September  2020, 25(9): 3305-3334. doi: 10.3934/dcdsb.2020063

Geometric method for global stability of discrete population models

School of Computing & Digital Media, London Metropolitan University, 166–220 Holloway Road, London N7 8DB, UK

* Corresponding author: Zhanyuan Hou

Received  February 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary fixed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by finding the eigenvalues of the Jacobian matrix at the fixed point. An intriguing question is proposed: Can a globally attracting fixed point induce a homoclinic cycle?

Citation: Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
References:
[1]

S. Ahmad and A. C. Lazer, One Species Extinction in an Autonomous Competitive Model, World Congress of Nonlinear Analysts, de Gruyter, Berlin, 1996, 359–368.

[2]

L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodifference equations applied to plant dispersal, competition, and control, in Differential Equations with Applications to Biology, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999, 15–30. doi: 10.1090/fic/021/02.

[3]

D. N. Atkinson, Mathematical models for plant competition and dispersal, Master's thesis, Texas Tech University in Lubbock, TX, 1997.

[4]

S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, J. Difference Equ. Appl., 23 (2017), 1378-1396.  doi: 10.1080/10236198.2017.1333116.

[5]

R. Bouyekhf and L. T. Gruyitch, An alternative approach for stability analysis of discrete time nonlinear dynamical systems, J. Difference Equ. Appl., 24 (2018), 68-81.  doi: 10.1080/10236198.2017.1391239.

[6]

E. Cabral BalreiraS. Elaydi and R. Luís, Global stability of higher dimensional monotone maps, J. Difference Equ. Appl., 23 (2017), 2037-2071.  doi: 10.1080/10236198.2017.1388375.

[7]

J. M. CushingS. LevargeN. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Equ. Appl., 10 (2004), 1139-1151.  doi: 10.1080/10236190410001652739.

[8]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, J. Math. Biol., 30 (1991), 161-168.  doi: 10.1007/BF00160333.

[9]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. Appl., 168 (1992), 385-400.  doi: 10.1016/0022-247X(92)90167-C.

[10]

M. GyllenbergJ. JiangL. Niu and P. Yan, On the classification of generalised competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discret. Contin. Dyn. Syst., 38 (2018), 615-650.  doi: 10.3934/dcds.2018027.

[11]

M. GyllenbergJ. Jiang and L. Niu, A note on global stability of three-dimensional Ricker models, J. Difference Equ. Appl., 25 (2019), 142-150.  doi: 10.1080/10236198.2019.1566459.

[12]

M. GyllenbergJ. JiangL. Niu and P. Yan, On the dynamics of multi-species Ricker models admitting a carrying simplex, J. Difference Equ. Appl., 25 (2019), 1489-1530.  doi: 10.1080/10236198.2019.1663182.

[13]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179.  doi: 10.1080/17513750801939236.

[14]

J. HofbauerV. Hutson and W. Jansen, Coexistence for systems generated by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570.  doi: 10.1007/BF00276199.

[15]

Z. Hou, Global attractor in autonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 127 (1999), 3633-3642.  doi: 10.1090/S0002-9939-99-05204-1.

[16]

Z. Hou, Global attractor in competitive Lotka-Volterra systems, Math. Nachr., 282 (2009), 995-1008.  doi: 10.1002/mana.200610785.

[17]

Z. Hou, Vanishing components in autonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 359 (2009), 302-310.  doi: 10.1016/j.jmaa.2009.05.054.

[18]

Z. Hou, Permanence, global attraction and stability, in Lotka-Volterra and Related Systems, Recent De Gruyter Ser. Math. Life Sci., 2, De Gruyter, Berlin, 2013, 1–62.

[19]

Z. Hou, Geometric method for global stability and repulsion in Kolmogorov systems, Dyn. Syst., 34 (2019), 456-483.  doi: 10.1080/14689367.2018.1554030.

[20]

J. JiangJ. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Difference Equations, 246 (2009), 1623-1672.  doi: 10.1016/j.jde.2008.10.008.

[21]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244.  doi: 10.3934/dcds.2016.36.217.

[22]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261.  doi: 10.1007/s00285-016-1052-y.

[23]

J. JiangL. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972.  doi: 10.1007/s00285-015-0920-1.

[24]

R. Kon, Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems, Proc. Amer. Math. Soc., 134 (2006), 3025-3034.  doi: 10.1090/S0002-9939-06-08309-2.

[25]

J. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976. doi: 10.1137/1.9781611970432.

[26]

F. Montes de Oca and M. L. Zeeman, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 192 (1995), 360-370.  doi: 10.1006/jmaa.1995.1177.

[27]

F. Montes de Oca and M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc, 124 (1996), 3677-3687.  doi: 10.1090/S0002-9939-96-03355-2.

[28] A. Pakes and R. Maller, Mathematical Ecology of Plant Species Competition. A Class of Deterministic Models for Binary Mixtures of Plant Genotypes, Cambridge Studies in Mathematical Biology, 10, Cambridge University Press, Cambridge, 1990.  doi: 10.1016/0020-7519(90)90100-2.
[29]

L. I. W. Roeger, Discrete May-Leonard competition models. II, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 841-860.  doi: 10.3934/dcdsb.2005.5.841.

[30]

L. I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models. I, J. Difference Equ. Appl., 10 (2004), 77-98.  doi: 10.1080/10236190310001603662.

[31]

A. Ruiz-Herrera, Topological criteria of global attraction with application in population dynamics, Nonlinearity, 25 (2012), 2823-2841.  doi: 10.1088/0951-7715/25/10/2823.

[32]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Equ. Appl., 19 (2013), 96-113.  doi: 10.1080/10236198.2011.628663.

[33]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.

[34]

H. L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equations Appl., 3 (1998), 335-357.  doi: 10.1080/10236199708808108.

[35]

Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Differential Equations, 176 (2001), 470-493.  doi: 10.1006/jdeq.2001.3989.

[36]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632.  doi: 10.1016/S0022-0396(02)00025-6.

[37]

M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.  doi: 10.1090/S0002-9939-1995-1264833-2.

show all references

References:
[1]

S. Ahmad and A. C. Lazer, One Species Extinction in an Autonomous Competitive Model, World Congress of Nonlinear Analysts, de Gruyter, Berlin, 1996, 359–368.

[2]

L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodifference equations applied to plant dispersal, competition, and control, in Differential Equations with Applications to Biology, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999, 15–30. doi: 10.1090/fic/021/02.

[3]

D. N. Atkinson, Mathematical models for plant competition and dispersal, Master's thesis, Texas Tech University in Lubbock, TX, 1997.

[4]

S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, J. Difference Equ. Appl., 23 (2017), 1378-1396.  doi: 10.1080/10236198.2017.1333116.

[5]

R. Bouyekhf and L. T. Gruyitch, An alternative approach for stability analysis of discrete time nonlinear dynamical systems, J. Difference Equ. Appl., 24 (2018), 68-81.  doi: 10.1080/10236198.2017.1391239.

[6]

E. Cabral BalreiraS. Elaydi and R. Luís, Global stability of higher dimensional monotone maps, J. Difference Equ. Appl., 23 (2017), 2037-2071.  doi: 10.1080/10236198.2017.1388375.

[7]

J. M. CushingS. LevargeN. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Equ. Appl., 10 (2004), 1139-1151.  doi: 10.1080/10236190410001652739.

[8]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, J. Math. Biol., 30 (1991), 161-168.  doi: 10.1007/BF00160333.

[9]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. Appl., 168 (1992), 385-400.  doi: 10.1016/0022-247X(92)90167-C.

[10]

M. GyllenbergJ. JiangL. Niu and P. Yan, On the classification of generalised competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discret. Contin. Dyn. Syst., 38 (2018), 615-650.  doi: 10.3934/dcds.2018027.

[11]

M. GyllenbergJ. Jiang and L. Niu, A note on global stability of three-dimensional Ricker models, J. Difference Equ. Appl., 25 (2019), 142-150.  doi: 10.1080/10236198.2019.1566459.

[12]

M. GyllenbergJ. JiangL. Niu and P. Yan, On the dynamics of multi-species Ricker models admitting a carrying simplex, J. Difference Equ. Appl., 25 (2019), 1489-1530.  doi: 10.1080/10236198.2019.1663182.

[13]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179.  doi: 10.1080/17513750801939236.

[14]

J. HofbauerV. Hutson and W. Jansen, Coexistence for systems generated by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570.  doi: 10.1007/BF00276199.

[15]

Z. Hou, Global attractor in autonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 127 (1999), 3633-3642.  doi: 10.1090/S0002-9939-99-05204-1.

[16]

Z. Hou, Global attractor in competitive Lotka-Volterra systems, Math. Nachr., 282 (2009), 995-1008.  doi: 10.1002/mana.200610785.

[17]

Z. Hou, Vanishing components in autonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 359 (2009), 302-310.  doi: 10.1016/j.jmaa.2009.05.054.

[18]

Z. Hou, Permanence, global attraction and stability, in Lotka-Volterra and Related Systems, Recent De Gruyter Ser. Math. Life Sci., 2, De Gruyter, Berlin, 2013, 1–62.

[19]

Z. Hou, Geometric method for global stability and repulsion in Kolmogorov systems, Dyn. Syst., 34 (2019), 456-483.  doi: 10.1080/14689367.2018.1554030.

[20]

J. JiangJ. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Difference Equations, 246 (2009), 1623-1672.  doi: 10.1016/j.jde.2008.10.008.

[21]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244.  doi: 10.3934/dcds.2016.36.217.

[22]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261.  doi: 10.1007/s00285-016-1052-y.

[23]

J. JiangL. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972.  doi: 10.1007/s00285-015-0920-1.

[24]

R. Kon, Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems, Proc. Amer. Math. Soc., 134 (2006), 3025-3034.  doi: 10.1090/S0002-9939-06-08309-2.

[25]

J. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976. doi: 10.1137/1.9781611970432.

[26]

F. Montes de Oca and M. L. Zeeman, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 192 (1995), 360-370.  doi: 10.1006/jmaa.1995.1177.

[27]

F. Montes de Oca and M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc, 124 (1996), 3677-3687.  doi: 10.1090/S0002-9939-96-03355-2.

[28] A. Pakes and R. Maller, Mathematical Ecology of Plant Species Competition. A Class of Deterministic Models for Binary Mixtures of Plant Genotypes, Cambridge Studies in Mathematical Biology, 10, Cambridge University Press, Cambridge, 1990.  doi: 10.1016/0020-7519(90)90100-2.
[29]

L. I. W. Roeger, Discrete May-Leonard competition models. II, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 841-860.  doi: 10.3934/dcdsb.2005.5.841.

[30]

L. I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models. I, J. Difference Equ. Appl., 10 (2004), 77-98.  doi: 10.1080/10236190310001603662.

[31]

A. Ruiz-Herrera, Topological criteria of global attraction with application in population dynamics, Nonlinearity, 25 (2012), 2823-2841.  doi: 10.1088/0951-7715/25/10/2823.

[32]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Difference Equ. Appl., 19 (2013), 96-113.  doi: 10.1080/10236198.2011.628663.

[33]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.

[34]

H. L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equations Appl., 3 (1998), 335-357.  doi: 10.1080/10236199708808108.

[35]

Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Differential Equations, 176 (2001), 470-493.  doi: 10.1006/jdeq.2001.3989.

[36]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632.  doi: 10.1016/S0022-0396(02)00025-6.

[37]

M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.  doi: 10.1090/S0002-9939-1995-1264833-2.

Figure 1.  A stability region for $ (\alpha, \beta) $ obtained by Theorems 3.3 and 3.5
Figure 2.  Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ for $ a_{ij} = 0 $, (a) $ a_{ji}>0 $, (b) $ a_{ji} = 0 $
Figure 3.  Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ when $ a_{ij}>0, a_{ji}>0 $ and $ [u, v]\cap\Gamma_j $ is on or below $ \Gamma_i $. (a) $ \Gamma_i = \Gamma_j $, (b) $ u_j\geq 0 $, (c) $ u_j = 0 $
Figure 4.  Configuration of $ h([u, v]\cap(\cup_{k\in \{i,j\}}(\Gamma_k\cup\Gamma^-_k))) $ when there is a fixed point $ x^* $ in the interior of $ [u, v] $
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