Article Contents
Article Contents

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

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

• 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.

Mathematics Subject Classification: 37G15, 39A28, 39A23.

 Citation:

• 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)

 $l$ $\nu_\ast^l$ $1$ $-0.201-0.980\iota$ $2$ $-0.919+0.393\iota$ $3$ $0.570+0.822\iota$ $4$ $0.691-0.723\iota$

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

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 $-0.428 + 0.904\iota$ $-0.633 - 0.774\iota$ $0.971 - 0.241\iota$ $-0.198 + 0.980\iota$
•  [1] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results, submitted, 2019. [2] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Applications and numerical results, submitted, 2019. [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. [4] B. Buffoni and  J. Toland,  Analytic Theory of Global Bifurcation–An Introduction, University Press, Princeton, NJ, 2003.  doi: 10.1515/9781400884339. [5] D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980. [6] S. Day, O. 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. [7] G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Heidelberg etc., 1996. doi: 10.1007/978-3-642-80043-6. [8] T. Faria, W. 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. [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. [10] G. Iooss, Bifurcation of maps and applications, Mathematics Studies, 36 (1979), North-Holland, Amsterdam etc. [11] H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester etc., 1982. [12] T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften, 132 (1980), Springer, Berlin etc. [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. [14] M. Kot, Diffusion-driven period-doubling bifurcations, BioSystems, 22 (1989), 279-287. [15] R. Kress, Linear Integral Equations ($3$rd ed.), Applied Mathematical Sciences, 82 (2014), Springer, Heidelberg etc. [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. [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. [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. [19] O. E. Lanford Ⅲ, Bifurcation of periodic solutions into invariant tori, Lect. Notes Math, 322 (1973), pp. 159–192, Springer, Berlin etc. [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. [21] R. Martin, Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, 11 (1976), John Wiley & Sons, Chichester etc. [22] M. Neubert, M. Kot and M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Popul. Biol, 48 (1995), 7-43. [23] R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. [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. [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. [26] C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, manuscript, (2019). [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. [28] G. Röst, Bifurcation of periodic delay differential equations at points of $1:4$ resonance, Functional Differential Equations, 13 (2006), 585-602. [29] D. Ruelle,  Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, Boston etc., 1989. [30] R. J. Sacker, Chapter 2 of authors's 1964 dissertation, J. Difference Equ. Appl., 15 (2009), 759-774.  doi: 10.1080/10236190802357735. [31] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Applied Mathematical Sciences, 109 (1995), Springer, Heidelberg etc.

Figures(8)

Tables(3)