The Discrete Dynamics of Monotonically Decomposable Maps

We extend results of Gouzé and Hadeler (in Nonlinear World 1:23–34, 1994) concerning the dynamics generated by a map on an ordered metric space that can be decomposed into increasing and decreasing parts. Our main results provide sufficient conditions for the existence of a globally asymptotically stable fixed point for the map. Applications to discrete-time, stage-structured population models are given.

[1]  R. Costantino,et al.  Experimentally induced transitions in the dynamic behaviour of insect populations , 1995, Nature.

[2]  R. Devault,et al.  On the recursive sequence , 2000 .

[3]  H. Thieme.  On the boundedness and the asymptotic behaviour of the non-negative solutions of Volterra-Hammerstein integral equations , 1980 .

[4]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

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

[6]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[7]  E. D. Sontagc,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2005 .

[8]  E. Camouzis,et al.  Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures , 2001 .

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

[10]  L. Collatz.  Functional analysis and numerical mathematics , 1968 .

[11]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[12]  U. Krause,et al.  Boundedness and Stability for Higher Order Difference Equations* , 2004 .

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

[14]  Eduardo Sontag,et al.  Global attractivity, I/O monotone small-gain theorems, and biological delay systems , 2005 .

[15]  Eduardo D. Sontag,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2006 .

[16]  J. Schröder.  Fehlerabschätzung bei linearen Gleichungssystemen mit dem Brouwerschen Fixpunktsatz , 1959 .

[17]  Convergence to equilibria in discrete population models , 2005 .

[18]  George Cosner.  Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models , 1997 .

[19]  E. D. Sontag,et al.  ON THE GLOBAL ATTRACTIVITY OF ABSTRACT DYNAMICAL SYSTEMS SATISFYING A SMALL GAIN HYPOTHESIS , WITH APPLICATIONS TO BIOLOGICAL DELAY SYSTEMS , 2022 .

[20]  R F Costantino,et al.  Nonlinear population dynamics: models, experiments and data. , 1998, Journal of theoretical biology.

[21]  Jim M Cushing,et al.  Chaos in Ecology: Experimental Nonlinear Dynamics , 2002 .

[22]  Alaa E. Hamza,et al.  On the recursive sequence x n+1 =.... , 2008 .

[23]  Horst R. Thieme,et al.  On a class of Hammerstein integral equations , 1979 .

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

[25]  Morris W. Hirsch,et al.  Monotone maps: A review , 2005 .

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

[27]  Jean-Luc Gouzé,et al.  A criterion of global convergence to equilibrium for differential systems. Application to Lotka-Volterra systems , 1988 .