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March  2022, 27(3): 1263-1284. doi: 10.3934/dcdsb.2021089

Bifurcation in the almost periodic $2$D Ricker map

 1 Department of Mathematics, California State University Bakersfield, Bakersfield, CA 93311-1022, USA 2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA

* Corresponding author: Brian Ryals

Received  August 2020 Revised  January 2021 Published  March 2022 Early access  March 2021

Fund Project: RJS is supported by a University of Southern California, Dornsife School of Letters Arts and Sciences Faculty Development Grant, 12-1855-0032

This paper studies bifurcations in the coupled $2$ dimensional almost periodic Ricker map. We establish criteria for stability of an almost periodic solution in terms of the Lyapunov exponents of a corresponding dynamical system and use them to find a bifurcation function. We find that if the almost periodic coefficients of all the maps are identical, then the bifurcation function is the same as the one obtained in the one dimensional case treated earlier, and that this result holds in $N$ dimension under modest coupling constraints. In the general two-dimensional case, we compute the Lyapunov exponents numerically and use them to examine the stability and bifurcations of the almost periodic solutions.

Citation: Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $2$D Ricker map. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1263-1284. doi: 10.3934/dcdsb.2021089
References:
 [1] A. Avila, J. Santamaria, M. Viana and A. Wilkinson, Cocycles over partially hyperbolic maps, Asterisque, 358 (2013), 1-12. [2] E. C. Balreira, S. Elaydi and R. Luis, Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.  doi: 10.3934/dcdsb.2014.19.323. [3] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. [4] J. W. S. Cassel, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957. [5] K. Chandrasekharan, Introduction to Analytic Number Theory, Number 148 in Die Grundlehren der matematishen Wissenshaft in Einzeldarstellung. Springer Verlag, New York, 1968. [6] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617. [7] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. [8] S. Elaydi and R. J. Sacker, Basin of attraction of periodic orbits of maps on the real line, J Difference Eq and Appl, 10 (2004), 881-888.  doi: 10.1080/10236190410001731443. [9] S. Gershgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. Sci. USSR. Classe des sci. math., 6 (1931), 749-754. [10] M. Keykobad, Positive solutions of positive linear systems, Lin. Alg. and its Appl., 64 (1985), 133-140.  doi: 10.1016/0024-3795(85)90271-X. [11] R. Luis, S. Elaydi and an d H. Oliveira, Stability of a ricker-type competition model and the competitive exclusion principle, J. of Biological Dynamics, 5 (2011), 636-660.  doi: 10.1080/17513758.2011.581764. [12] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math Soc., 19 (1968), 197-231. [13] W. E. Ricker, Stock and recruitment, J. Fisheries Research Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039. [14] B. Ryals, Dynamics of the Degenerate 2D Ricker Equation, Math. Methods Appl. Sci., 42 (2019), 553-566.  doi: 10.1002/mma.5360. [15] B. Ryals, A sufficient condition for stability using slopes of isoclines in planar mappings, J. Difference Eq. and Appl., 26 (2020), 370-383.  doi: 10.1080/10236198.2020.1737034. [16] B. Ryals and R. J. Sacker, Global stability in the 2-D Ricker equation, J Difference Eq and Appl, 21 (2015), 1068-1081.  doi: 10.1080/10236198.2015.1065825. [17] B. Ryals and R. J. Sacker, Global stability in the 2D Ricker equation revisited, Discrete and Continuous Dynam. Syst.-B, 22 (2017), 585-604.  doi: 10.3934/dcdsb.2017028. [18] R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.  doi: 10.1080/10236190601008752. [19] R. J. Sacker, Bifurcation in the almost periodic Ricker map, J. Difference Eq. and Appl., 25 (2019), 599-618.  doi: 10.1080/10236198.2019.1604696. [20] R. J. Sacker and G. R. Sell, Almost periodicity, Ricker map, Beverton-Holt map and others, a general method, J Difference Eq and Appl, 23 (2017), 1286-1297.  doi: 10.1080/10236198.2017.1320397. [21] R. J. Sacker and G. R. Sell, Corrigendum, J. Difference Eq. Appl., 24 (2018), 164. doi: 10.1080/10236198.2017.1379183. [22] A. Wilkinson, What are Lyapunov exponents, and why are they interesting?, Bull. Amer. Math. Soc., 54 (2017), 79-105.  doi: 10.1090/bull/1552. [23] L. -S. Young, Mathematical theory of Lyapunov exponents, J. Phys. A: Mathematical and Theoretical, 46 (2013), 254001, 17pp. doi: 10.1088/1751-8113/46/25/254001.

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References:
 [1] A. Avila, J. Santamaria, M. Viana and A. Wilkinson, Cocycles over partially hyperbolic maps, Asterisque, 358 (2013), 1-12. [2] E. C. Balreira, S. Elaydi and R. Luis, Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.  doi: 10.3934/dcdsb.2014.19.323. [3] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. [4] J. W. S. Cassel, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957. [5] K. Chandrasekharan, Introduction to Analytic Number Theory, Number 148 in Die Grundlehren der matematishen Wissenshaft in Einzeldarstellung. Springer Verlag, New York, 1968. [6] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617. [7] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. [8] S. Elaydi and R. J. Sacker, Basin of attraction of periodic orbits of maps on the real line, J Difference Eq and Appl, 10 (2004), 881-888.  doi: 10.1080/10236190410001731443. [9] S. Gershgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. Sci. USSR. Classe des sci. math., 6 (1931), 749-754. [10] M. Keykobad, Positive solutions of positive linear systems, Lin. Alg. and its Appl., 64 (1985), 133-140.  doi: 10.1016/0024-3795(85)90271-X. [11] R. Luis, S. Elaydi and an d H. Oliveira, Stability of a ricker-type competition model and the competitive exclusion principle, J. of Biological Dynamics, 5 (2011), 636-660.  doi: 10.1080/17513758.2011.581764. [12] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math Soc., 19 (1968), 197-231. [13] W. E. Ricker, Stock and recruitment, J. Fisheries Research Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039. [14] B. Ryals, Dynamics of the Degenerate 2D Ricker Equation, Math. Methods Appl. Sci., 42 (2019), 553-566.  doi: 10.1002/mma.5360. [15] B. Ryals, A sufficient condition for stability using slopes of isoclines in planar mappings, J. Difference Eq. and Appl., 26 (2020), 370-383.  doi: 10.1080/10236198.2020.1737034. [16] B. Ryals and R. J. Sacker, Global stability in the 2-D Ricker equation, J Difference Eq and Appl, 21 (2015), 1068-1081.  doi: 10.1080/10236198.2015.1065825. [17] B. Ryals and R. J. Sacker, Global stability in the 2D Ricker equation revisited, Discrete and Continuous Dynam. Syst.-B, 22 (2017), 585-604.  doi: 10.3934/dcdsb.2017028. [18] R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.  doi: 10.1080/10236190601008752. [19] R. J. Sacker, Bifurcation in the almost periodic Ricker map, J. Difference Eq. and Appl., 25 (2019), 599-618.  doi: 10.1080/10236198.2019.1604696. [20] R. J. Sacker and G. R. Sell, Almost periodicity, Ricker map, Beverton-Holt map and others, a general method, J Difference Eq and Appl, 23 (2017), 1286-1297.  doi: 10.1080/10236198.2017.1320397. [21] R. J. Sacker and G. R. Sell, Corrigendum, J. Difference Eq. Appl., 24 (2018), 164. doi: 10.1080/10236198.2017.1379183. [22] A. Wilkinson, What are Lyapunov exponents, and why are they interesting?, Bull. Amer. Math. Soc., 54 (2017), 79-105.  doi: 10.1090/bull/1552. [23] L. -S. Young, Mathematical theory of Lyapunov exponents, J. Phys. A: Mathematical and Theoretical, 46 (2013), 254001, 17pp. doi: 10.1088/1751-8113/46/25/254001.
The stability region of the coexistence fixed point of equation (4) is shown in the $p$-$q$ plane for the coupling values $a = 0.5$ and $b = 0.7$. The geometry is similar for all $ab < 1$, with the stability region bounded by two straight lines and a piece of a hyperbola
Plot of the Bifurcation Equation $B(\gamma,\gamma,b) = 0$. A bifurcation takes place as the parameter pair $(b,\log{\gamma})$ crosses from the stability region ($B<0$) to the instability region ($B>0$)
Plots of the values of the two Lyapunov Exponents $\chi_1$ (left plot) and $\chi_2$ (right plot) as functions of the parameter $0\leq b \leq 0.99$. Values used were $\gamma_{01} = e^{1.2}$, $\gamma_{02} = e^{1.6}$, $a_{12} = 0.6$, $a_{21} = 0.8$, $G_1(\theta) = \sin(2\pi \theta)$, $G_2(\theta) = \sin(2\pi\theta)\cos(2\pi\theta)$, and $\omega = \frac{e^{\pi}}{25}\approx 0.8984$. Both are decreasing functions with respect to $b$. The $b$ values used were $0.01m$ for $m = 0, 1, \cdots 99$
The border of the stability region for an almost periodic solution is shown for the values $b = 0$, $b = 0.2$, $b = 0.4$, and $b = 0.6$, in a neighborhood of $(e^2,e^2)$. Here the $x$-axis is $\log(\gamma_{01})$ and the $y$-axis is $\log(\gamma_{02})$. The curves are ordered from bottom left to top right in order of increasing $b$, so that the stability region is growing with $b$. Values used were $a_{12} = 0.5$, $a_{21} = 0.7$, $G_1(\theta) = \sin(2\pi \theta)$, $G_2(\theta) = \sin(2\pi\theta)\cos(2\pi\theta)$, and $\omega = \frac{e^{\pi}}{25}\approx 0.8984$
The top left image shows the long-term dynamics of $(x_1(t),x_2(t),\gamma_1(t))$ for $b = 0.4$, while the top right shows the long-term dynamics of $(x_1(t),x_2(t),\gamma_2(t))$ for the same value of $b$. The bottom row uses $b = 0.2$ instead where the AP solution is unstable. Values used were $a_{12} = 0.5$, $a_{21} = 0.7$ and the almost periodic sequences are $G_1(\theta_k) = \sin(2\pi \theta_k)$, $G_2(\theta_k) = \sin(2\pi\theta_k)\cos(2\pi\theta_k)$ where $\theta_k = \theta+k\omega$ with $\omega = \frac{e^{\pi}}{25}\approx 0.8984$
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