# American Institute of Mathematical Sciences

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

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

 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.

Citation: Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete and Continuous Dynamical Systems, 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. [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. [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. [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. [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. [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. [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. [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. [9] M. Chipot, D. Kinderlehrer and M. Kowalczyk, A variational principle for molecular motors, Meccanica, 38 (2003), 505-518.  doi: 10.1023/A:1024719028273. [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. [11] J. H. Espenson, Chemical Kinetics and Reaction Mechanisms, 2nd edition, McGraw-Hill, New York, 1995. [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. [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. [14] M. H. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. [15] J.-F. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 1738-1744.  doi: 10.1137/S003614109326063X. [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. [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. [18] H. Matano and T. Ogiwara, Convergence results for general cooperative systems with mass conservation, work in progress. [19] J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642-646.  doi: 10.1137/0518049. [20] J. Mierczyński, Cooperative irreducible systems of ordinary differential equations with first integral, arXiv: 1208.4697 [math.CA], 2012. [21] F. Nakajima, Periodic time dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), 421-427.  doi: 10.1137/0136032. [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. [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. [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. [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. [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. [27] H. H. Schaefer, Banach Lattice and Positive Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [9] M. Chipot, D. Kinderlehrer and M. Kowalczyk, A variational principle for molecular motors, Meccanica, 38 (2003), 505-518.  doi: 10.1023/A:1024719028273. [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. [11] J. H. Espenson, Chemical Kinetics and Reaction Mechanisms, 2nd edition, McGraw-Hill, New York, 1995. [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. [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. [14] M. H. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. [15] J.-F. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 1738-1744.  doi: 10.1137/S003614109326063X. [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. [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. [18] H. Matano and T. Ogiwara, Convergence results for general cooperative systems with mass conservation, work in progress. [19] J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642-646.  doi: 10.1137/0518049. [20] J. Mierczyński, Cooperative irreducible systems of ordinary differential equations with first integral, arXiv: 1208.4697 [math.CA], 2012. [21] F. Nakajima, Periodic time dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), 421-427.  doi: 10.1137/0136032. [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. [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. [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. [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. [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. [27] H. H. Schaefer, Banach Lattice and Positive Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974. [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. [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. [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. [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.
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