May  2022, 27(5): 2917-2934. doi: 10.3934/dcdsb.2021166

Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition

1. 

Department of Economics, Social Sciences University of Ankara, Ulus-Ankara, Turkey

2. 

Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

* Corresponding author

Received  October 2020 Revised  May 2021 Published  May 2022 Early access  June 2021

Fund Project: This research of YK is partially funded by the NSF-DMS (Award Number 1716802); the NSFIOS/DMS (Award Number 1558127); DARPA-SBIR 2016.2 SB162-005; and the James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472)

The paper studies the pattern formation dynamics of a discrete in time and space model with nonlocal resource competition and dispersal. Our model is generalized from the metapopulation model proposed by Doebeli and Killingback [2003. Theor. Popul. Biol. 64, 397-416] in which competition for resources occurs only between neighboring populations. Our study uses symmetric discrete probability kernels to model nonlocal interaction and dispersal. A linear stability analysis of the model shows that solutions to this equation exhibits pattern formation when the dispersal rate is sufficiently small and the discrete interaction kernel satisfies certain conditions. Moreover, a weakly nonlinear analysis is used to approximate stationary patterns arising from the model. Numerical solutions to the model and the approximations obtained through the weakly nonlinear analysis are compared.

Citation: Ozgur Aydogmus, Yun Kang. Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2917-2934. doi: 10.3934/dcdsb.2021166
References:
[1]

L. J. Allen, Y. Lou and A. L. Nevai, Spatial patterns in a discrete-time SIS patch model, J. Math. Biol., 58, (2009), 339-375. doi: 10.1007/s00285-008-0194-y.

[2]

O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Modell. Nat. Phenom., 10 (2015), 17-29.  doi: 10.1051/mmnp/201510603.

[3]

O. Aydogmus, Discovering the effect of nonlocal payoff calculation on the stabilty of ess: Spatial patterns of hawk-dove game in metapopulations, J. Theor. Biol., 442 (2018), 87-97.  doi: 10.1016/j.jtbi.2018.01.016.

[4]

O. Aydogmus, Phase transitions in a logistic metapopulation model with nonlocal interactions, Bull. Math. Biol., 80 (2018), 228-253.  doi: 10.1007/s11538-017-0373-3.

[5]

O. AydogmusY. KangM. E. Kavgaci and H. Bereketoglu, Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complexity, 31 (2017), 88-95.  doi: 10.1016/j.ecocom.2017.04.001.

[6]

M. BeekmanD. Sumpter and F. Ratnieks, Phase transition between disordered and ordered foraging in pharaoh's ants, Proc. Natl. Acad. Sci. U.S.A, 98 (2015), 9703-9706.  doi: 10.1073/pnas.161285298.

[7]

N. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.

[8]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.

[9]

C. CobboldF. Lutscher and J. Sherratt, Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes, Ecol. Complexity, 24 (2015), 69-81.  doi: 10.1016/j.ecocom.2015.10.001.

[10]

N. B. Davies, Dunnock Behaviour and Social Evolution, 3, Oxford University Press, 1992.

[11]

M. Doebeli and T. Killingback, Metapopulation dynamics with quasi-local competition, Theor. Popul. Biol., 64 (2003), 397-416.  doi: 10.1016/S0040-5809(03)00106-0.

[12]

R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Popul. Biol., 46 (1994), 363-394.  doi: 10.1006/tpbi.1994.1032.

[13]

R. EftimieG. de Vries and M. Lewis, Weakly nonlinear analysis of a hyperbolic model for animal group formation, J. Math. Biol., 59 (2009), 37-74.  doi: 10.1007/s00285-008-0209-8.

[14]

M. Fuentes, M. Kuperman and V. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics, Phys. Rev. Lett., 91 (2003), 158104. doi: 10.1103/PhysRevLett.91.158104.

[15]

M. FuentesM. Kuperman and V. Kenkre, Analytical considerations in the study of spatial patterns arising from nonlocal interaction effects, J. Phys. Chem. B, 108 (2004), 10505-10508.  doi: 10.1021/jp040090k.

[16]

G. GambinoM. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion, Math. Comput. Simul., 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.

[17]

G. GambinoM. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in a 2d domain, Nonlinear Anal. Real World Appl., 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.

[18]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Modell. Nat. Phenom., 1 (2006), 63-80.  doi: 10.1051/mmnp:2006004.

[19] M. Gilpin and I. Hanski, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, 1997. 
[20]

M. GyllenbergG. Söderbacka and S. Ericsson, Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model, Math. Biosci., 118 (1993), 25-49.  doi: 10.1016/0025-5564(93)90032-6.

[21]

I. Hanski, A practical model of metapopulation dynamics, J. Anim. Ecol., (1994), 151-162. doi: 10.2307/5591.

[22]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101. doi: 10.1103/PhysRevLett.87.198101.

[23]

M. P. Hassell, N. H. Comins and R. M. May, Spatial structure and chaos in insect population dynamics, Nature 353, (1991) 255-258. doi: 10.1038/353255a0.

[24]

M. H. Holmes, Introduction to Perturbation Methods, 20, Springer Science & amp; Business Media, 2012. doi: 10.1007/978-1-4614-5477-9.

[25]

R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.  doi: 10.1007/BF02462004.

[26]

S. Levin, Dispersion and population interactions, Am. Nat., (1974), 207-228. doi: 10.1086/282900.

[27]

R. Levins, Extinction, in Some Mathematical Questions in Biology, American Mathematical Society, Providence, RI.

[28]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[29]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer International Publishing, 2019. doi: 10.1007/978-3-030-29294-2.

[30]

N. MadrasJ. Wu and X. Zou, Local-nonlocal interaction and spatial-temporal patterns in single species population over a patchy environment, Canad. Appl. Math. Q., 4 (1996), 109-133. 

[31]

M. Mandal and A. Asif, Continuous and Discrete Time Signals and Systems, Cambridge University Press, 2007.

[32]

Y. E. Maruvka and N. M. Shnerb, Nonlocal competition and logistic growth: Patterns, defects, and fronts, Phys. Rev. E, 73 (2006), 011903. doi: 10.1103/PhysRevE.73.011903.

[33]

J. Murray, Mathematical biology ii: Spatial models and biomedical applications, Springer, 2003.

[34]

M. NeubertM. Kot and M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Pop. Biol., 48 (1995), 7-43.  doi: 10.1006/tpbi.1995.1020.

[35]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, 14, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-4978-6.

[36]

L. A. D. RodriguesD. C. Mistro and S. Petrovskii, Pattern formation in a space-and time-discrete predator-prey system with a strong allee effect, Theor. Ecol., 5 (2012), 341-362.  doi: 10.1007/s11538-010-9593-5.

[37]

A. Sasaki, Clumped distribution by neighbourhood competition, J. Theor. Biol., 186 (1997), 415-430.  doi: 10.1006/jtbi.1996.0370.

[38]

J. Smith, Mathematics of the Discrete Fourier Transform (DFT): With Audio Applicaitons, W3K Publishing, 2007.

[39]

J. Stuart, On the non-linear mechanism of wave disturbances in stable and unstable parallel flows. part i, J. Fluid Mech., 9 (1960), 152-171.  doi: 10.1017/S002211206000116X.

[40]

C. TopazA. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[41]

A. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. London, Ser. B., 237 (1952), 37-72. 

show all references

References:
[1]

L. J. Allen, Y. Lou and A. L. Nevai, Spatial patterns in a discrete-time SIS patch model, J. Math. Biol., 58, (2009), 339-375. doi: 10.1007/s00285-008-0194-y.

[2]

O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Modell. Nat. Phenom., 10 (2015), 17-29.  doi: 10.1051/mmnp/201510603.

[3]

O. Aydogmus, Discovering the effect of nonlocal payoff calculation on the stabilty of ess: Spatial patterns of hawk-dove game in metapopulations, J. Theor. Biol., 442 (2018), 87-97.  doi: 10.1016/j.jtbi.2018.01.016.

[4]

O. Aydogmus, Phase transitions in a logistic metapopulation model with nonlocal interactions, Bull. Math. Biol., 80 (2018), 228-253.  doi: 10.1007/s11538-017-0373-3.

[5]

O. AydogmusY. KangM. E. Kavgaci and H. Bereketoglu, Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complexity, 31 (2017), 88-95.  doi: 10.1016/j.ecocom.2017.04.001.

[6]

M. BeekmanD. Sumpter and F. Ratnieks, Phase transition between disordered and ordered foraging in pharaoh's ants, Proc. Natl. Acad. Sci. U.S.A, 98 (2015), 9703-9706.  doi: 10.1073/pnas.161285298.

[7]

N. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.

[8]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.

[9]

C. CobboldF. Lutscher and J. Sherratt, Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes, Ecol. Complexity, 24 (2015), 69-81.  doi: 10.1016/j.ecocom.2015.10.001.

[10]

N. B. Davies, Dunnock Behaviour and Social Evolution, 3, Oxford University Press, 1992.

[11]

M. Doebeli and T. Killingback, Metapopulation dynamics with quasi-local competition, Theor. Popul. Biol., 64 (2003), 397-416.  doi: 10.1016/S0040-5809(03)00106-0.

[12]

R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Popul. Biol., 46 (1994), 363-394.  doi: 10.1006/tpbi.1994.1032.

[13]

R. EftimieG. de Vries and M. Lewis, Weakly nonlinear analysis of a hyperbolic model for animal group formation, J. Math. Biol., 59 (2009), 37-74.  doi: 10.1007/s00285-008-0209-8.

[14]

M. Fuentes, M. Kuperman and V. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics, Phys. Rev. Lett., 91 (2003), 158104. doi: 10.1103/PhysRevLett.91.158104.

[15]

M. FuentesM. Kuperman and V. Kenkre, Analytical considerations in the study of spatial patterns arising from nonlocal interaction effects, J. Phys. Chem. B, 108 (2004), 10505-10508.  doi: 10.1021/jp040090k.

[16]

G. GambinoM. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion, Math. Comput. Simul., 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.

[17]

G. GambinoM. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in a 2d domain, Nonlinear Anal. Real World Appl., 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.

[18]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Modell. Nat. Phenom., 1 (2006), 63-80.  doi: 10.1051/mmnp:2006004.

[19] M. Gilpin and I. Hanski, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, 1997. 
[20]

M. GyllenbergG. Söderbacka and S. Ericsson, Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model, Math. Biosci., 118 (1993), 25-49.  doi: 10.1016/0025-5564(93)90032-6.

[21]

I. Hanski, A practical model of metapopulation dynamics, J. Anim. Ecol., (1994), 151-162. doi: 10.2307/5591.

[22]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101. doi: 10.1103/PhysRevLett.87.198101.

[23]

M. P. Hassell, N. H. Comins and R. M. May, Spatial structure and chaos in insect population dynamics, Nature 353, (1991) 255-258. doi: 10.1038/353255a0.

[24]

M. H. Holmes, Introduction to Perturbation Methods, 20, Springer Science & amp; Business Media, 2012. doi: 10.1007/978-1-4614-5477-9.

[25]

R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.  doi: 10.1007/BF02462004.

[26]

S. Levin, Dispersion and population interactions, Am. Nat., (1974), 207-228. doi: 10.1086/282900.

[27]

R. Levins, Extinction, in Some Mathematical Questions in Biology, American Mathematical Society, Providence, RI.

[28]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[29]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer International Publishing, 2019. doi: 10.1007/978-3-030-29294-2.

[30]

N. MadrasJ. Wu and X. Zou, Local-nonlocal interaction and spatial-temporal patterns in single species population over a patchy environment, Canad. Appl. Math. Q., 4 (1996), 109-133. 

[31]

M. Mandal and A. Asif, Continuous and Discrete Time Signals and Systems, Cambridge University Press, 2007.

[32]

Y. E. Maruvka and N. M. Shnerb, Nonlocal competition and logistic growth: Patterns, defects, and fronts, Phys. Rev. E, 73 (2006), 011903. doi: 10.1103/PhysRevE.73.011903.

[33]

J. Murray, Mathematical biology ii: Spatial models and biomedical applications, Springer, 2003.

[34]

M. NeubertM. Kot and M. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Pop. Biol., 48 (1995), 7-43.  doi: 10.1006/tpbi.1995.1020.

[35]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, 14, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-4978-6.

[36]

L. A. D. RodriguesD. C. Mistro and S. Petrovskii, Pattern formation in a space-and time-discrete predator-prey system with a strong allee effect, Theor. Ecol., 5 (2012), 341-362.  doi: 10.1007/s11538-010-9593-5.

[37]

A. Sasaki, Clumped distribution by neighbourhood competition, J. Theor. Biol., 186 (1997), 415-430.  doi: 10.1006/jtbi.1996.0370.

[38]

J. Smith, Mathematics of the Discrete Fourier Transform (DFT): With Audio Applicaitons, W3K Publishing, 2007.

[39]

J. Stuart, On the non-linear mechanism of wave disturbances in stable and unstable parallel flows. part i, J. Fluid Mech., 9 (1960), 152-171.  doi: 10.1017/S002211206000116X.

[40]

C. TopazA. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[41]

A. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. London, Ser. B., 237 (1952), 37-72. 

Figure 1.  The patchy environment $ S $ and neighborhoods of a patch $ q = (q_1, q_2). $ In Panels (a) and (b), von Neumann and Moore neighborhoods of patch $ (q_1, q_2) $ (colored in red) are determined by patches colored in gray, respectively
Figure 2.  Dispersion relations for the examples E1-4
Figure 3.  Comparison between numerical solutions to the CML (on the left) for examples E1 and E3, and the weakly nonlinear first-order approximations of these solutions (on the right). Panel (a) illustrates stationary waves in a 1-dimensional habitat for E1. Similarly, panel (c) shows stationary patterns in a 2-dimensional habitat for E3. In panels (c) and (d), colors represent the population size and approximated population size, respectively
Figure 4.  Comparison between numerical solutions to CML (on the left) and the weakly nonlinear first order approximation of these solutions (on the right). Panel (a) illustrates stationary waves in a 1-dimensional habitat for the parameters given in example E2
Figure 5.  Comparison between numerical solution to CML (on the left) and the weakly nonlinear first order approximation of this solutions (on the right). Panel (a) illustrates stationary waves in a 2-dimensional habitat for E4. In panels (a) and (b), colors represent the population size and approximated population size, respectively
[1]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[2]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087

[3]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[4]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036

[5]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339

[6]

Xavier Blanc, Claude Le Bris, Frédéric Legoll, Tony Lelièvre. Beyond multiscale and multiphysics: Multimaths for model coupling. Networks and Heterogeneous Media, 2010, 5 (3) : 423-460. doi: 10.3934/nhm.2010.5.423

[7]

Suman Ganguli, David Gammack, Denise E. Kirschner. A Metapopulation Model Of Granuloma Formation In The Lung During Infection With Mycobacterium Tuberculosis. Mathematical Biosciences & Engineering, 2005, 2 (3) : 535-560. doi: 10.3934/mbe.2005.2.535

[8]

Irada Dzhalladova, Miroslava Růžičková. Simplification of weakly nonlinear systems and analysis of cardiac activity using them. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3435-3453. doi: 10.3934/dcdsb.2021191

[9]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021

[10]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

[11]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[12]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

[13]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

[14]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609

[15]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

[16]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010

[17]

Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103

[18]

Marco Scianna, Luca Munaron. Multiscale model of tumor-derived capillary-like network formation. Networks and Heterogeneous Media, 2011, 6 (4) : 597-624. doi: 10.3934/nhm.2011.6.597

[19]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111

[20]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (390)
  • HTML views (465)
  • Cited by (0)

Other articles
by authors

[Back to Top]