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

Christian Aarset; Christian Pötzsche

doi:10.3934/cpaa.2020081

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2020, Volume 19, Issue 4: 1847-1874. Doi: 10.3934/cpaa.2020081

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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

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Abstract

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.

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

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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 $

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Figure 2. Absolute value, real part and imaginary part of $ \nu_+(\alpha) $

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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

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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

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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

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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)

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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] $

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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

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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 $

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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

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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 $

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