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The discrete dynamics of monotonically decomposable maps. (English) Zbl 1118.65057
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.
MSC:
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
92D25Population dynamics (general)
References:
[1]Angeli D., Sontag E.D. (2003) Monotone Control Systems. IEEE Trans Autom Control 48, 1684–1698 · doi:10.1109/TAC.2003.817920
[2]Berman A., Plemmons R. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic, New York
[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
[4]Collatz L. (1966) Functional Analysis and Numerical Mathematics. Academic, NY
[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 · doi:10.1038/375227a0
[6]Cushing J.M. (1988) Nonlinear matrix models and population dynamics. Nat Resour Model 2, 539–580
[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 · doi:10.1006/jtbi.1998.0736
[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 · doi:10.1080/10236190512331319334
[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)
[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)
[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
[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
[15]Hirsch M.W., Smith H.L. (2005) Monotone Maps: a review. J Differ Eqns Appl 11, 379–398 · Zbl 1080.37016 · doi:10.1080/10236190412331335445
[16]Kulenović M., Ladas G. (2002). Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC, Boca Raton
[17]Kulenović M., Ladas G., Sizer W. (1998). On the recursive sequence x n+1 = (αx n + βx n-1)/(γx n + δx n-1). Math Sci Res Hot-Line 2(5): 1–16
[18]Krause U., Pituk M. (2004) Boundedness and stability for higher order difference equations. J Differ Eqns Appl 10, 343–356 · Zbl 1049.39006 · doi:10.1080/1023619031000115377
[19]Schröder J. (1959) Fehlerabschätzungen bei linearen Gleichungssystemen mit dem Brouwerschen Fixpunktssatz. Arch Rat Mech Anal 3, 28–44 · Zbl 0099.11002 · doi:10.1007/BF00284162
[20]Smith H.L. (1998) Planar competitive and cooperative difference equations. J Differ Eqns Appl 3, 335–357 · Zbl 0907.39004 · doi:10.1080/10236199708808108
[21]Thieme H.R. (1979) On a class of hammerstein integral equations. Manuscripta Math 29, 49–84 · Zbl 0417.45003 · doi:10.1007/BF01309313
[22]Thieme H.R. (1980) On a class of hammerstein integral equations. Manuscripta Math 31, 379–412 · Zbl 0454.45003 · doi:10.1007/BF02320701