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On the non-existence of periodic solutions of the reactive-diffusive volterra system of equations
J. theor. Biol. (1980) 82, 537-540
LETTERS TO THE EDITOR
On the Non-existence of Periodic Solutions of the Reactive-diffusive Volterra System of Equations Recently Bhargava & Saxena (1977) have investigated the two species Volterra type non-linear reactive diffusive growth equations with diffusion on one dimensional infinite interval. They have claimed that they have found periodic solutions which oscillate about the constant (space independent) equilibrium solution; such solutions are also claimed to be stable with respect to nonlinear perturbations for small values of the diffusion constants. We show here that temporally fluctuating periodic solutions to the type of equations considered by Bhargava & Saxena (1977) cannot exist when the diffusion medium is any arbitrary but finite fixed one-dimensional interval; the arbitrariness of the finite interval implies in the limit as the length of the interval expands to infinity, the result of Bhargava & Saxena (1977) cannot hold. In fact we will show that the constant equilibrium solution is globally asymptotically stable. Such a stability precludes the possibility of temporal, spatial or spatio temporal fluctuations in the solutions as time t + co. We consider the following system: aN1
at-
a2Nl
-61
-+NI[~I-RNI--PINzI ax2
xE(-L,L), t>co (1) a2N2 at’ -+NzC--~~+PZN-Y~NZI ax2 where Si, ai, pi, yi (i = 1,2) are positive constants; L is an arbitrary fixed positive constant. The system (1) has a non-trivial equilibrium at (NlO, NzO) where aN2
-s
Nlo=(~~~2+~2P~)l(~1~2+P1P2)
(2) N20
Positivity
= hP2
- a2~1)l(y1~2
+ Pd32).
of Nlo and Nzo demands that we assume hlrd>
bzIP2).
(3) we add the following initial and
In order to make the system (1) determinate boundary conditions. Nib, 0) = QIl(x), N(x, 0) = 42(x), aN1 aN2 z+O and ~+0 asx+*L
-LzGxlL
foralltz0
(4) (5)
537
0022-5193/80/030537+04
$02.00/O
@ 1980 Academic Press Inc. (London)Ltd.
538
K.
GOPALSAMY
AND
B.
D.
AGGARWALA
where +r and & are non-negative continuous functions which are not identically zero on any subinterval of [-L, L] having vanishing derivatives at x=*L.
The local existence of positive classical solutions of (1 k(5) for I E [O. T) for some positive T follows from a result of Williams & Chow (1978). From the results of Conway & Smoller (19774b) one can derive apriori bounds for the solutions of (l)-(5) which are uniform with respect to both spaceand time. By continuation this will imply that positive solutions of (l)-(5) exist for all t 2 0 on E-L, L]. We assume that N,(x, t) and NZ(x, l) have been continued for all t 2 0. The non-existence of periodic solutions can be proved as follows. Using the positivity of N*(x, t), P&(x, t) for all (x, t) E t--L, L) x (0,001 we define a functional
It is easy to see that the integrand in (6) is non-negative for all positive Ni, N2 and the integrand vanishes only when Ni(x, f)=N,o (i = 1,2). Calculating the rate of change of V’ along the solutions of ( 1). dV’ (Nl, Nz) dt 7)
+I
:~(N,-N,o~[-y,(N,-N,o)B,(IV,--N,,,)ldx
+
I
-1; V2 - N~o)[PzWI - NIO)- ~02 - No)1 dx.
(9)
LETTERS
TO
By the no flux boundary conditions
THE
539
EDITOR
(9) simplifies to
dv’ WI, N2) = t)-No12+
dt
(Y2/P2)W2k
t) -N2d2
+(yq(“Nk’,“.$&-J2
d2NlO + (
alv,(x, >(
p2
t) ax
1 N2b,
>I 2
t)
dx
00)
It follows from (10) that V’( . ) is monotonically decreasing as t increases and since it is bounded below by zero, V’(Nl, N2)+ 0 as t -0;). From the continuity of Ni(x, t) and the form of the integrand in (6), it will follow that
Nib, O-+NlO N2b,
0 +
N20
as t+co.
One can show that the convergence to constant equilibrium uniform with respect to x on every finite interval. Similarly we can consider the competition system aN1
-*
a2N -+
ataN2 at-
’ ax2 -
62
a2N2 + ax2
Nl(al-
YINI
-
in (11) is in fact
-PJG)
-L
(11)
I
P2N
-
t>O
(12)
y2N2)
with the initial and boundary conditions of the type in (4) and (5); again Si, (Yi, &, yi (i = 1,2) are positive constants. The constant nontrivial equilibrium solution of (12) is given by (Nio, N20) where N~o=((Y~Y~-P~cY~)/(Y~Y~-P~P~)
(13) N20
To ensure the positivity
=
(n(~2-~11P2)/(~1~1-P1P2).
of Nlo and Nzo we shall assume &
a2
(14) P2’
The global existence of positive solutions-of (12) with (14) and conditions of the type (4) and (5) follow from the already quoted references. For (12) again using the positive of Ni(x, t), N2(x, t) for all x E (-L, L) and t 5 0 we
K. GOPALSAMY
540
AND
B. D. AGGARWALA
define a functional V’(N1, iv*) = f-1 iC,[Nl(X, t) -N,,,-
N,cJlog (Nl(X. f)liV,,,)l
+ c2[N(x, 1)- N2o- Nzolog WAX, t )/‘NxdlI dx
(15)
where the positive constants cl and c2 will be selected suitably. Calculating dV’jdt along the system (12) and using the no flux boundary conditions,
(16) If we select cl = l/p1 and c, = 1/p2, then the quadratic form clylwl
-NuY
+-CCIP,+ CZP?)(NI -Nw)N,
- Iv2,,) + c2ydRi2 -Nd
is positive definite when (14) holds and hence as before V’(NI, Nz) + 0 as t + ~0. For such a choice of ci and ~2, V’ is non-negative with a global zero minimum at (N,(x, t), N2(x, t)) = (Nlo, NZO)and hence by the continuity of the solutions (Nr(x, t). N2(x, t)) it will follow that (N,(x. t),
~VZ(X,
t)) + (NIu, N,,,)
(17)
as t + a.
Such a convergence precludes the fluctuations in the solutions for t + ,X K. GOPALSAMY-
School of Mathematics Fiinders University Bedford Park, S.A., 5042. Australia
B. D. AGGARWALA
Department of Mathematics The University of Calgary 2920-24th Av. N. W. Calgary, Alberta Canada T2N IN4 (Received 27 August
1979, and in revised form 13 September 1979)
REFERENCES BHARGAVA,S.C.& SAXENA,R.P. (1977). J. theor. Biol. 67,399. CONWAY,E.& SMOLLER,J.(~~~~U). Comm. in part. diff.eqns.2,679. CONWAY, E. & SMOLLER, J. (19776). SZAMJ. app. Math. 33,673. WILLIAMS, S. A. & CHOW,P. L.(1978). J. Math. Anal. Appl.62, 157.
LETTERS TO THE EDITOR
On the Non-existence of Periodic Solutions of the Reactive-diffusive Volterra System of Equations Recently Bhargava & Saxena (1977) have investigated the two species Volterra type non-linear reactive diffusive growth equations with diffusion on one dimensional infinite interval. They have claimed that they have found periodic solutions which oscillate about the constant (space independent) equilibrium solution; such solutions are also claimed to be stable with respect to nonlinear perturbations for small values of the diffusion constants. We show here that temporally fluctuating periodic solutions to the type of equations considered by Bhargava & Saxena (1977) cannot exist when the diffusion medium is any arbitrary but finite fixed one-dimensional interval; the arbitrariness of the finite interval implies in the limit as the length of the interval expands to infinity, the result of Bhargava & Saxena (1977) cannot hold. In fact we will show that the constant equilibrium solution is globally asymptotically stable. Such a stability precludes the possibility of temporal, spatial or spatio temporal fluctuations in the solutions as time t + co. We consider the following system: aN1
at-
a2Nl
-61
-+NI[~I-RNI--PINzI ax2
xE(-L,L), t>co (1) a2N2 at’ -+NzC--~~+PZN-Y~NZI ax2 where Si, ai, pi, yi (i = 1,2) are positive constants; L is an arbitrary fixed positive constant. The system (1) has a non-trivial equilibrium at (NlO, NzO) where aN2
-s
Nlo=(~~~2+~2P~)l(~1~2+P1P2)
(2) N20
Positivity
= hP2
- a2~1)l(y1~2
+ Pd32).
of Nlo and Nzo demands that we assume hlrd>
bzIP2).
(3) we add the following initial and
In order to make the system (1) determinate boundary conditions. Nib, 0) = QIl(x), N(x, 0) = 42(x), aN1 aN2 z+O and ~+0 asx+*L
-LzGxlL
foralltz0
(4) (5)
537
0022-5193/80/030537+04
$02.00/O
@ 1980 Academic Press Inc. (London)Ltd.
538
K.
GOPALSAMY
AND
B.
D.
AGGARWALA
where +r and & are non-negative continuous functions which are not identically zero on any subinterval of [-L, L] having vanishing derivatives at x=*L.
The local existence of positive classical solutions of (1 k(5) for I E [O. T) for some positive T follows from a result of Williams & Chow (1978). From the results of Conway & Smoller (19774b) one can derive apriori bounds for the solutions of (l)-(5) which are uniform with respect to both spaceand time. By continuation this will imply that positive solutions of (l)-(5) exist for all t 2 0 on E-L, L]. We assume that N,(x, t) and NZ(x, l) have been continued for all t 2 0. The non-existence of periodic solutions can be proved as follows. Using the positivity of N*(x, t), P&(x, t) for all (x, t) E t--L, L) x (0,001 we define a functional
It is easy to see that the integrand in (6) is non-negative for all positive Ni, N2 and the integrand vanishes only when Ni(x, f)=N,o (i = 1,2). Calculating the rate of change of V’ along the solutions of ( 1). dV’ (Nl, Nz) dt 7)
+I
:~(N,-N,o~[-y,(N,-N,o)B,(IV,--N,,,)ldx
+
I
-1; V2 - N~o)[PzWI - NIO)- ~02 - No)1 dx.
(9)
LETTERS
TO
By the no flux boundary conditions
THE
539
EDITOR
(9) simplifies to
dv’ WI, N2) = t)-No12+
dt
(Y2/P2)W2k
t) -N2d2
+(yq(“Nk’,“.$&-J2
d2NlO + (
alv,(x, >(
p2
t) ax
1 N2b,
>I 2
t)
dx
00)
It follows from (10) that V’( . ) is monotonically decreasing as t increases and since it is bounded below by zero, V’(Nl, N2)+ 0 as t -0;). From the continuity of Ni(x, t) and the form of the integrand in (6), it will follow that
Nib, O-+NlO N2b,
0 +
N20
as t+co.
One can show that the convergence to constant equilibrium uniform with respect to x on every finite interval. Similarly we can consider the competition system aN1
-*
a2N -+
ataN2 at-
’ ax2 -
62
a2N2 + ax2
Nl(al-
YINI
-
in (11) is in fact
-PJG)
-L
(11)
I
P2N
-
t>O
(12)
y2N2)
with the initial and boundary conditions of the type in (4) and (5); again Si, (Yi, &, yi (i = 1,2) are positive constants. The constant nontrivial equilibrium solution of (12) is given by (Nio, N20) where N~o=((Y~Y~-P~cY~)/(Y~Y~-P~P~)
(13) N20
To ensure the positivity
=
(n(~2-~11P2)/(~1~1-P1P2).
of Nlo and Nzo we shall assume &
a2
(14) P2’
The global existence of positive solutions-of (12) with (14) and conditions of the type (4) and (5) follow from the already quoted references. For (12) again using the positive of Ni(x, t), N2(x, t) for all x E (-L, L) and t 5 0 we
K. GOPALSAMY
540
AND
B. D. AGGARWALA
define a functional V’(N1, iv*) = f-1 iC,[Nl(X, t) -N,,,-
N,cJlog (Nl(X. f)liV,,,)l
+ c2[N(x, 1)- N2o- Nzolog WAX, t )/‘NxdlI dx
(15)
where the positive constants cl and c2 will be selected suitably. Calculating dV’jdt along the system (12) and using the no flux boundary conditions,
(16) If we select cl = l/p1 and c, = 1/p2, then the quadratic form clylwl
-NuY
+-CCIP,+ CZP?)(NI -Nw)N,
- Iv2,,) + c2ydRi2 -Nd
is positive definite when (14) holds and hence as before V’(NI, Nz) + 0 as t + ~0. For such a choice of ci and ~2, V’ is non-negative with a global zero minimum at (N,(x, t), N2(x, t)) = (Nlo, NZO)and hence by the continuity of the solutions (Nr(x, t). N2(x, t)) it will follow that (N,(x. t),
~VZ(X,
t)) + (NIu, N,,,)
(17)
as t + a.
Such a convergence precludes the fluctuations in the solutions for t + ,X K. GOPALSAMY-
School of Mathematics Fiinders University Bedford Park, S.A., 5042. Australia
B. D. AGGARWALA
Department of Mathematics The University of Calgary 2920-24th Av. N. W. Calgary, Alberta Canada T2N IN4 (Received 27 August
1979, and in revised form 13 September 1979)
REFERENCES BHARGAVA,S.C.& SAXENA,R.P. (1977). J. theor. Biol. 67,399. CONWAY,E.& SMOLLER,J.(~~~~U). Comm. in part. diff.eqns.2,679. CONWAY, E. & SMOLLER, J. (19776). SZAMJ. app. Math. 33,673. WILLIAMS, S. A. & CHOW,P. L.(1978). J. Math. Anal. Appl.62, 157.