BULLETIN OF MATHEMATICAL BIOLOGY
vOLU~rE39, 1977
LETTEI~ TO THE EDITOI~ A N EXACTLY SOLUBLE NoN-LINEAI~ DYNAMICAL SYSTEM
The question whether non-linear, non-equilibrium systems oscillate and are stable is of great biological interest and has been dealt with extensively in the literature. In particular, reaction models with diffusion (Nicolis, Prigogine and Glansdorff, 1975) or without (Goodwin 1963; Walter 1968, 1970; Meyer 1973; Viniegra-Gonzalez 1973; Pavlidis 1973) have been intensely studied, largely by computer simulation. The investigations are concerned, in general, with systems of equations of the form S k = F ~ ( S i , S j ) - G~(S~, S j ) ,
i, j , k = 1 . . . n,
with F~ and G~ a priori non-linear, which m a y be considered as generalizations of the Lotka-Volterra equations and as such, encompass a very wide range of biological phenomena, from controlled biochemical reactions to ecology. It may be instructive to re-examine the problem of oscillations and stability analytically, at least in some very simple cases, as the mathematical difficulties escalate rapidly with the degree of complexity of the system. This is done here, also by means of a model, which is, however, susceptible of exact analytic solutions, and which probably represents, for non-linear systems, what the simple harmonic oscillator is for linear ones. It follows, in particular, that within the frame of the model oscillations and stability are indeed possible. The model in question, corresponds to the chain of reactions a~-t-b~ v-%a~-i-b]~+l ...b~+l
~-~a,~+a~ ~ - b k
~-a~_l+a~_l...]c
= 1.....
n
(1)
which, to first order in ~, when y ~ p, yield the system of non-linear differential equations 5~ = ~a~(b~+ 1 - b~), (2) =
(3)
where as and b~ denote concentrations. Alternatively, if a# represents the amount of information stored in element/c of the system, be and b~+l the rates 129
130
LETTER TO TIlE EDITOR
at which information is subtracted or respectively added to element /c, then (2) and (3) may represent the evolution of a certain ecological system. Equations (2) and (3) are integrable exactly. This can be seen b y integrating (2) ak = ½Cexp { - ~ / S ( b ~ - b ~ + l ) d t } ,
(4)
where C is an integration constant, and then b y substituting (4) into (3). The result is Q~ = #C2{exp [ - ~(Q~- Q~-l)] - exp [ - ~(Q~+I - Qk)]},
(5)
where Q~ - - 2 St blc dt. Equation (5) is the equation of motion of the completely integrable exponential lattice first introduced b y Toda. A very detailed analysis of this interesting system is contained in Toda's excellent review article (Toda, 1975). It is sufficient here, to recall some of its most remarkable properties. The motion of the Toda lattice is stable and possesses as many conserved quantities as the lattice elements. If the number of elements becomes infinite, so does the number of integrals of the motion in agreement with the fact that (5) reduces to the Korteweg-de Vries equation in the continuum limit. For an infinitely long lattice (5) admits the solitary wave (or "sol±ton") solution exp { - y ( Q ~ - Q k - 1 ) } - 1 = ~#C 2 sinh~ ~ sech2(a]c ± t%/(~#)C sinh~), where sinh2a and a-1 represent height and width of the pulse respectively and is arbitrary. Multi-sol±ton solutions can also be found (Hirota, 1973). Moreover, the infinitely long lattice admits the periodic solution exp ( - ~ ] ( Q ~ - Q ~ - I ) } - I
= (2Kv)2{dn2[2K(k/2 ± v t ) ] - E / K } ,
(6)
where the frequency v and the wave length 2 satisfy the relation 2Kv =
1
[sn2( - 2K/A) - 1 + (E/K)] -1/2
and K and E are the complete elliptic integrals of the first and second kind. I f the lattice is cyclic and has finite length (0 < /c < N) such that N / 2 = integer, then a periodic wave solution still exists and is given b y (6) with ;t = N. Finally, the lattice covers both the harmonic (~ --> 0, ~#C 2 finite) and the strongest anharmonic limits (n -+ oo) which, from the point of view of the applications, looks very attractive. This work was supported b y the National l%esearch Council of Canada. Department of Physics and Astronomy, University of Regina, Regina, Saskatchewan $4S 0A2, Canada
G. PAPINI
L E T T E R TO THE EDITOI{
131
LITERATURE Goodwin, B. C. 1963. Temporal Organization in Cells. New Y o r k : Academm Press. Hlrota, 1%. 1973. " E x a c t N-Soliton Solution of a Nonlinear Lumped Network Equation. °' J. Phys. Soc., Japan, 35, 286-288. Meyer, D. H. 1973. "Two-Dimensional Analysis of Chemical Oscillators." I n Biological and Biochemical Oscillators, B. Chance, E. K. Pye, A. K. Ghosh and B. Hess, eds, pp. 31-40. New York: Academic Press. Nleolis, G., Prigogine, i. and Glansdorff, P. 1975. " O n the Mechanism of Instabilities in Nonlinear Systems." Adv. Chem. Phys., 32, 1-11. This article contains also references to earlier contributions of the Brussels school. Pavhdis, Th. 1973. Biological Oscillators: Their ~Iathematical Analysis. New York: Academic Press. Toda, M. 1975. "Studies of a Non-Linear Lattice." Phys. Rep. 18C, 1-29. This review article contains m a n y references to earlier related papers. Vimegra-Gonzalez, G. 1973. Stability Properties of Metabolic Pathways with Feedback Interactions." I n Biological and Biochemical Oscillators, B. Chance, E. X. Pye, A. K. Ghosh and ]3. Hess, eds, pp. 41-59. New York: Academic Press. Walter, C. F. 1968. "Oscillations in Controlled Biochemical Systems." Biophys. J., 9, 863-872. Walter, C. 1% 1970. "The Occurrence and the Significance of Limit Cycle Beh aw o r in Controlled Biochemical Systems." J. Theor. Biol., 27, 259-272.
vOLU~rE39, 1977
LETTEI~ TO THE EDITOI~ A N EXACTLY SOLUBLE NoN-LINEAI~ DYNAMICAL SYSTEM
The question whether non-linear, non-equilibrium systems oscillate and are stable is of great biological interest and has been dealt with extensively in the literature. In particular, reaction models with diffusion (Nicolis, Prigogine and Glansdorff, 1975) or without (Goodwin 1963; Walter 1968, 1970; Meyer 1973; Viniegra-Gonzalez 1973; Pavlidis 1973) have been intensely studied, largely by computer simulation. The investigations are concerned, in general, with systems of equations of the form S k = F ~ ( S i , S j ) - G~(S~, S j ) ,
i, j , k = 1 . . . n,
with F~ and G~ a priori non-linear, which m a y be considered as generalizations of the Lotka-Volterra equations and as such, encompass a very wide range of biological phenomena, from controlled biochemical reactions to ecology. It may be instructive to re-examine the problem of oscillations and stability analytically, at least in some very simple cases, as the mathematical difficulties escalate rapidly with the degree of complexity of the system. This is done here, also by means of a model, which is, however, susceptible of exact analytic solutions, and which probably represents, for non-linear systems, what the simple harmonic oscillator is for linear ones. It follows, in particular, that within the frame of the model oscillations and stability are indeed possible. The model in question, corresponds to the chain of reactions a~-t-b~ v-%a~-i-b]~+l ...b~+l
~-~a,~+a~ ~ - b k
~-a~_l+a~_l...]c
= 1.....
n
(1)
which, to first order in ~, when y ~ p, yield the system of non-linear differential equations 5~ = ~a~(b~+ 1 - b~), (2) =
(3)
where as and b~ denote concentrations. Alternatively, if a# represents the amount of information stored in element/c of the system, be and b~+l the rates 129
130
LETTER TO TIlE EDITOR
at which information is subtracted or respectively added to element /c, then (2) and (3) may represent the evolution of a certain ecological system. Equations (2) and (3) are integrable exactly. This can be seen b y integrating (2) ak = ½Cexp { - ~ / S ( b ~ - b ~ + l ) d t } ,
(4)
where C is an integration constant, and then b y substituting (4) into (3). The result is Q~ = #C2{exp [ - ~(Q~- Q~-l)] - exp [ - ~(Q~+I - Qk)]},
(5)
where Q~ - - 2 St blc dt. Equation (5) is the equation of motion of the completely integrable exponential lattice first introduced b y Toda. A very detailed analysis of this interesting system is contained in Toda's excellent review article (Toda, 1975). It is sufficient here, to recall some of its most remarkable properties. The motion of the Toda lattice is stable and possesses as many conserved quantities as the lattice elements. If the number of elements becomes infinite, so does the number of integrals of the motion in agreement with the fact that (5) reduces to the Korteweg-de Vries equation in the continuum limit. For an infinitely long lattice (5) admits the solitary wave (or "sol±ton") solution exp { - y ( Q ~ - Q k - 1 ) } - 1 = ~#C 2 sinh~ ~ sech2(a]c ± t%/(~#)C sinh~), where sinh2a and a-1 represent height and width of the pulse respectively and is arbitrary. Multi-sol±ton solutions can also be found (Hirota, 1973). Moreover, the infinitely long lattice admits the periodic solution exp ( - ~ ] ( Q ~ - Q ~ - I ) } - I
= (2Kv)2{dn2[2K(k/2 ± v t ) ] - E / K } ,
(6)
where the frequency v and the wave length 2 satisfy the relation 2Kv =
1
[sn2( - 2K/A) - 1 + (E/K)] -1/2
and K and E are the complete elliptic integrals of the first and second kind. I f the lattice is cyclic and has finite length (0 < /c < N) such that N / 2 = integer, then a periodic wave solution still exists and is given b y (6) with ;t = N. Finally, the lattice covers both the harmonic (~ --> 0, ~#C 2 finite) and the strongest anharmonic limits (n -+ oo) which, from the point of view of the applications, looks very attractive. This work was supported b y the National l%esearch Council of Canada. Department of Physics and Astronomy, University of Regina, Regina, Saskatchewan $4S 0A2, Canada
G. PAPINI
L E T T E R TO THE EDITOI{
131
LITERATURE Goodwin, B. C. 1963. Temporal Organization in Cells. New Y o r k : Academm Press. Hlrota, 1%. 1973. " E x a c t N-Soliton Solution of a Nonlinear Lumped Network Equation. °' J. Phys. Soc., Japan, 35, 286-288. Meyer, D. H. 1973. "Two-Dimensional Analysis of Chemical Oscillators." I n Biological and Biochemical Oscillators, B. Chance, E. K. Pye, A. K. Ghosh and B. Hess, eds, pp. 31-40. New York: Academic Press. Nleolis, G., Prigogine, i. and Glansdorff, P. 1975. " O n the Mechanism of Instabilities in Nonlinear Systems." Adv. Chem. Phys., 32, 1-11. This article contains also references to earlier contributions of the Brussels school. Pavhdis, Th. 1973. Biological Oscillators: Their ~Iathematical Analysis. New York: Academic Press. Toda, M. 1975. "Studies of a Non-Linear Lattice." Phys. Rep. 18C, 1-29. This review article contains m a n y references to earlier related papers. Vimegra-Gonzalez, G. 1973. Stability Properties of Metabolic Pathways with Feedback Interactions." I n Biological and Biochemical Oscillators, B. Chance, E. X. Pye, A. K. Ghosh and ]3. Hess, eds, pp. 41-59. New York: Academic Press. Walter, C. F. 1968. "Oscillations in Controlled Biochemical Systems." Biophys. J., 9, 863-872. Walter, C. 1% 1970. "The Occurrence and the Significance of Limit Cycle Beh aw o r in Controlled Biochemical Systems." J. Theor. Biol., 27, 259-272.
