Smith, H. L. The discrete dynamics of monotonically decomposable maps. (English) Zbl 1118.65057 J. Math. Biol. 53, No. 4, 747-758 (2006); erratum ibid. 57, No. 2, 309-310 (2008). Summary: We extend results of J.-L. Gouzé and K. P. Hadeler [Nonlinear World 1, 23–34 (1994; Zbl 0803.65076)] 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. Cited in 1 ReviewCited in 22 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 47H10 Fixed-point theorems 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 92D25 Population dynamics (general) Keywords:ordered metric space; fixed point; population models Citations:Zbl 0803.65076 PDF BibTeX XML Cite \textit{H. L. Smith}, J. Math. Biol. 53, No. 4, 747--758 (2006; Zbl 1118.65057) Full Text: DOI OpenURL References: [1] Angeli D., Sontag E.D. (2003) Monotone Control Systems. IEEE Trans Autom Control 48, 1684–1698 · Zbl 1364.93326 [2] Berman A., Plemmons R. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic, New York · Zbl 0484.15016 [3] Cosner C. (1997) Comparison Principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models. Dyn Contin Discrete Impuls Syst 3, 283–303 · Zbl 0885.35015 [4] Collatz L. (1966) Functional Analysis and Numerical Mathematics. Academic, NY · Zbl 0148.39002 [5] Costantino R.F., Cushing J.M., Dennis B., Desharnais R.A. (1995) Experimentally induced transitions in the dynamic behavior of insect populations. Nature 375, 227–230 [6] Cushing J.M. (1988) Nonlinear matrix models and population dynamics. Nat Resour Model 2, 539–580 · Zbl 0850.92069 [7] Cushing J.M., Costantino R.F., Dennis B., Desharnais R.A., Henson S.M. (1998) Nonlinear population dynamics: models, experiments and data. J Theor Biol 194, 1–9 · Zbl 0973.92034 [8] Cushing J.M., Costantino R.F., Dennis B., Desharnais R.A., Henson S.M. (2003) Chaos in Ecology, Experimental Nonlinear Dynamics. Academic, New York [9] El-Morshedy H.A., Liz E. (2005) Convergence to equilibria in discrete population models. J Differ Eqns Appl 11, 117–131 · Zbl 1070.39022 [10] Enciso, G.A., Sontag, E.D. On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems. Discrete Contin Dyn Syst (to appear) · Zbl 1111.93071 [11] Enciso, G.A., Smith, H.L., Sontag, E.D. Non-monotone systems decomposable into monotone systems with negative feedback. J Diff Eqns (to appear) · Zbl 1103.34021 [12] Gouzé, J.-L. A criterion of global convergence to equilibrium for differential systems with an application to Lotka-Volterra systems. Rapport de Recherche 894, INRIA (1988) [13] Gouzé J.-L., Hadeler K.P. (1994) Monotone flows and order intervals. Nonlinear World 1, 23–34 · Zbl 0803.65076 [14] Hirsch M.W., Smith H.L. (2005). Monotone dynamical systems. In: Canada A., Drabek P., Fonda A. (eds). Handbook of Differential Equations, Ordinary Differential Equations, vol. 2, Elsevier, Amsterdam, pp. 239-357 · Zbl 1094.34003 [15] Hirsch M.W., Smith H.L. (2005) Monotone Maps: a review. J Differ Eqns Appl 11, 379–398 · Zbl 1080.37016 [16] Kulenović M., Ladas G. (2002). Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC, Boca Raton · Zbl 0981.39011 [17] Kulenović M., Ladas G., Sizer W. (1998). On the recursive sequence x n+1 = ({\(\alpha\)}x n + {\(\beta\)}x n-1)/({\(\gamma\)}x n + {\(\delta\)}x n-1). Math Sci Res Hot-Line 2(5): 1–16 · Zbl 0960.39502 [18] Krause U., Pituk M. (2004) Boundedness and stability for higher order difference equations. J Differ Eqns Appl 10, 343–356 · Zbl 1049.39006 [19] Schröder J. (1959) Fehlerabschätzungen bei linearen Gleichungssystemen mit dem Brouwerschen Fixpunktssatz. Arch Rat Mech Anal 3, 28–44 · Zbl 0099.11002 [20] Smith H.L. (1998) Planar competitive and cooperative difference equations. J Differ Eqns Appl 3, 335–357 · Zbl 0907.39004 [21] Thieme H.R. (1979) On a class of hammerstein integral equations. Manuscripta Math 29, 49–84 · Zbl 0417.45003 [22] Thieme H.R. (1980) On a class of hammerstein integral equations. Manuscripta Math 31, 379–412 · Zbl 0454.45003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.