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Bifurcations in periodic integrodifference equations in \(C(\Omega)\): II. Discrete torus bifurcations. (English) Zbl 1441.39011

The main purpose of this paper is to provide a Neimark-Sacker bifurcation result for time-periodic difference equations in arbitrary Banach spaces. The authors study the bifurcation of discrete invariant tori caused by a pair of complex-conjugated Floquet multipliers crossing the complex unit circle. This criterion is made explicit for integro-difference equations defining infinite-dimensional discrete dynamical systems used in theoretical ecology to describe the temporal evolution and spatial dispersion of populations with nonoverlapping generations.
As an application, the authors combine analytical and numerical tools for a detailed bifurcation analysis of a spatial predator-prey model. Since such realistic models can typically only be studied numerically, the assumptions are formulated to allow numerically stable verifications.

MSC:

39A28 Bifurcation theory for difference equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
39A23 Periodic solutions of difference equations
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
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References:

[1] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in \(C(\Omega)\) I: Analytical results, submitted, 2019. · Zbl 1473.37063
[2] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in \(C(\Omega)\) I: Applications and numerical results, submitted, 2019. · Zbl 1473.37063
[3] J. Bramburger; F. Lutscher, Analysis of integrodifference equations with a separable dispersal kernel, Acta Applicandae Mathematicae, 161, 127-151 (2019) · Zbl 1414.37036
[4] D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.
[5] S. Day; O. Junge; K. Mischaikow, A rigerous numerical method for the global dynamics of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3, 117-160 (2004) · Zbl 1059.37068
[6] G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Heidelberg etc., 1996. · Zbl 0857.65003
[7] T. Faria; W. Huang; J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces, SIAM J. Math. Anal., 34, 173-203 (2002) · Zbl 1085.34064
[8] M. E. Hochstenbach, A Jacobi-Davidson type method for the product eigenvalue problem, J. Computational and Applied Mathematics, 212, 46-62 (2008) · Zbl 1139.65028
[9] G. Iooss, Bifurcation of maps and applications, Mathematics Studies, 36 (1979), North-Holland, Amsterdam etc. · Zbl 0408.58019
[10] H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester etc., 1982. · Zbl 0465.47001
[11] T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften, 132 (1980), Springer, Berlin etc. · Zbl 0435.47001
[12] M. Kot; W. Schaffer, Discrete-time growth-dispersal models, Math. Biosc, 80, 109-136 (1986) · Zbl 0595.92011
[13] M. Kot, Diffusion-driven period-doubling bifurcations, BioSystems, 22, 279-287 (1989)
[14] R. Kress, Linear Integral Equations \((3\) rd ed.), Applied Mathematical Sciences, 82 (2014), Springer, Heidelberg etc.
[15] D. Kressner, The periodic QR algorithm is a disguised QR algorithm, Linear Algebra and its Applications, 417, 423-433 (2006) · Zbl 1119.65023
[16] T. Krisztin, H.-O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of An Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, 11 (1999), AMS, Providence, RI. · Zbl 1004.34002
[17] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, (3rd ed.), Applied Mathematical Sciences, 112 (2004), Springer, Berlin etc.
[18] O. E. Lanford Ⅲ, Bifurcation of periodic solutions into invariant tori, Lect. Notes Math, 322 (1973), pp. 159-192, Springer, Berlin etc.
[19] D. Lay, Characterizations of the essential spectrum of F. E. Browder, Bull. Am. Math. Soc, 74, 246-248 (1968) · Zbl 0157.45103
[20] R. Martin, Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, 11 (1976), John Wiley & Sons, Chichester etc.
[21] M. Neubert; M. Kot; M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Popul. Biol, 48, 7-43 (1995) · Zbl 0863.92016
[22] R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37, 473-478 (1970) · Zbl 0216.41602
[23] C. Pötzsche, Bifurcations in a periodic discrete-time environment, Nonlin. Analysis: Real World Applications, 14, 53-82 (2013) · Zbl 1278.39023
[24] C. Pötzsche, Numerical dynamics of integrodifference equations: Basics and discretization errors in a \(C^0\)-setting, Applied Mathematics and Computation, 354, 422-443 (2019) · Zbl 1429.65322
[25] C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, manuscript, (2019).
[26] G. Röst, Neimark-Sacker bifurcation for periodic delay differential equations, Nonlin. Analysis (TMA), 60, 1025-1044 (2005) · Zbl 1064.34058
[27] G. Röst, Bifurcation of periodic delay differential equations at points of \(1:4\) resonance, Functional Differential Equations, 13, 585-602 (2006)
[28] R. J. Sacker, Chapter 2 of authors’s 1964 dissertation, J. Difference Equ. Appl., 15, 759-774 (2009) · Zbl 1188.37018
[29] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Applied Mathematical Sciences, 109 (1995), Springer, Heidelberg etc. · Zbl 0834.46003
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