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Necessary condition for the existence of periodic solutions to systems of reaction-diffusion equations
MATHEMATICAL
BIOSCIENCES
345
21, 345-350 (1974)
Necessary Condition for the Existence of Periodic Solutions to Systems of Reaction-Diffusion Equations GERALD ROSEN Department of Physics, Drexel University Philadelphia, Penmylvania 19104 Communicated
by C. Newton
ABSTRACT It is shown that periodic solutions ci= ci(x,l)m c,(x, t + T) to systems of reactiondiffusion equations of the form &,/at = DiV2ci+ Q,(c) are such that the average over a period of a certain functional of the ci’s vanishes if the boundary conditions are such that each ci is independent of time or its normal derivative vanishes at all boundary points. Sufficient to preclude a periodic solution immediately if gQ,(c)/ &, = aQj(c)/i3c,, this necessary condition for the existence of periodic solutions also provides a useful criterion in the general case. A special application of the necessary condition is presented for periodic running wave solutions with ci= ci(t -f(x))= ci(t -f(x)+ T).
1.
INTRODUCTION
Considerable interest has been attached diffusion equations of the form &,/dt=
recently
to systems of reaction-
(1)
DiV2ci + Q,(c),
where the enumerator index i runs from 1 to k, ci= ci(x,t) denotes the concentration of a molecular or biological species, Di is the diffusivity constant for the Ph species, and Qi(c), a generic algebraic function of the concentration k-tuple c = (c i, . . . ,qJ, describes the local rate of production of the i* species due to chemical reactions or biological processes [l-7]. Usually a system of reaction-diffusion equations (1) is considered to govern the concentration functions through a connected spatial region R with either ci a prescribed function of x or II. Vc,=O at spatial points on the boundary surface of R (ndenoting a unit vector normal to the surface at the 0 American
Elsevier
Publishing
Company,
Inc., 1974
346
G. ROSEN
boundary
point);
thus, one usually has F(n.Vci(n,t))=Cl
for
XE~R,
(2)
as a consequence of the boundary conditions for each value of i= 1 to k. A solution to the system of Eq. (1) is said to be periodic if there exists a positive constant T such that ci(x, t + T) E ci(x, t) for all x E R and all t > 0. The purpose of the present communication is to report a necessary condition for the existence of periodic solutions to a system of Eq. (1) supplemented with boundary conditions which imply (2). Sufficient to preclude a periodic solution immediately if k= 1 or if aQ,(c)/ac, = aQ,(c)/ac,, the necessary condition takes the form
2 s[m-
,ig)]d3x=0,
(3)
R
where the bars denote averages over a period, A (x,t)
-f
/‘+ =A (x,s)ds,
(4)
I
and the quantity
W,= Wi(c) is defined
by
(5) where F,=F&+Qi(k)dX
(6)
0
with h a (I-tuple) 2.
DERIVATION
parameter. OF THE NECESSARY
In order to derive (3), it is convenient
H=H(& i-i
CONDITION
to introduce
the functional
j-[@ilVci]2-ciF,]d3~ R
in which Fi is given in terms of Qi by (6). By integrating
(7) (2) over the
ON SYSTEMS OF REACTION-DIFFUSION
boundary
EQUATIONS
surface of R and using Gauss’ theorem, it follows that
and hence the time derivative
of (7) is
(9)
where (1) and the identity
are employed, and the definition (5) is evoked. Now for a periodic solution ci(x, t) G c,(x, t + T), the functional (7) is periodic in t, H(t+
and therefore 3.
H(t)=
OF THE NECESSARY
-0,
(11)
CONDITION
as a k-dimensional vector field Q(c)= (Q,(c), . . . ,Qk(c)) in kc-space, the curl of Q(c) has the components
Qtdc) The quantity
T(dH(t)/dt)
the average of (9) over a period yields (3).
IMPLICATIONS
Viewed dimensional
T)-
G
--aQ,W
acj
aQ,(c) 7’
(12)
(5) can be expressed in terms of (12) as
IV, = i j-1
/ ‘Q~i,,l(hC)hdhC/.
(13)
0
Thus, if the curl of Q(c) defined by (12) vanishes identically, it follows from (13) that Wi vanishes identically, and hence (3) implies that the system (1)
348
G. ROSEN
does not admit a periodic solution: Vorticity in Q(c) is a prerequisite for a periodic solution to the system of Eq. (1). In fact, the Schwarz inequality applied to the
lI’i(aci/&)
terms in (3) produces
(14)
For k = 1, (13) yields I+‘, = 0, and no periodic solutions exists, irrespective of the form taken by Q,(c,). For k =2, the components of (13) are proportional to the integral of hQ,,,&c) over A, and the latter quantity must not vanish for periodic solutions, Of particular interest is the k=2 model theory studied extensively by Prigogine and others [l-4]; for this special case the production rate terms are Q, = A + c:c2 - Bc, - c, (15) Q2 = Bc, - c;cz with A, B positive that
constant
parameters,
W,/c,=
- W,/c,
and it follows from (13) with (15)
= +c,(c, +2c,) - fB
(16)
is indefinite in sign, enabling (3) to be satisfied by periodic solutions for suitable values of B. Finally, for spatially homogeneous periodic solutions (independent of x), (3) implies a general “virial theorem” for c(t) -c(t + T),
-&z?wmq=o,
(17)
i=l
which augments the well-known criteria and nonexistence Poincare, Bendixon, Floquet, and others for periodic solutions first-order autonomous ordinary differential equations. 4.
PERIODIC
RUNNING
theorems of to systems of
WAVE SOLUTIONS
It has been demonstrated experimentally [8,9] that certain systems of the form (1) supplemented with boundary condition that imply (2) admit periodic running wave solutions, for which ci=ci(r)=ci(r+
T) with ~-t--f(x)
in a suitable region R. In order for the system (1) to admit such a periodic running wave solution, the wave-front function j= j(x) must satisfy the
ON SYSTEMS
OF REACTION-DIFFUSION
349
EQUATIONS
conditions
IVf12=a ( =positive constant)
(18a)
V2f= b (= constant), and ci must satisfy the system of second-order
(18b) ordinary
differential
equa-
tions (19) That the quantities a and b defined by (18a) and (18b) must necessarily be constants for periodic wave solutions follows from the assumption that ci depends exclusively on r and hence satisfies Eq. (19), an equation involving only 7~ t-f(x) as an independent variable. Integrating Eq. (19) over a period yields Qi(c) =0, which implies that a(c) must be indefinite in sign for all values of i. Multiplying (19) by ci and integrating the resulting equation over a period yields the necessary condition for a periodic running wave solution + Qi(C)ci ~0, which implies that Qi(c) must be predominantly positive for large values of ci for all values of i. Finally, multiplying (19) by (dci/dr) and integrating the resulting equation over a period yields the necessary condition - (1 + bD,) (dci/dT)2
+ m
-0,
(21)
which implies that ci = c,(r) is a predominantly increasing function of r in the region of c-space where Q,(c) .is positive (negative) if the quantity (1 + bDi) is positive (negative). Now since time-averages over a period are independent of x for a periodic running wave solution, (3) becomes
2 [gpq=o,
(22)
and in view of (21) it follows that k-
c i-l
CJ$
= T-L$U.dc=O,
(23)
350
G. ROSEN
where
u,= U,(c)=(l
+bD,)-‘Q&)-
W,(c).
Components of the curl of U(c) must be indefinite in sign through the c-space region of the solution in order for (23) to be satisfied, because the closed path-integral in c-space can be expressed as the surface integral of the curl of U(c) by the generalized Stokes’ theorem. REFERENCES I
6
P. Glansdorff and I. Prigogine, Thermo&zamic Theo/y of Structure, Stabi& and Fluctuations, Wiley, New York, 1971. XIV and XV. I. Prigogine and G. Nicolis, Biological order, structure and instabilities, Quart. Reu. Biophys. 4, 107-148 (1971). I. Prigogine, G. Nicolis, and A. Babloyantz, Thermodynamics of evolution, Physics Today 25, 23-28 (Nov. 1972). H. M. Martinez, Morphogenesis and chemical dissipative structures, J. Theor. Biol. 36, 479-501 (1972). E. W. Montroll, Lectures on nonlinear rate equations, especially those with quadratic nonlinearities, in Lectures in Theoretical Physics, (A. 0. Barut and W. E. B&ton, Eds.) Gordon and Breach, New York, 1968, pp. 531-573. H. A. Schwarz, Applications of the spur diffusion model to the radiation chemistry of
7 8
aqueous solutions, J. Physical Chem. 73, 1928-1953 (1969). B. M. Cherkas, On nonlinear diffusion equations, J. Diff: Equations 11, 284-299 (1972). M. Herschkowitz-Kaufman, Structures dissipatives dans une reaction chimique
2 3 4 5
homogene, 9
Comptes Rendus 27OC, 1049-1052 (1970).
A. T. Winfree,
Spiral waves of chemical
activity,
Science 175, 634-635
(1972).
BIOSCIENCES
345
21, 345-350 (1974)
Necessary Condition for the Existence of Periodic Solutions to Systems of Reaction-Diffusion Equations GERALD ROSEN Department of Physics, Drexel University Philadelphia, Penmylvania 19104 Communicated
by C. Newton
ABSTRACT It is shown that periodic solutions ci= ci(x,l)m c,(x, t + T) to systems of reactiondiffusion equations of the form &,/at = DiV2ci+ Q,(c) are such that the average over a period of a certain functional of the ci’s vanishes if the boundary conditions are such that each ci is independent of time or its normal derivative vanishes at all boundary points. Sufficient to preclude a periodic solution immediately if gQ,(c)/ &, = aQj(c)/i3c,, this necessary condition for the existence of periodic solutions also provides a useful criterion in the general case. A special application of the necessary condition is presented for periodic running wave solutions with ci= ci(t -f(x))= ci(t -f(x)+ T).
1.
INTRODUCTION
Considerable interest has been attached diffusion equations of the form &,/dt=
recently
to systems of reaction-
(1)
DiV2ci + Q,(c),
where the enumerator index i runs from 1 to k, ci= ci(x,t) denotes the concentration of a molecular or biological species, Di is the diffusivity constant for the Ph species, and Qi(c), a generic algebraic function of the concentration k-tuple c = (c i, . . . ,qJ, describes the local rate of production of the i* species due to chemical reactions or biological processes [l-7]. Usually a system of reaction-diffusion equations (1) is considered to govern the concentration functions through a connected spatial region R with either ci a prescribed function of x or II. Vc,=O at spatial points on the boundary surface of R (ndenoting a unit vector normal to the surface at the 0 American
Elsevier
Publishing
Company,
Inc., 1974
346
G. ROSEN
boundary
point);
thus, one usually has F(n.Vci(n,t))=Cl
for
XE~R,
(2)
as a consequence of the boundary conditions for each value of i= 1 to k. A solution to the system of Eq. (1) is said to be periodic if there exists a positive constant T such that ci(x, t + T) E ci(x, t) for all x E R and all t > 0. The purpose of the present communication is to report a necessary condition for the existence of periodic solutions to a system of Eq. (1) supplemented with boundary conditions which imply (2). Sufficient to preclude a periodic solution immediately if k= 1 or if aQ,(c)/ac, = aQ,(c)/ac,, the necessary condition takes the form
2 s[m-
,ig)]d3x=0,
(3)
R
where the bars denote averages over a period, A (x,t)
-f
/‘+ =A (x,s)ds,
(4)
I
and the quantity
W,= Wi(c) is defined
by
(5) where F,=F&+Qi(k)dX
(6)
0
with h a (I-tuple) 2.
DERIVATION
parameter. OF THE NECESSARY
In order to derive (3), it is convenient
H=H(& i-i
CONDITION
to introduce
the functional
j-[@ilVci]2-ciF,]d3~ R
in which Fi is given in terms of Qi by (6). By integrating
(7) (2) over the
ON SYSTEMS OF REACTION-DIFFUSION
boundary
EQUATIONS
surface of R and using Gauss’ theorem, it follows that
and hence the time derivative
of (7) is
(9)
where (1) and the identity
are employed, and the definition (5) is evoked. Now for a periodic solution ci(x, t) G c,(x, t + T), the functional (7) is periodic in t, H(t+
and therefore 3.
H(t)=
OF THE NECESSARY
-0,
(11)
CONDITION
as a k-dimensional vector field Q(c)= (Q,(c), . . . ,Qk(c)) in kc-space, the curl of Q(c) has the components
Qtdc) The quantity
T(dH(t)/dt)
the average of (9) over a period yields (3).
IMPLICATIONS
Viewed dimensional
T)-
G
--aQ,W
acj
aQ,(c) 7’
(12)
(5) can be expressed in terms of (12) as
IV, = i j-1
/ ‘Q~i,,l(hC)hdhC/.
(13)
0
Thus, if the curl of Q(c) defined by (12) vanishes identically, it follows from (13) that Wi vanishes identically, and hence (3) implies that the system (1)
348
G. ROSEN
does not admit a periodic solution: Vorticity in Q(c) is a prerequisite for a periodic solution to the system of Eq. (1). In fact, the Schwarz inequality applied to the
lI’i(aci/&)
terms in (3) produces
(14)
For k = 1, (13) yields I+‘, = 0, and no periodic solutions exists, irrespective of the form taken by Q,(c,). For k =2, the components of (13) are proportional to the integral of hQ,,,&c) over A, and the latter quantity must not vanish for periodic solutions, Of particular interest is the k=2 model theory studied extensively by Prigogine and others [l-4]; for this special case the production rate terms are Q, = A + c:c2 - Bc, - c, (15) Q2 = Bc, - c;cz with A, B positive that
constant
parameters,
W,/c,=
- W,/c,
and it follows from (13) with (15)
= +c,(c, +2c,) - fB
(16)
is indefinite in sign, enabling (3) to be satisfied by periodic solutions for suitable values of B. Finally, for spatially homogeneous periodic solutions (independent of x), (3) implies a general “virial theorem” for c(t) -c(t + T),
-&z?wmq=o,
(17)
i=l
which augments the well-known criteria and nonexistence Poincare, Bendixon, Floquet, and others for periodic solutions first-order autonomous ordinary differential equations. 4.
PERIODIC
RUNNING
theorems of to systems of
WAVE SOLUTIONS
It has been demonstrated experimentally [8,9] that certain systems of the form (1) supplemented with boundary condition that imply (2) admit periodic running wave solutions, for which ci=ci(r)=ci(r+
T) with ~-t--f(x)
in a suitable region R. In order for the system (1) to admit such a periodic running wave solution, the wave-front function j= j(x) must satisfy the
ON SYSTEMS
OF REACTION-DIFFUSION
349
EQUATIONS
conditions
IVf12=a ( =positive constant)
(18a)
V2f= b (= constant), and ci must satisfy the system of second-order
(18b) ordinary
differential
equa-
tions (19) That the quantities a and b defined by (18a) and (18b) must necessarily be constants for periodic wave solutions follows from the assumption that ci depends exclusively on r and hence satisfies Eq. (19), an equation involving only 7~ t-f(x) as an independent variable. Integrating Eq. (19) over a period yields Qi(c) =0, which implies that a(c) must be indefinite in sign for all values of i. Multiplying (19) by ci and integrating the resulting equation over a period yields the necessary condition for a periodic running wave solution + Qi(C)ci ~0, which implies that Qi(c) must be predominantly positive for large values of ci for all values of i. Finally, multiplying (19) by (dci/dr) and integrating the resulting equation over a period yields the necessary condition - (1 + bD,) (dci/dT)2
+ m
-0,
(21)
which implies that ci = c,(r) is a predominantly increasing function of r in the region of c-space where Q,(c) .is positive (negative) if the quantity (1 + bDi) is positive (negative). Now since time-averages over a period are independent of x for a periodic running wave solution, (3) becomes
2 [gpq=o,
(22)
and in view of (21) it follows that k-
c i-l
CJ$
= T-L$U.dc=O,
(23)
350
G. ROSEN
where
u,= U,(c)=(l
+bD,)-‘Q&)-
W,(c).
Components of the curl of U(c) must be indefinite in sign through the c-space region of the solution in order for (23) to be satisfied, because the closed path-integral in c-space can be expressed as the surface integral of the curl of U(c) by the generalized Stokes’ theorem. REFERENCES I
6
P. Glansdorff and I. Prigogine, Thermo&zamic Theo/y of Structure, Stabi& and Fluctuations, Wiley, New York, 1971. XIV and XV. I. Prigogine and G. Nicolis, Biological order, structure and instabilities, Quart. Reu. Biophys. 4, 107-148 (1971). I. Prigogine, G. Nicolis, and A. Babloyantz, Thermodynamics of evolution, Physics Today 25, 23-28 (Nov. 1972). H. M. Martinez, Morphogenesis and chemical dissipative structures, J. Theor. Biol. 36, 479-501 (1972). E. W. Montroll, Lectures on nonlinear rate equations, especially those with quadratic nonlinearities, in Lectures in Theoretical Physics, (A. 0. Barut and W. E. B&ton, Eds.) Gordon and Breach, New York, 1968, pp. 531-573. H. A. Schwarz, Applications of the spur diffusion model to the radiation chemistry of
7 8
aqueous solutions, J. Physical Chem. 73, 1928-1953 (1969). B. M. Cherkas, On nonlinear diffusion equations, J. Diff: Equations 11, 284-299 (1972). M. Herschkowitz-Kaufman, Structures dissipatives dans une reaction chimique
2 3 4 5
homogene, 9
Comptes Rendus 27OC, 1049-1052 (1970).
A. T. Winfree,
Spiral waves of chemical
activity,
Science 175, 634-635
(1972).