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doi: 10.3934/dcdsb.2021166
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

[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.   Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

[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.   Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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
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