April  2020, 19(4): 1847-1874. doi: 10.3934/cpaa.2020081

Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations

Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria

C.P. dedicates this paper to Professor Tomás Caraballo - friend and colleague - on the occasion of his 60th birthday

Received  April 2019 Revised  October 2019 Published  January 2020

We provide a convenient Neimark-Sacker bifurcation result for time-periodic difference equations in arbitrary Banach spaces. It ensures 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 integrodifference equations, which are infinite-dimensional discrete dynamical systems popular in theoretical ecology, and are used to describe the temporal evolution and spatial dispersal of populations with nonoverlapping generations. As an application, we combine analytical and numerical tools for a detailed bifurcation analysis of a spatial predator-prey model. Since such realistic models can frequently only be studied numerically, we formulate our assumptions in such a fashion as to allow for numerically stable verification.

Citation: Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1847-1874. doi: 10.3934/cpaa.2020081
References:
[1]

C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results, submitted, 2019. Google Scholar

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C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Applications and numerical results, submitted, 2019. Google Scholar

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J. Bramburger and F. Lutscher, Analysis of integrodifference equations with a separable dispersal kernel, Acta Applicandae Mathematicae, 161 (2019), 127-151.  doi: 10.1007/s10440-018-0207-9.  Google Scholar

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G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Heidelberg etc., 1996. doi: 10.1007/978-3-642-80043-6.  Google Scholar

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T. FariaW. Huang and J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces, SIAM J. Math. Anal., 34 (2002), 173-203.  doi: 10.1137/S0036141001384971.  Google Scholar

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M. E. Hochstenbach, A Jacobi-Davidson type method for the product eigenvalue problem, J. Computational and Applied Mathematics, 212 (2008), 46-62.  doi: 10.1016/j.cam.2006.11.020.  Google Scholar

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G. Iooss, Bifurcation of maps and applications, Mathematics Studies, 36 (1979), North-Holland, Amsterdam etc.  Google Scholar

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H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester etc., 1982.  Google Scholar

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T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften, 132 (1980), Springer, Berlin etc.  Google Scholar

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M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosc, 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

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M. Kot, Diffusion-driven period-doubling bifurcations, BioSystems, 22 (1989), 279-287.   Google Scholar

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R. Kress, Linear Integral Equations ($3$rd ed.), Applied Mathematical Sciences, 82 (2014), Springer, Heidelberg etc. Google Scholar

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D. Kressner, The periodic QR algorithm is a disguised QR algorithm, Linear Algebra and its Applications, 417 (2006), 423-433.  doi: 10.1016/j.laa.2003.06.014.  Google Scholar

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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.  Google Scholar

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O. E. Lanford Ⅲ, Bifurcation of periodic solutions into invariant tori, Lect. Notes Math, 322 (1973), pp. 159–192, Springer, Berlin etc. Google Scholar

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D. Lay, Characterizations of the essential spectrum of F. E. Browder, Bull. Am. Math. Soc, 74 (1968), 246-248.  doi: 10.1090/S0002-9904-1968-11905-6.  Google Scholar

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R. Martin, Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, 11 (1976), John Wiley & Sons, Chichester etc.  Google Scholar

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M. NeubertM. Kot and M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Popul. Biol, 48 (1995), 7-43.   Google Scholar

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R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478.   Google Scholar

[24]

C. Pötzsche, Bifurcations in a periodic discrete-time environment, Nonlin. Analysis: Real World Applications, 14 (2013), 53-82.  doi: 10.1016/j.nonrwa.2012.05.002.  Google Scholar

[25]

C. Pötzsche, Numerical dynamics of integrodifference equations: Basics and discretization errors in a $C^0$-setting, Applied Mathematics and Computation, 354 (2019), 422-443.  doi: 10.1016/j.amc.2019.02.033.  Google Scholar

[26]

C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, manuscript, (2019). Google Scholar

[27]

G. Röst, Neimark-Sacker bifurcation for periodic delay differential equations, Nonlin. Analysis (TMA), 60 (2005), 1025-1044.  doi: 10.1016/j.na.2004.08.043.  Google Scholar

[28]

G. Röst, Bifurcation of periodic delay differential equations at points of $1:4$ resonance, Functional Differential Equations, 13 (2006), 585-602.   Google Scholar

[29] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, Boston etc., 1989.   Google Scholar
[30]

R. J. Sacker, Chapter 2 of authors's 1964 dissertation, J. Difference Equ. Appl., 15 (2009), 759-774.  doi: 10.1080/10236190802357735.  Google Scholar

[31]

E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Applied Mathematical Sciences, 109 (1995), Springer, Heidelberg etc.  Google Scholar

show all references

References:
[1]

C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results, submitted, 2019. Google Scholar

[2]

C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Applications and numerical results, submitted, 2019. Google Scholar

[3]

J. Bramburger and F. Lutscher, Analysis of integrodifference equations with a separable dispersal kernel, Acta Applicandae Mathematicae, 161 (2019), 127-151.  doi: 10.1007/s10440-018-0207-9.  Google Scholar

[4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation–An Introduction, University Press, Princeton, NJ, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[5]

D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.  Google Scholar

[6]

S. DayO. Junge and K. Mischaikow, A rigerous numerical method for the global dynamics of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3 (2004), 117-160.  doi: 10.1137/030600210.  Google Scholar

[7]

G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Heidelberg etc., 1996. doi: 10.1007/978-3-642-80043-6.  Google Scholar

[8]

T. FariaW. Huang and J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces, SIAM J. Math. Anal., 34 (2002), 173-203.  doi: 10.1137/S0036141001384971.  Google Scholar

[9]

M. E. Hochstenbach, A Jacobi-Davidson type method for the product eigenvalue problem, J. Computational and Applied Mathematics, 212 (2008), 46-62.  doi: 10.1016/j.cam.2006.11.020.  Google Scholar

[10]

G. Iooss, Bifurcation of maps and applications, Mathematics Studies, 36 (1979), North-Holland, Amsterdam etc.  Google Scholar

[11]

H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester etc., 1982.  Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften, 132 (1980), Springer, Berlin etc.  Google Scholar

[13]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosc, 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[14]

M. Kot, Diffusion-driven period-doubling bifurcations, BioSystems, 22 (1989), 279-287.   Google Scholar

[15]

R. Kress, Linear Integral Equations ($3$rd ed.), Applied Mathematical Sciences, 82 (2014), Springer, Heidelberg etc. Google Scholar

[16]

D. Kressner, The periodic QR algorithm is a disguised QR algorithm, Linear Algebra and its Applications, 417 (2006), 423-433.  doi: 10.1016/j.laa.2003.06.014.  Google Scholar

[17]

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.  Google Scholar

[18]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, (3rd ed.), Applied Mathematical Sciences, 112 (2004), Springer, Berlin etc. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[19]

O. E. Lanford Ⅲ, Bifurcation of periodic solutions into invariant tori, Lect. Notes Math, 322 (1973), pp. 159–192, Springer, Berlin etc. Google Scholar

[20]

D. Lay, Characterizations of the essential spectrum of F. E. Browder, Bull. Am. Math. Soc, 74 (1968), 246-248.  doi: 10.1090/S0002-9904-1968-11905-6.  Google Scholar

[21]

R. Martin, Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, 11 (1976), John Wiley & Sons, Chichester etc.  Google Scholar

[22]

M. NeubertM. Kot and M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Popul. Biol, 48 (1995), 7-43.   Google Scholar

[23]

R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478.   Google Scholar

[24]

C. Pötzsche, Bifurcations in a periodic discrete-time environment, Nonlin. Analysis: Real World Applications, 14 (2013), 53-82.  doi: 10.1016/j.nonrwa.2012.05.002.  Google Scholar

[25]

C. Pötzsche, Numerical dynamics of integrodifference equations: Basics and discretization errors in a $C^0$-setting, Applied Mathematics and Computation, 354 (2019), 422-443.  doi: 10.1016/j.amc.2019.02.033.  Google Scholar

[26]

C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, manuscript, (2019). Google Scholar

[27]

G. Röst, Neimark-Sacker bifurcation for periodic delay differential equations, Nonlin. Analysis (TMA), 60 (2005), 1025-1044.  doi: 10.1016/j.na.2004.08.043.  Google Scholar

[28]

G. Röst, Bifurcation of periodic delay differential equations at points of $1:4$ resonance, Functional Differential Equations, 13 (2006), 585-602.   Google Scholar

[29] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, Boston etc., 1989.   Google Scholar
[30]

R. J. Sacker, Chapter 2 of authors's 1964 dissertation, J. Difference Equ. Appl., 15 (2009), 759-774.  doi: 10.1080/10236190802357735.  Google Scholar

[31]

E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Applied Mathematical Sciences, 109 (1995), Springer, Heidelberg etc.  Google Scholar

Figure 1.  Supercritical discrete torus bifurcation from a branch of $ \theta $-periodic solutions $ \phi(\alpha) $ (dotted) to (△α) into an $ \theta $-periodic invariant set $ {\mathcal T}_\alpha\subset {\mathcal U} $ (solid lines), where $ \theta = 4 $
Figure 2.  Absolute value, real part and imaginary part of $ \nu_+(\alpha) $
Figure 3.  Invariant circles displaying total populations from a Neimark-Sacker bifurcation in the autonomous IDE (△α) with right-hand side (5.1) at $ \alpha^\ast = \sqrt{3} $ (left) and $ \alpha^\ast = -\sqrt{3} $ (right). Attractive objects are in green, repulsive ones in red
Figure 4.  Schematic bifurcation diagram for the predator-prey model (△α) given by (5.3). For instance, non-primary bifurcations along the trivial solution are ignored
Figure 5.  $ 4 $-periodic solution branch $ \phi(\alpha) $ to the IDE (△α) with right-hand side (5.3) for $ \alpha\in[0.5, 2.3] $. The distribution of the prey $ \phi^1(\alpha) $ is marked in green, while the predators $ \phi^2(\alpha) $ vary from blue to yellow
Figure 6.  $ 4 $-periodic invariant circles displaying total populations from a Neimark-Sacker bifurcation in the IDE (△α) with right-hand side (5.3) for $ \alpha = 0.9 $ (top), $ \alpha = 0.95 $ (center), $ \alpha = 1 $ (bottom)
Figure 7.  Floquet multipliers $ \lambda^i(\alpha) $ along the $ 4 $-periodic coexistence solution branch $ \phi(\alpha) $ of (△α) indicating three critical parameter values $ \alpha_i^\ast $ in the interval $ [0.5, 2.3] $
Figure 8.  Assumptions on the spectrum $ \sigma(D_1\Pi(0, \alpha^\ast))\subset {\mathbb C} $ with essential radius $ r_0 $ in Thm. A.1
Table 1.  The powers of $ \nu_\ast $ are verifying the nonresonance condition 4.2(ⅰ)
$ l $ $ \nu_\ast^l $
$ 1 $ $ -0.201-0.980\iota $
$ 2 $ $ -0.919+0.393\iota $
$ 3 $ $ 0.570+0.822\iota $
$ 4 $ $ 0.691-0.723\iota $
$ l $ $ \nu_\ast^l $
$ 1 $ $ -0.201-0.980\iota $
$ 2 $ $ -0.919+0.393\iota $
$ 3 $ $ 0.570+0.822\iota $
$ 4 $ $ 0.691-0.723\iota $
Table 2.  Critical parameters $ \alpha_i^\ast $ where Floquet multipliers along $ \phi(\alpha) $ cross $ {\mathbb S}^1 $, the transversality condition $ \rho_i^\ast $ and the bifurcation indicator $ \delta_i^\ast $
$ i $ $ \alpha_i^\ast $ $ \rho_i^\ast $ $ \delta_i^\ast $
1 0.91831 1.9260 -0.859
2 1.28936 1.5721 -0.395
3 2.17617 1.0357 -0.318
$ i $ $ \alpha_i^\ast $ $ \rho_i^\ast $ $ \delta_i^\ast $
1 0.91831 1.9260 -0.859
2 1.28936 1.5721 -0.395
3 2.17617 1.0357 -0.318
Table 3.  The powers of $ \lambda_+(\alpha_i^\ast) $, verifying the nonresonance condition in Thm. 4.2(ⅲ)
$ i $ $ \lambda_+(\alpha_i^\ast) $ $ \lambda_+(\alpha_i^\ast)^2 $ $ \lambda_+(\alpha_i^\ast)^3 $ $ \lambda_+(\alpha_i^\ast)^4 $
1 $ -0.937 + 0.350\iota $ $ 0.755 - 0.656\iota $ $ -0.478 + 0.878\iota $ $ 0.140 - 0.990\iota $
2 $ -0.970 + 0.243\iota $ $ 0.881 - 0.472\iota $ $ -0.740 + 0.673\iota $ $ 0.554 - 0.833\iota $
3 $ -0.428 + 0.904\iota $ $ -0.633 - 0.774\iota $ $ 0.971 - 0.241\iota $ $ -0.198 + 0.980\iota $
$ i $ $ \lambda_+(\alpha_i^\ast) $ $ \lambda_+(\alpha_i^\ast)^2 $ $ \lambda_+(\alpha_i^\ast)^3 $ $ \lambda_+(\alpha_i^\ast)^4 $
1 $ -0.937 + 0.350\iota $ $ 0.755 - 0.656\iota $ $ -0.478 + 0.878\iota $ $ 0.140 - 0.990\iota $
2 $ -0.970 + 0.243\iota $ $ 0.881 - 0.472\iota $ $ -0.740 + 0.673\iota $ $ 0.554 - 0.833\iota $
3 $ -0.428 + 0.904\iota $ $ -0.633 - 0.774\iota $ $ 0.971 - 0.241\iota $ $ -0.198 + 0.980\iota $
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