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 & 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. Google Scholar

[2]

S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167.Google Scholar

[3]

E. Cabral BalreiraS. 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. Google Scholar

[4]

P. Cull, Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75. doi: 10.1016/S0092-8240(88)90016-X. Google Scholar

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003.Google Scholar

[6]

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005.Google Scholar

[7]

S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[12]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010.Google Scholar

[13]

W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. Google Scholar

[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. Google Scholar

[15]

R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92. doi: 10.1080/10236190601008752. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[19]

H. Smith, Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. Google Scholar

show all references

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. Google Scholar

[2]

S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167.Google Scholar

[3]

E. Cabral BalreiraS. 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. Google Scholar

[4]

P. Cull, Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75. doi: 10.1016/S0092-8240(88)90016-X. Google Scholar

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003.Google Scholar

[6]

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005.Google Scholar

[7]

S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[12]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010.Google Scholar

[13]

W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. Google Scholar

[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. Google Scholar

[15]

R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92. doi: 10.1080/10236190601008752. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[19]

H. Smith, Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. Google Scholar

Figure 1.  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 $T(C)$
Figure 2.  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 $p,q=2$
Figure 3.  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 $[1-ab,1]$
Figure 4.  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
Figure 5.  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
Figure 6.  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 $(x,y)$
Figure 7.  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
Figure 9.  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 $y=\frac{y^*}{x^*}x$
Figure 8.  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
[1]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323

[2]

Patrick J. Johnson, Mark E. Burke. An investigation of the global properties of a two-dimensional competing species model. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 109-128. doi: 10.3934/dcdsb.2008.10.109

[3]

Elissar Nasreddine. Two-dimensional individual clustering model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 307-316. doi: 10.3934/dcdss.2014.7.307

[4]

Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559

[5]

Claude-Michel Brauner, Xinyue Fan, Luca Lorenzi. Two-dimensional stability analysis in a HIV model with quadratic logistic growth term. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1813-1844. doi: 10.3934/cpaa.2013.12.1813

[6]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

[7]

Yunshyong Chow, Kenneth Palmer. On a discrete three-dimensional Leslie-Gower competition model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4367-4377. doi: 10.3934/dcdsb.2019123

[8]

Michael Herty, Adrian Fazekas, Giuseppe Visconti. A two-dimensional data-driven model for traffic flow on highways. Networks & Heterogeneous Media, 2018, 13 (2) : 217-240. doi: 10.3934/nhm.2018010

[9]

Tibye Saumtally, Jean-Patrick Lebacque, Habib Haj-Salem. A dynamical two-dimensional traffic model in an anisotropic network. Networks & Heterogeneous Media, 2013, 8 (3) : 663-684. doi: 10.3934/nhm.2013.8.663

[10]

Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090

[11]

Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009

[12]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715

[13]

Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099

[14]

Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109

[15]

Scott Gordon. Nonuniformity of deformation preceding shear band formation in a two-dimensional model for Granular flow. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1361-1374. doi: 10.3934/cpaa.2008.7.1361

[16]

Zuguo Chen, Yonggang Li, Xiaofang Chen, Chunhua Yang, Weihua Gui. Anode effect prediction based on collaborative two-dimensional forecast model in aluminum electrolysis production. Journal of Industrial & Management Optimization, 2019, 15 (2) : 595-618. doi: 10.3934/jimo.2018060

[17]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[18]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173

[19]

Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739

[20]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (23)
  • HTML views (90)
  • Cited by (1)

Other articles
by authors

[Back to Top]