# American Institute of Mathematical Sciences

March  2017, 22(2): 585-604. doi: 10.3934/dcdsb.2017028

## Global stability in the 2D Ricker equation revisited

 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: R. J. Sacker

Received  February 2016 Revised  August 2016 Published  December 2016

We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.

Citation: Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028
##### References:
 [1] A.S. Ackleh and P.L. Salceanu, Competitive exclusion and coexistence in an n-species Ricker model, J Biological Dynamics, 9 (2015), 321-331.  doi: 10.1080/17513758.2015.1020576. [2] S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167. [3] E. Cabral 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. [4] P. Cull, Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75.  doi: 10.1016/S0092-8240(88)90016-X. [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003. [6] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. [7] S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008. [8] H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573-596.  doi: 10.1007/BF00275495. [9] J. Li, Simple mathematical models for mosquito populations with genetically altered mosquitos, Math. Bioscience, 189 (2004), 39-59.  doi: 10.1016/j.mbs.2004.01.001. [10] E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynam. Syst.-B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191. [11] C. Mira, L. Gardini, A. Barugola and J. -C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, volume 20 of Series in Nonlinear Sciences, World Scientific, Tokyo, Japan, 1996. [12] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. [13] W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. [14] 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. [15] R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.  doi: 10.1080/10236190601008752. [16] R. J. Sacker and H. F. von Bremen, Global asymptotic stability in the Jia Li model for genetically altered mosquitos, In Linda J. S. Allen-et. al. , editor, Difference Equations and Discrete Dynamical Systems, Proc. 9th Internat. Conf. on Difference Equations and Appl. (2004), pages 87-100. World Scientific, 2005. [17] R. J. Sacker and H. F. von Bremen, Dynamic reduction with applications to mathematical biology and other areas, J. Biological Dynamics, 1 (2007), 437-453.  doi: 10.1080/17513750701605572. [18] A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Federenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers Group, Dordrecht, Netherlands, 1997. [19] H. Smith, Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357.  doi: 10.1080/10236199708808108.

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##### References:
 [1] A.S. Ackleh and P.L. Salceanu, Competitive exclusion and coexistence in an n-species Ricker model, J Biological Dynamics, 9 (2015), 321-331.  doi: 10.1080/17513758.2015.1020576. [2] S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167. [3] E. Cabral 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. [4] P. Cull, Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75.  doi: 10.1016/S0092-8240(88)90016-X. [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003. [6] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. [7] S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008. [8] H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573-596.  doi: 10.1007/BF00275495. [9] J. Li, Simple mathematical models for mosquito populations with genetically altered mosquitos, Math. Bioscience, 189 (2004), 39-59.  doi: 10.1016/j.mbs.2004.01.001. [10] E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynam. Syst.-B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191. [11] C. Mira, L. Gardini, A. Barugola and J. -C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, volume 20 of Series in Nonlinear Sciences, World Scientific, Tokyo, Japan, 1996. [12] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. [13] W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. [14] 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. [15] R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.  doi: 10.1080/10236190601008752. [16] R. J. Sacker and H. F. von Bremen, Global asymptotic stability in the Jia Li model for genetically altered mosquitos, In Linda J. S. Allen-et. al. , editor, Difference Equations and Discrete Dynamical Systems, Proc. 9th Internat. Conf. on Difference Equations and Appl. (2004), pages 87-100. World Scientific, 2005. [17] R. J. Sacker and H. F. von Bremen, Dynamic reduction with applications to mathematical biology and other areas, J. Biological Dynamics, 1 (2007), 437-453.  doi: 10.1080/17513750701605572. [18] A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Federenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers Group, Dordrecht, Netherlands, 1997. [19] H. Smith, Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357.  doi: 10.1080/10236199708808108.
The curves $T^k(C)$ for $k=0,1,2,3$ are shown, as well as the unstable manifolds (thicker lines) from $(p,0)$ and $(0,q)$. The unstable manifolds intersect at the coexistence fixed point. The curves, ordered from bottom left to top right, go $C, T^2(C)$, the unstable manifolds, $T^3(C)$, and then finally $T(C)$
The figure shows the upper bounds implied by Theorems 2.3, 3.2, and Conjecture 1, respectively. In the left column, we have the upper bounds for $p$ and in the right column we have the upper bounds for $q$. We have capped the upper bounds at 2 for the plots since the fixed point loses stability past $p,q=2$
The left branch of a typical graph of $V$ versus $\sigma$. For small $t$ the graph may lie completely above the $\sigma$-axis on the interval $[1-ab,1]$
In the left figure, we show the isocline $L_p$ and the curve $y=-\frac{1}{a}\ln\left(\frac{2x^*}{x}-1\right)-\frac{x-p}{a}$ by solid lines. The shaded regions are where the function moves closer in the $x$ coordinate. On the right, the isocline $L_q$ and the curve $x=-\frac{1}{b}\ln\left(\frac{2y^*}{y}-1\right)-\frac{y-q}{b}$ are shown as solid lines. The shaded regions are where the $y$ coordinate moves closer. The union of the two regions is the entire plane
A graph of the function $G(x,y)$ is shown for parameters $p=1.8$, $q=1.9$, $a=0.2$, $b=0.3$. We observe that this appears to be a concave function with a maximum at the fixed point illustrated by the vertical line
A graph of the entry $G_{xx}$ is shown for parameters $p=1.8$, $q=1.9$, $a=0.2$, $b=0.3$. The entry is clearly negative for all $(x,y)$
A graph of the determinant of the Hessian is shown for parameters $p=1.8$, $q=1.9$, $a=0.2$, $b=0.3$. The determinant is clearly positive
A plot of the isoclines of $T^2$ relative to the isoclines of $T$ as proved in Lemma5.4, see also Figure 8. The straight dashed line is $y=\frac{y^*}{x^*}x$
The plane is divided into six regions $H_n$ by the isoclines $L_p$, $L_q$ (in the figure $L_p$ is the solid line from the top middle to the bottom, and $L_q$ is the other solid line) and the line $y=\frac{y^*}{x^*}x$ (shown dashed
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