O,to].
Proof. - The first part of the theorem follows from Lemma 4.1 if one takes into account Proposition 3.3. The second part follows from Theorem 3.3 in [6] and from the pro£erties of the spectrum o(A+Jf(n*)) [9].
~
5 t =2.5
o,
t-- 5
O~
t= 7.5
0
12
dl=dz-- 0.1
14
~6
a,z: 4
al 1. azz. 1
J8
I10
Ah:o.5 0=0
Remark 4.3 - If we denote by i
Fig. 2
*
R 2 := {ueX+I N ~u}
I
and
/t = 15
' t=]O
R~ := {u~X+Jn~1*} then, thanks to Theorem 4.2 we can state that
(R uR ) = Further properties of the domains of attraction dom (0) and dom (¢*) for the parabolic system (2.4) are given by the following theorem, which is a direct consequence of Lemma 4.1. Theorem 4.4 - Under the assumptions of Theorem o 4:2, if u c X + is majorized by a ~ d o m ( 0 ) , t h e n
t=3
0.8
t=l 0.6 0.4 t=O
0.2 0
lim IIU(t)u~I = O t++~
!
0 all:d2:0.01
I
2
4
a,f a22=1
a~2' 4
I
6
At :0.5
8
10
(~=0
o
If u EX+ is minorized by a ~°c dom (~*) Fig. 3
then
lim [IU(t)u°-~*ll= 0 . 3.5. It is important to make the following remark.
,~
Remark 4.5 - With homogeneous Neumann boundary conditions the results already obtained for the ODE system are reproduced for system (2.4) as far as the equilibrium solutions are concerned; the same is true for the asymptotic behaviour of solutions whose initial condition is eompara ble with elements of dom (O) or of dom (~*) (see Theorem 4.4). If this is not the case for u°~X+, it is rather difficult to give an analytlc answer about the qualitative behaviour of the solution U(t)u o. Anyhow numerical simulations have been carried out to show the behaviour of the solution in the particularly interesting case in
3. 2.5 2 1.5. 10.5_ 0
0
dl:az= 0.01
i
16
'4
a11. a22 ~ 1 a12s
4
Fig. 4
li
At =0.5 0=0
~10
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction- diffusion system The figures show the different behaviour of the solutions of system (2.4) for the Neumann (Figs.l,2),and Dirichlet (Fig.3) boundary conditions.In all cases the range of the values of the initial condition u°(x)#0 belongs in part to dom(O) and in part to dom(~*). Fig.l shows a ten dency toward O, while Fig.2 shows a tendency toward ~*. Fig.3 shows the eventual tendency of the solution toward a nontrivial nonhomogeneous equi librium state; initially the trivial solution "attracts" the lower parts of u°(x).
545
Let now ~M and ~M denote the first eigenvalue and the corresponding eigenfunction, spectively, of problem (4.5) when ~-~M = max max ~. (x) x~3~ ic{l,2} I ~---Bm= min xe~
re-
(4.10)
min B. (x) it{l,2} I
Theorem 4.7 - If g' (O) a12
(+)
@M :=
4.2. Boundary conditions of the third type.
> 1
(4.11)
(all+d IIM ) (a22+d21M)
When the coefficients ~.(.) in (2.1b) are 1 not both identically zero, the only spatially homogeneous equilibrium solution of system (2.4) is the origin. We give here conditions for the asymptotic stability of this point; i.e. conditions for the eventual extinction of the epidemic. We denote by Xm the first eigenvalue asso
Moreover two nontrivial equilibrium solutions
ciated with the boundary value problem
O<<~_~<~+ exist such that for any u°~X+-{O}:
A~ + I~ = 0 ~ ~-f~+
m
that for any u°~X+, u°#O: lim sup IIU(t)u~I >~ k t->+~
in on ~
= min min ~.(x) x~3g iE{l,2} l
(4.12)
lim dist(U(t)u °,[~_,~+])=O t->~
(4.5)
a ~ = O
.
(4.13)
Here dist denotes the distance associated with the sup norm.
when ~
then the trivial solution is an unstable equili brium solution; in particular a k>O exists such
(4.6)
B~BM= max max B.(x) x~3~ i~{1,2} I Theorem 4.6 - If
Proof. - Equation (4.11) implies in particular that (3.5) holds. Then, as stated in Proposition 3.2, system (3.1) admits two equilibrium solutions in the po sitive cone _Kc_R2. They are given by the origin-O=(O,O)' and-the only nontrivial solution =(~I,~2)'EK of the system f(¢)=O. Due to the
g(z) < (all+dlXm)(a22+d21m) , for any z>O (4.7) z a12 then the trivial solution is globally asymptot! cally stable in X+ for system (2.4). Proof. - If B I is defined as in (3.3) and if Tm(t) , t~O, denotes the evolution operator of the linear system d u(t) = AmU (t) +BlU(t) d-~
' t>O
(4.8)
where A m defined as in (2.2) with 6i(x)=B M and ~i(x)=~m
assumptions made on g, under condition (3.5) it is possible to find another function g:R+~R+ such that (i), (ii), (iii) hold for it; g is strictly concave in R+; in addition to this we may impose that
a)
g'(O)
b)
g(~l ) ~ g(~l )
for any ~I
c)
~(~i ) = g(~l )
for any ~ i ~ I
= g'(O)
Let
(i=1,2; xc3~), then f(O:=(-all~l+al2~21,
U(t)u°~Tm(t)u°
, for any u°eX+, and t>O.
~ ~[~i I c ~
(4.14)
(4.9) ~g (~I)-a22 ~2 /
SiNce, under the assumptions made, the origin is globally asymptotically stable for system (4.8), the theorem follows from (4.9) (see [13; Cor. 3]). (+) For homogeneous Dirichlet boundary conditions the space X has to be suitable modified; this does not affect the statements of the following theorems [4].
\~2/
and let U(t) denote the evolution operator of the following system d u(t) = Au(t)+F[u(t) ] , t>O d--t-
(4.15)
where F is the Nemitskii operator generated by Due to b) and c), by comparison theorems it is clear that
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system
546
U(t) ~ U(t)
, t~O
(4.16)
If we now proceed as in [3], it can be shown that for system (4.15) the origin is unstable and (4.12) holds for U(t), and hence for U(t). Moreover a unique nontrivial equilibrium solution 4>>O exists for system (4.15), which is globally asymptotically stable in X+-{O}. Furthermore for any choice of uOeX+-{O} a subsolution ! and a supersolution ~ * stem
of sy-
A~ + FIe] = 0
(4.17)
exists such that u o ~<¢Mn. In correspondence of ~, an c>O exists such that f(¢)~
for any ~EK
s.t. ~,
(4.23)
Due to (4.23) we have that g U(t)u ° .< TM(t)¢Mn
(4.24)
where T~(t) is the evolution operator of system (4.22). Since TM(t)~M~ monotonically
tends to zero
the theorem follows. can be found such that ¢~U(~)u°$~ for a suitable ~>O.
Observe now that since fsf in K is also a subsolution of
then
A¢ + FIe] = O Moreover
(4.18)
since $ ~ * ,
and f(~)=f(~)
for
~ ¢ * than ~ is also a supersolution of (4.18). Due to monotonicity arguments [3] we can then state that a ¢_>>O, and a ¢+~¢_ exist such that (4.13) holds. Theorem 4.8 - If @M
<
(all+dl%M)(a22+d2XM)
z
(4.19)
a12
is not satisfied for any z>O, then the trivial solution is locally asymptotically stable. Moreover if n** denotes the saddle point of the following ODE system d d-~ then,
= -%MD¢ + f(~)
for
any u°eX+,
,
t>O
(4.20)
o ** u <¢M ~ , we h a v e
Proof. - Consider the matrix
=
-all a12 ) g'(O) +Ct~ -a22 !
(4.21)
Since @MO such that the matrix
(-XMD+Bc) has both negative eigenvalues,
and then the trivial solution is globally asymptotically stable for the linear system d
d--t v(t) = AMV(t)
+ B v(t)
, t>O
(4.22)
where A M is defined as in (2.2) with 6i(x)=B m and ~i(x)=~M
(i=1,2; x e ~ ) .
Let u°~X+, u°<¢Mn
lim IIu(t)u°-¢+II = o t~++~
(4.25)
References [1] R.M. Anderson, The dynamics and control of direct life cycle helminth parasites, in Vito Volterra Symposium on Math. Models in Biology (C. Barigozzi, ed.) Lect. Notes in Biomath., 39, (1980), 278-322. [2] V. Capasso and L. Maddalena, A non linear diffusion system modelling the spread of oro faecal diseases, in Nonlinear Phenomena in Mathematical Sciences (V. Lakshmikantham, ed.), Ac. Press., New York, 1981. [3] V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biolosy, 13, (1981), 173-184. [4] V. Capasso and L. Maddalena, Periodic Solutions for a reaction-diffusion system model llng the spread of a class of epidemics, SIAM J. Appl. Math., to appear.
lim llUCt)u°II = O t~+~
B
Remark 4.9 - It can be easily shown that under the assumptions of Theorem 4.8 an equilibrium solution ¢+EX+ exists such that if u°EX+ u°>.¢+ then
. Then an ne~, ~
[5] V. Capasso and S.L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidem. et Sant~ Publ., 27, (1979). 121-132. Errata, Ibidem 28, (1980), 330. [6] N. Chafee, Behaviour of solutions leaving the neighborhood of a saddle point for a non linear evolution equation, J. Math. Anal. Appl., 58, (1977), 312-325. [7] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969. [8] L. Galeone, Positivity and asymptotic behaviour of the numerical solution of weakly coupled systems of semilinear parabolic dif ferential equations, Proc. IOth. I M A C S W o r n Congress, Montreal, 1982. [9] K.P. Hadeler, Diffusion
equations
in biolo-
V. Capasso, L. Maddalena / Saddle point behaviour for a reaction-diffusion system gy, in Mathematics of Biology, C.I.M.E. L~ guori Editore, 1979, Napoli. [IO] J.H. Hale, Ordinary Differential Equations, Wiley-lnterscience, New York, 1969. [Ii] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. [12] V. Lakshmikantham, Comparison results for reaction-diffusion equations in a Banach space, Conf. Sem. Mat. Univ. Bari, 161, (1979). [13] R.H. Martin, Asymptotic stability and critical points for nonlinear quasimonotone p~ rabolic systems, J. Diff. Equations 30, (1978), 391-423. [14] J. Rauch and J.A. Smoller, Qualitative theo ry of the Fitzhugh-Nagumo equations, Advan ~ ces in Math. 27 (1978), 12-44. [15] J.F. Selgrade, Mathematical analysis of a cellular control process with positive feed back, SlAM J. Appl. Math., 36, (1979), 219-229. [16] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 229-310. [17] H.H. Wilson, Ordinary Differential Equations, Addison-Wesley Publ. Comp., London, 1971.
Ackowled~ement. It is a great pleasure to acknowledge the useful contribution of Prof.L.Galeone whose numerical work on this subject lead to Figs. 2-4.(See [8]7.
547