Monotone maps: A review

The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics.

[1]  P. Polácik,et al.  Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations , 1993 .

[2]  Sze-Bi Hsu,et al.  Competitive exclusion and coexistence for competitive systems on ordered Banach spaces , 1996 .

[3]  R. Nussbaum.  Iterated nonlinear maps and Hilbert’s projective metric. II , 1989 .

[4]  Hal L. Smith.  Invariant curves for mappings , 1986 .

[5]  Peter Takáč,et al.  Convergence in the part metric for discrete dynamical systems in ordered topological cones , 1996 .

[6]  Jifa Jiang,et al.  The General Properties of Discrete-Time Competitive Dynamical Systems , 2001 .

[7]  Jiang Jifa,et al.  Stable Cycles for Attractors of Strongly Monotone Discrete-Time Dynamical Systems , 1996 .

[8]  Nonlinear functional analysis and its applications, part I: Fixed-point theorems , 1991 .

[9]  R. Nussbaum.  Hilbert's Projective Metric and Iterated Nonlinear Maps , 1988 .

[10]  Bingtuan Li,et al.  Analysis of linear determinacy for spread in cooperative models , 2002, Journal of mathematical biology.

[11]  Hiroshi Matano,et al.  Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations , 1989 .

[12]  Chris Cosner,et al.  Book Review: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems , 1996 .

[13]  Positive Solutions of Some Nonlinear Eigenvalue Problems , 1969 .

[14]  J. Cushing.  Periodic Lotka-Volterra competition equations , 1986, Journal of mathematical biology.

[15]  Hal L. Smith.  Complicated dynamics for low-dimensional strongly monotone maps , 1997 .

[16]  F. M. Arscott,et al.  PERIODIC‐PARABOLIC BOUNDARY VALUE PROBLEMS AND POSITIVITY , 1992 .

[17]  Ulrich Krause,et al.  A limit set trichotomy for monotone nonlinear dynamical systems , 1992 .

[18]  J. K. Hale,et al.  Competition for fluctuating nutrient , 1983 .

[19]  Jim M Cushing,et al.  Some Discrete Competition Models and the Competitive Exclusion Principle† , 2004 .

[20]  Alan C. Lazer,et al.  On an abstract competition model and applications , 1991 .

[21]  Horst R. Thieme,et al.  Quasi convergence and stability for strongly order-preserving semiflows , 1990 .

[22]  O. Perron.  Zur Theorie der Matrices , 1907 .

[23]  R. Lui.  A nonlinear integral operator arising from a model in population genetics IV. clines , 1986 .

[24]  S. M. Verduyn Lunel,et al.  Lower and upper bounds for ω-limit sets of nonexpansive maps , 2001 .

[25]  Robert Jentzsch.  Über Integralgleichungen mit positivem Kern. , 1912 .

[26]  M. Mackey,et al.  Probabilistic properties of deterministic systems , 1985, Acta Applicandae Mathematicae.

[27]  Attractors for discrete-time monotone dynamical systems in strongly ordered spaces , 1985 .

[28]  M. Hirsch.  Systems of di erential equations which are competitive or cooperative I: limit sets , 1982 .

[29]  Robert H. Gardner,et al.  A spatial model for the spread of invading organisms subject to competition , 1997 .

[30]  M. J.,et al.  Two Species Competition in a Periodic Environment * , 2004 .

[31]  Roger D. Nussbaum,et al.  Estimates of the periods of periodic points for nonexpansive operators , 1991 .

[32]  G. Birkhoff.  Extensions of Jentzsch’s theorem , 1957 .

[33]  Jim M Cushing,et al.  NONLINEAR MATRIX MODELS AND POPULATION DYNAMICS , 1988 .

[34]  P. Lancaster,et al.  Zur Theorie der ?-Matrizen , 1975 .

[35]  Roger D. Nussbaum,et al.  A limit set trichotomy for self-mappings of normal cones in banach spaces , 1993 .

[36]  B. M. Fulk.  MATH , 1992 .

[37]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .

[38]  UNICITE DE LA SOLUTION D'UN SYSTEME NON LINEAIRE STRICTEMENT COOPERATIF , 1994 .

[39]  P. Takáč.  Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups , 1990 .

[40]  L. Allen,et al.  Dispersal and competition models for plants , 1996 .

[41]  John C. Gower,et al.  The properties of a stochastic model for two competing species , 1958 .

[42]  P. Polácik,et al.  Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems , 1992 .

[43]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[44]  J. F. Selgrade,et al.  Convergence to equilibrium in a genetic model with differential viability between the sexes , 1987, Journal of mathematical biology.

[45]  Hal L. Smith.  Periodic solutions of periodic competitive and cooperative systems , 1986 .

[46]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[47]  P. Brunovský,et al.  Convergence in general periodic parabolic equations in one space dimension , 1992 .

[48]  Jifa Jiang,et al.  The long-run behavior of periodic competitive Kolmogorov systems , 2002 .

[49]  P. H. Leslie.  On the use of matrices in certain population mathematics. , 1945, Biometrika.

[50]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[51]  P. de Mottoni,et al.  Competition systems with periodic coefficients: A geometric approach , 1981 .

[52]  R. Nussbaum.  Iterated nonlinear maps and Hilbert's projective metric: a summary , 1987 .

[53]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[54]  Hiroshi Matano,et al.  Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems , 1984 .

[55]  Peter Takáč,et al.  Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems , 1992 .

[56]  F. Browder.  Nonlinear functional analysis , 1970 .

[57]  R. Courant,et al.  Methoden der mathematischen Physik , .

[58]  A. C. Thompson.  ON CERTAIN CONTRACTION MAPPINGS IN A PARTIALLY ORDERED VECTOR SPACE , 1963 .

[59]  R. Lui.  A Nonlinear Integral Operator Arising from a Model in Population Genetics I. Monotone Initial Data , 1982 .

[60]  M. Slemrod,et al.  Asymptotic behavior of nonlinear contraction semigroups , 1973 .

[61]  Heteroclinic Orbits and Convergence of Order- Preserving Set-Condensing Semiflows with Applications to Integrodifferential Equations , 1995 .

[62]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[63]  Hans F. Weinberger,et al.  Long-Time Behavior of a Class of Biological Models , 1982 .

[64]  P. Polácik.  Existence of unstable sets for invariant sets in compact semiflows. Applications in order-preserving semiflows , 1990 .

[65]  E. N. Dancer,et al.  Stability of fixed points for order-preserving discrete-time dynamical systems. , 1991 .

[66]  Jifa Jiang,et al.  On the finite-dimensional dynamical systems with limited competition , 2002 .

[67]  Discrete monotone dynamics and time-periodic competition between two species , 1997 .

[68]  Nonlinear parabolic equations: Qualitative properties of solutions , 1990 .

[69]  H. Keller,et al.  Some Positone Problems Suggested by Nonlinear Heat Generation , 1967 .

[70]  H. B. Keller,et al.  Elliptic boundary value problems suggested by nonlinear diffusion processes , 1969 .

[71]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[72]  F. Zanolin.  Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems , 1992 .

[73]  Horst R. Thieme,et al.  Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread , 1979 .

[74]  Peter Takác.  Domains of Attraction of Generic $\omega$-Limit Sets for Strongly Monotone Semiflows , 1991 .

[75]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[76]  H. Amann.  Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces , 1976 .

[77]  P. Takáč.  Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. , 1992 .

[78]  Hal L. Smith.  Periodic competitive differential equations and the discrete dynamics of competitive maps , 1986 .

[79]  Global attractivity and stability in some monotone discrete dynamical systems , 1996, Bulletin of the Australian Mathematical Society.

[80]  $p$-arcs in strongly monotone discrete-time dynamical systems , 1994 .

[81]  Chun Chor Litwin 鄭振初 Cheng,et al.  An extension of the results of hirsch on systems of differential equations which are competitive or cooperative , 1996 .

[82]  H. Amann.  On the number of solutions of nonlinear equations in ordered Banach spaces , 1972 .

[83]  N. Alikakos,et al.  Discrete order preserving semigroups and stability for periodic parabolic differential equations , 1989 .

[84]  E. N. Dancer.  Some remarks on a boundedness assumption for monotone dynamical systems , 1998 .

[85]  J. Hale.  Dynamics of a Scalar Parabolic Equation , 1999 .

[86]  M. A. Krasnoselʹskii.  The operator of translation along the trajectories of differential equations , 1968 .

[87]  M. Hirsch.  Stability and convergence in strongly monotone dynamical systems. , 1988 .

[88]  Peter Takáč,et al.  Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology , 1990 .

[89]  Peter Hess,et al.  Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems , 1993 .

[90]  H. Amann.  On the Existence of Positive Solutions of Nonlinear Elliptic Boundary value Problems , 1971 .

[91]  P. Polácik,et al.  Chapter 16 - Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds , 2002 .

[92]  Jack K. Hale,et al.  Structural stability for time-periodic one-dimensional parabolic equations , 1992 .

[93]  Horst R. Thieme,et al.  Stable Coexistence and Bi-stability for Competitive Systems on Ordered Banach Spaces , 2001 .

[94]  V. Hutson,et al.  The evolution of dispersal rates in a heterogeneous time-periodic environment , 2001, Journal of mathematical biology.

[95]  Hal L. Smith.  Cooperative systems of differential equations with concave nonlinearities , 1986 .

[96]  Jifa Jiang,et al.  Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems $ , 2002 .

[97]  P. Hess,et al.  Periodic-Parabolic Boundary Value Problems and Positivity , 1991 .

[98]  Hal L. Smith.  Planar competitive and cooperative difference equations , 1998 .