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June  2020, 40(6): 3883-3907. doi: 10.3934/dcds.2020129

Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Toshiko Ogiwara 1, , Danielle Hilhorst 2, and  Hiroshi Matano 3,,

1. 

Department of Mathematics, Josai University, 1-1 Keyakidai, Sakado, Saitama 350-0295, Japan
2. 

CNRS, Laboratoire de Mathématiques, Université de Paris-Sud, F-91405 Orsay Cedex, France
3. 

Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

* Corresponding author: Hiroshi Matano

Received  June 2019 Published  February 2020

Fund Project: This work was supported by the CNRS GDRI ReaDiNet and by JSPS KAKENHI Grant Numbers 26610028, 16H02151

We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which $ 0 $ is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.

Keywords: Order-preserving dynamical systems, the comparison principle, convergence, mass conservation, first integral, fixed point, time-periodic solution.
Mathematics Subject Classification: Primary: 34C12, 34K13, 35B40; Secondary: 35B51.
Citation: Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
References:
[1]

N. D. Alikakos, P. Hess and H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations, 82 (1989), 322-341.  doi: 10.1016/0022-0396(89)90136-8.  Google Scholar

[2]

O. Arino, Monotone semi-flows which have a monotone first integral, in Delay differential equations and dynamical systems, Lecture Notes in Math., 1475 (1991), Springer, Berlin, 64–75. doi: 10.1007/BFb0083480.  Google Scholar

[3]

M. Banaji and D. Angeli, Convergence in strongly monotone systems with an increasing first integral, SIAM J. Math., 42 (2010), 334-353.  doi: 10.1137/090760751.  Google Scholar

[4]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135.  doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[5]

D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection, J. Differential Equations, 193 (2003), 27-48.  doi: 10.1016/S0022-0396(03)00148-7.  Google Scholar

[6]

D. Bothe and M. Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 49-59.  doi: 10.3934/dcdss.2012.5.49.  Google Scholar

[7]

M. Chipot, S. Hastings and D. Kinderlehrer, Transport in a molecular motor system, M2AN Math. Model. Numer. Anal., 38 (2004), 1011-1034.  doi: 10.1051/m2an:2004048.  Google Scholar

[8]

M. Chipot, D. Hilhorst, D. Kinderlehrer and M. Olech, Contraction in L1 for a system arising in chemical reactions and molecular motors, Differ. Equ. Appl., 1 (2009), 139-151.  doi: 10.7153/dea-01-07.  Google Scholar

[9]

M. Chipot, D. Kinderlehrer and M. Kowalczyk, A variational principle for molecular motors, Meccanica, 38 (2003), 505-518.  doi: 10.1023/A:1024719028273.  Google Scholar

[10]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing ratchet, Partial differential equations and inverse problems, Contemp. Math., 362 (2004), Amer. Math. Soc., Providence, RI, 167–175. doi: 10.1090/conm/362/06611.  Google Scholar

[11]

J. H. Espenson, Chemical Kinetics and Reaction Mechanisms, 2nd edition, McGraw-Hill, New York, 1995. Google Scholar

[12]

S. Hastings, D. Kinderlehrer and J. B. McLeod, Diffusion mediated transport in multiple state systems, SIAM J. Math. Anal., 39 (2008), 1208-1230.  doi: 10.1137/060650994.  Google Scholar

[13]

S. Hastings, D. Kinderlehrer and J. B. McLeod, Diffusion mediated transport with a look at motor proteins, in, Recent Advances in Nonlinear Analysis, World Sci. Publ., Hackensack, NJ, (2008), 95–111. doi: 10.1142/9789812709257_0006.  Google Scholar

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M. H. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.   Google Scholar

[15]

J.-F. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 1738-1744.  doi: 10.1137/S003614109326063X.  Google Scholar

[16]

D. Kinderlehrer and M. Kowalczyk, Diffusion-mediated transport and the flashing ratchet, Arch. Ration. Mech. Anal., 161 (2002), 149-179.  doi: 10.1007/s002050100173.  Google Scholar

[17]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645-673.   Google Scholar

[18]

H. Matano and T. Ogiwara, Convergence results for general cooperative systems with mass conservation, work in progress. Google Scholar

[19]

J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642-646.  doi: 10.1137/0518049.  Google Scholar

[20]

J. Mierczyński, Cooperative irreducible systems of ordinary differential equations with first integral, arXiv: 1208.4697 [math.CA], 2012. Google Scholar

[21]

F. Nakajima, Periodic time dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), 421-427.  doi: 10.1137/0136032.  Google Scholar

[22]

P. Poláčik, Convergence in Smooth Strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eqns., 79 (1989), 89-110.  doi: 10.1016/0022-0396(89)90115-0.  Google Scholar

[23]

P. Poláčik and E. Yanagida, Existence of stable subharmonic solutions for reaction-diffusion equations, Special issue in celebration of Jack K. Hale's 70th birthday, Part 4 (Atlanta, GA/Lisbon, 1998), J. Differential Equations, 169 (2001), 255-280.  doi: 10.1006/jdeq.2000.3899.  Google Scholar

[24]

P. Poláčik and E. Yanagida, Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain, Discrete Contin. Dyn. Syst., 8 (2002), 209-218.  doi: 10.3934/dcds.2002.8.209.  Google Scholar

[25]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[26]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA Journal of Applied Mathematics, 48 (1992), 249-264.  doi: 10.1093/imamat/48.3.249.  Google Scholar

[27]

H. H. Schaefer, Banach Lattice and Positive Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974.  Google Scholar

[28]

G. R. Sell and F. Nakajima, Almost periodic gross-substitute dynamical systems, Tôhoku Math. J., 32 (1980), 255–263. doi: 10.2748/tmj/1178229641.  Google Scholar

[29]

H. L. Smith and H. R. Thieme, Convergence for strongly ordered preserving semi-flows, SIAM J. Math. Anal., 22 (1991), 1081-1101.  doi: 10.1137/0522070.  Google Scholar

[30]

P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244.  doi: 10.1016/0022-247X(90)90040-M.  Google Scholar

[31]

B. Tang, Y. Kuang and H. Smith, Strictly nonautonomous cooperative system with a first integral, SIAM J. Math. Anal., 24 (1993), 1331-1339.  doi: 10.1137/0524076.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, P. Hess and H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations, 82 (1989), 322-341.  doi: 10.1016/0022-0396(89)90136-8.  Google Scholar

[2]

O. Arino, Monotone semi-flows which have a monotone first integral, in Delay differential equations and dynamical systems, Lecture Notes in Math., 1475 (1991), Springer, Berlin, 64–75. doi: 10.1007/BFb0083480.  Google Scholar

[3]

M. Banaji and D. Angeli, Convergence in strongly monotone systems with an increasing first integral, SIAM J. Math., 42 (2010), 334-353.  doi: 10.1137/090760751.  Google Scholar

[4]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135.  doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[5]

D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection, J. Differential Equations, 193 (2003), 27-48.  doi: 10.1016/S0022-0396(03)00148-7.  Google Scholar

[6]

D. Bothe and M. Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 49-59.  doi: 10.3934/dcdss.2012.5.49.  Google Scholar

[7]

M. Chipot, S. Hastings and D. Kinderlehrer, Transport in a molecular motor system, M2AN Math. Model. Numer. Anal., 38 (2004), 1011-1034.  doi: 10.1051/m2an:2004048.  Google Scholar

[8]

M. Chipot, D. Hilhorst, D. Kinderlehrer and M. Olech, Contraction in L1 for a system arising in chemical reactions and molecular motors, Differ. Equ. Appl., 1 (2009), 139-151.  doi: 10.7153/dea-01-07.  Google Scholar

[9]

M. Chipot, D. Kinderlehrer and M. Kowalczyk, A variational principle for molecular motors, Meccanica, 38 (2003), 505-518.  doi: 10.1023/A:1024719028273.  Google Scholar

[10]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing ratchet, Partial differential equations and inverse problems, Contemp. Math., 362 (2004), Amer. Math. Soc., Providence, RI, 167–175. doi: 10.1090/conm/362/06611.  Google Scholar

[11]

J. H. Espenson, Chemical Kinetics and Reaction Mechanisms, 2nd edition, McGraw-Hill, New York, 1995. Google Scholar

[12]

S. Hastings, D. Kinderlehrer and J. B. McLeod, Diffusion mediated transport in multiple state systems, SIAM J. Math. Anal., 39 (2008), 1208-1230.  doi: 10.1137/060650994.  Google Scholar

[13]

S. Hastings, D. Kinderlehrer and J. B. McLeod, Diffusion mediated transport with a look at motor proteins, in, Recent Advances in Nonlinear Analysis, World Sci. Publ., Hackensack, NJ, (2008), 95–111. doi: 10.1142/9789812709257_0006.  Google Scholar

[14]

M. H. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.   Google Scholar

[15]

J.-F. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 1738-1744.  doi: 10.1137/S003614109326063X.  Google Scholar

[16]

D. Kinderlehrer and M. Kowalczyk, Diffusion-mediated transport and the flashing ratchet, Arch. Ration. Mech. Anal., 161 (2002), 149-179.  doi: 10.1007/s002050100173.  Google Scholar

[17]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645-673.   Google Scholar

[18]

H. Matano and T. Ogiwara, Convergence results for general cooperative systems with mass conservation, work in progress. Google Scholar

[19]

J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642-646.  doi: 10.1137/0518049.  Google Scholar

[20]

J. Mierczyński, Cooperative irreducible systems of ordinary differential equations with first integral, arXiv: 1208.4697 [math.CA], 2012. Google Scholar

[21]

F. Nakajima, Periodic time dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), 421-427.  doi: 10.1137/0136032.  Google Scholar

[22]

P. Poláčik, Convergence in Smooth Strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eqns., 79 (1989), 89-110.  doi: 10.1016/0022-0396(89)90115-0.  Google Scholar

[23]

P. Poláčik and E. Yanagida, Existence of stable subharmonic solutions for reaction-diffusion equations, Special issue in celebration of Jack K. Hale's 70th birthday, Part 4 (Atlanta, GA/Lisbon, 1998), J. Differential Equations, 169 (2001), 255-280.  doi: 10.1006/jdeq.2000.3899.  Google Scholar

[24]

P. Poláčik and E. Yanagida, Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain, Discrete Contin. Dyn. Syst., 8 (2002), 209-218.  doi: 10.3934/dcds.2002.8.209.  Google Scholar

[25]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[26]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA Journal of Applied Mathematics, 48 (1992), 249-264.  doi: 10.1093/imamat/48.3.249.  Google Scholar

[27]

H. H. Schaefer, Banach Lattice and Positive Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974.  Google Scholar

[28]

G. R. Sell and F. Nakajima, Almost periodic gross-substitute dynamical systems, Tôhoku Math. J., 32 (1980), 255–263. doi: 10.2748/tmj/1178229641.  Google Scholar

[29]

H. L. Smith and H. R. Thieme, Convergence for strongly ordered preserving semi-flows, SIAM J. Math. Anal., 22 (1991), 1081-1101.  doi: 10.1137/0522070.  Google Scholar

[30]

P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244.  doi: 10.1016/0022-247X(90)90040-M.  Google Scholar

[31]

B. Tang, Y. Kuang and H. Smith, Strictly nonautonomous cooperative system with a first integral, SIAM J. Math. Anal., 24 (1993), 1331-1339.  doi: 10.1137/0524076.  Google Scholar

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