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May  2021, 26(5): 2721-2737. doi: 10.3934/dcdsb.2020202

Properties of basins of attraction for planar discrete cooperative maps

M. R. S. Kulenović , , J. Marcotte and  O. Merino

Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816, USA

* Corresponding author: M. R. S. Kulenović

Received  January 2020 Revised  March 2020 Published  May 2021 Early access  June 2020

It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction $ \mathcal{B} $ with relatively simple geometry: the boundary of each component of $ \mathcal{B} $ consists of the union of two unordered curves, and the components of $ \mathcal{B} $ are not comparable as sets. The boundary curves are Lipschitz if the map is of class $ C^1 $. Further, if a periodic point is in $ \partial \mathcal{B} $, then $ \partial\mathcal{B} $ is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

Keywords: Basin of attraction, competitive, cooperative, difference equation, invariant sets, stable manifold, unstable manifold.
Mathematics Subject Classification: Primary: 37C70, 37D05, 37D10; Secondary: 39A23, 39A30.
Citation: M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202
References:
[1]

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A. Brett and M. R. S. Kulenović, Basins of attraction of equilibrium points of monotone difference equations, Sarajevo J. Math., 5 (2009), 211-233.   Google Scholar

[3]

A. Cima, A. Gasull and V. Mañosa, Basin of attraction of triangular maps with applications, J. Difference Equ. Appl., 20 (2014), 423-437.  doi: 10.1080/10236198.2013.852187.  Google Scholar

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S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect III: Multispecies, J. Biol. Dyn., 12 (2018), 271-287.  doi: 10.1080/17513758.2018.1439537.  Google Scholar

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M. Garić-Demirović, M. R. S. Kulenović and M. Nurkanović, Basins of attraction of certain homogeneous second order quadratic fractional difference equation, J. Concr. Appl. Math., 13 (2015), 35-50.   Google Scholar

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J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.  Google Scholar

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M. W. Hirsch and H. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier B. V., Amsterdam, 2 (2005), 239-357.   Google Scholar

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M. W. Hirsch and H. Smith, Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398.  doi: 10.1080/10236190412331335445.  Google Scholar

[11]

G. L. Karakostas, The dynamics of a cooperative difference system with coefficient a Metzler matrix, J. Difference Equ. Appl., 20 (2014), 685-693.  doi: 10.1080/10236198.2013.799153.  Google Scholar

[12]

M. R. S. Kulenović and O. Merino, Global bifurcations for competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.  doi: 10.3934/dcdsb.2009.12.133.  Google Scholar

[13]

M. R. S. Kulenović and O. Merino, Invariant manifolds for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915.  doi: 10.1080/10236198.2018.1438418.  Google Scholar

[14]

M. R. S. Kulenović, O. Merino and M. Nurkanović, Global dynamics of certain competitive system in the plane, J. Difference Equ. Appl., 18 (2012), 1951-1966.  doi: 10.1080/10236198.2011.605357.  Google Scholar

[15]

M. R. S. Kulenović, S. Moranjkić and Z. Nurkanović, Global dynamics and bifurcation of a perturbed sigmoid Beverton-Holt difference equation, Math. Methods Appl. Sci., 39 (2016), 2696-2715.  doi: 10.1002/mma.3722.  Google Scholar

[16]

G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi and D. Ribble, Competition models with the Allee effect, J. Difference Equ. Appl., 20 (2015), 1127-1151.  doi: 10.1080/10236198.2014.897341.  Google Scholar

[17]

S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, Phys. D, 17 (1985), 125-153.  doi: 10.1016/0167-2789(85)90001-6.  Google Scholar

[18]

J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815. Google Scholar

[19]

H. E. Nusse and J. A. Yorke, Basins of attraction, Science, 271 (1996), 1376-1380.  doi: 10.1126/science.271.5254.1376.  Google Scholar

[20]

H. E. Nusse and J. A. Yorke, The structure of basins of attraction and their trapping regions, Ergodic Theory Dynam. Systems, 17 (1997), 463-481.  doi: 10.1017/S0143385797069782.  Google Scholar

[21]

H. E. Nusse and J. A. Yorke, Characterizing the basins with the most entangled boundaries, Ergodic Theory Dynam. Systems, 23 (2003), 895-906.  doi: 10.1017/S0143385702001360.  Google Scholar

[22]

H. E. Nusse and J. A. Yorke, Bifurcations of basins of attraction from the view point of prime ends, Topology Appl., 154 (2007), 2567-2579.  doi: 10.1016/j.topol.2006.07.019.  Google Scholar

[23]

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

show all references

References:
[1]

A. Berger and A. Duh, Global saddle-type dynamics for convex second-order difference equations, J. Difference Equ. Appl., 23 (2017), 1807-1823.  doi: 10.1080/10236198.2017.1367390.  Google Scholar

[2]

A. Brett and M. R. S. Kulenović, Basins of attraction of equilibrium points of monotone difference equations, Sarajevo J. Math., 5 (2009), 211-233.   Google Scholar

[3]

A. Cima, A. Gasull and V. Mañosa, Basin of attraction of triangular maps with applications, J. Difference Equ. Appl., 20 (2014), 423-437.  doi: 10.1080/10236198.2013.852187.  Google Scholar

[4]

S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect III: Multispecies, J. Biol. Dyn., 12 (2018), 271-287.  doi: 10.1080/17513758.2018.1439537.  Google Scholar

[5]

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

[6]

M. Garić-Demirović, M. R. S. Kulenović and M. Nurkanović, Basins of attraction of certain homogeneous second order quadratic fractional difference equation, J. Concr. Appl. Math., 13 (2015), 35-50.   Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.  Google Scholar

[8]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[9]

M. W. Hirsch and H. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier B. V., Amsterdam, 2 (2005), 239-357.   Google Scholar

[10]

M. W. Hirsch and H. Smith, Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398.  doi: 10.1080/10236190412331335445.  Google Scholar

[11]

G. L. Karakostas, The dynamics of a cooperative difference system with coefficient a Metzler matrix, J. Difference Equ. Appl., 20 (2014), 685-693.  doi: 10.1080/10236198.2013.799153.  Google Scholar

[12]

M. R. S. Kulenović and O. Merino, Global bifurcations for competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.  doi: 10.3934/dcdsb.2009.12.133.  Google Scholar

[13]

M. R. S. Kulenović and O. Merino, Invariant manifolds for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915.  doi: 10.1080/10236198.2018.1438418.  Google Scholar

[14]

M. R. S. Kulenović, O. Merino and M. Nurkanović, Global dynamics of certain competitive system in the plane, J. Difference Equ. Appl., 18 (2012), 1951-1966.  doi: 10.1080/10236198.2011.605357.  Google Scholar

[15]

M. R. S. Kulenović, S. Moranjkić and Z. Nurkanović, Global dynamics and bifurcation of a perturbed sigmoid Beverton-Holt difference equation, Math. Methods Appl. Sci., 39 (2016), 2696-2715.  doi: 10.1002/mma.3722.  Google Scholar

[16]

G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi and D. Ribble, Competition models with the Allee effect, J. Difference Equ. Appl., 20 (2015), 1127-1151.  doi: 10.1080/10236198.2014.897341.  Google Scholar

[17]

S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, Phys. D, 17 (1985), 125-153.  doi: 10.1016/0167-2789(85)90001-6.  Google Scholar

[18]

J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815. Google Scholar

[19]

H. E. Nusse and J. A. Yorke, Basins of attraction, Science, 271 (1996), 1376-1380.  doi: 10.1126/science.271.5254.1376.  Google Scholar

[20]

H. E. Nusse and J. A. Yorke, The structure of basins of attraction and their trapping regions, Ergodic Theory Dynam. Systems, 17 (1997), 463-481.  doi: 10.1017/S0143385797069782.  Google Scholar

[21]

H. E. Nusse and J. A. Yorke, Characterizing the basins with the most entangled boundaries, Ergodic Theory Dynam. Systems, 23 (2003), 895-906.  doi: 10.1017/S0143385702001360.  Google Scholar

[22]

H. E. Nusse and J. A. Yorke, Bifurcations of basins of attraction from the view point of prime ends, Topology Appl., 154 (2007), 2567-2579.  doi: 10.1016/j.topol.2006.07.019.  Google Scholar

[23]

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

Figure 1.  (a) The basin of attraction $ \mathcal{B} $ of the zero fixed point $ \mathrm o $ of the map $ T(x, y) = (y, x^3+y^3) $. Note that $ \mathcal{B} $ is unbounded, and $ \partial \mathcal{B} $ contains two fixed points $ \mathrm{p}_1 $ and $ \mathrm{p}_2 $ which are saddle points. The union of the stable manifolds of $ \mathrm{p}_1 $ and $ \mathrm{p}_2 $ gives $ \partial \mathcal{B} $. (b) The basin of attraction $ \mathcal{B} $ of the zero fixed point of the map $ T(x, y) = (y^3, x^3+y^3) $. The set $ \mathcal{B} $ is bounded, and $ \partial \mathcal{B} $ contains two fixed points $ \mathrm{p}_1 $ and $ \mathrm{p}_2 $ (saddles) and a repelling minimal period-two point $ {\mathrm q}_1 $ and $ {\mathrm q}_2 $. The union of the stable manifolds of $ \mathrm{p}_1 $ and $ \mathrm{p}_2 $ gives $ \partial \mathcal{B} $
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Figure 2.  (a) the basin $ \mathcal{B} $ of the fixed point $ \bar{\mathrm{x}} $ has three components $ \mathcal{B}^\prime $, $ \mathcal{B}_* $, $ \mathcal{B}^{\prime\prime} $ whose closure is in $ \mbox{int}(\mathcal{R}) $ and such that $ \mathcal{B}^\prime <\!\!<_{se}\mathcal{B}_* <\!\!<_{se}\mathcal{B}^{\prime\prime} $. Each component has boundary curves $ \mathcal{C}_+ $ and $ \mathcal{C}_- $. (b) The set $ \mathcal{B} $ has only one component, which has part of its boundary in $ \partial \mathcal{R} $. Also, $ \bar{\mathrm{x}} \in \partial \mathcal{R} $. The point $ \mathrm{p} $ is an endpoint of both boundary curves $ \mathcal{C}_- $ and $ \mathcal{C}_+ $. The point $ \mathrm{p} $ is a fixed point of $ T $.
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Figure 3.  Basin of attraction of the origin $ \mathrm{o} $ for the map $ U $ in (8)]Basin of attraction of the origin $ \mathrm{o} $ for the map $ U $ in (8). The points $ \mathrm{p} $ and $ \mathrm{q} $ are saddle fixed points.
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Figure 4.  Graphs of $ \phi $ from (12) and the identity function on the nonnegative semi axis.]Graphs of $ \phi $ from (12) and the identity function on the nonnegative semi axis. $ \phi $ has locally asymptotically stable fixed points $ 0 $, $ b = 2.06 $, and a repelling fixed point $ a = 0.95 $. The real numbers $ c = 6.03 $ and $ d = 12.80 $ are pre-images of $ a $. The basin of attraction of $ 0 $ on the semi-axis consists of the intervals $ 0\leq t < a $ and $ c< t < d $. All decimal numbers have been rounded to two decimals
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Figure 5.  The partial derivatives of $ V(x, y) $
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Figure 6.  (a) Three components of the basin of attraction of the zero fixed point $ \mathrm o $ of the map $ V(x, y) $ in Example 1. Here $ \mathrm r $, $ \mathrm s $ are saddle fixed points, $ {\mathrm p}_1 $ and $ {\mathrm q}_1 $ are a saddle period-two point, $ {\mathrm p}_2 $ and $ {\mathrm q}_2 $ are repelling fixed points, and $ {\mathrm p}_3 $, $ {\mathrm p}_4 $, $ {\mathrm q}_3 $, $ {\mathrm q}_4 $ are eventual period-two points. The boundary of the invariant part of the basin of attraction consist of stable manifolds of saddle fixed points with a period-two endpoints. In addition, there are two eventually period-two points which are end points of another piece of the basin of attraction which is mapped into the invariant part. (b) The invariant component of the basin of attraction of the origin $ \mathrm o $.
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Figure 7.  Global dynamics for map (13) as given in Proposition 1. Here $ \alpha = 0.4 $ and $ \delta = 0.7 $
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Figure 8.  The four cases in the definition of $ \mathcal{C}_\pm $
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