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On the steady-states of a two-species non-local cross-diffusion model

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.
Wydawca
Rocznik
Strony
1--19
Opis fizyczny
Bibliogr. 24 poz., wykr.
Twórcy
  • Department of Applied Mathematics, University of Colorado Boulder, Engineering Center, ECOT 225, 526 UCB, Boulder, CO 80309-0526, USA
autor
  • Department of Electrical and Computer Engineering, Carnegie Mellon University Silicon Valley, 870 E El Camino Real, APT 171, Mountain View, CA 94040, USA
Bibliografia
  • [1] A. W. Bateman, M. A. Lewis, G. Gall, M. B. Manser and T. H. Clutton-Brock, Territoriality and home-range dynamics in meerkats, Suricata suricatta: A mechanistic modelling approach, J. Animal Ecol. 84 (2015), no. 1, 260-271.
  • [2] J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett. 24 (2011), no. 11, 1927-1932.
  • [3] J. Bedrossian, N. Rodríguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity 24 (2011), no. 6, 1683-1714.
  • [4] A. L. Bertozzi and D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal. 9 (2010), no. 6, 1617-1637.
  • [5] M. Burger, M. di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci. 11 (2013), no. 3, 709-738.
  • [6] S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol. 16 (1983), no. 2, 181-198.
  • [7] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys. 17 (2015), no. 1, 233-258.
  • [8] N. S. Clayton, D. P. Griffiths, N. J. Emery and A. Dickinson, Elements of episodic-like memory in animals, Philos. Trans. Roy. Soc. B 356(1413) (2001), 1483-1491.
  • [9] J. M. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science, Santa Fe Inst. Stud. Sci. Complex. Lecture Notes IV, Addison-Wesley, Reading, 1997.
  • [10] M. E. Gurtin and A. C. Pipkin, A note on interacting populations that disperse to avoid crowding, Quart. Appl. Math. 42 (1984), no. 1, 87-94.
  • [11] G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal. 49 (2017), no. 1, 272-296.
  • [12] I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal. 44 (2012), no. 2, 568-602.
  • [13] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349-374.
  • [14] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 45-121.
  • [15] P. R. Moorcoft, M. A. Lewis and R. L. Crabree, Home range analysis using a mechanistic home range model, Ecology 80 (1999), no. 5, 1656-1665.
  • [16] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol. 50 (2005), no. 1, 49-66.
  • [17] M. Morisita, Habitat preference and evaluation of environment of an animal. Experimental studies on the population density of an ant-lion, Glenuroides japonicus M’L, Phisol. Ecolg. Japan 1 (1952).
  • [18] M. Morisita, Measuring of habitat value by the “environmental density” method, Spatial Ecology 1 (1971), 379-401.
  • [19] N. Rodríguez and L. Ryzhik, Exploring the effects of social preference, economic disparity, and heterogeneous environments on segregation, Commun. Math. Sci. 14 (2016), no. 2, 363-387.
  • [20] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83-99.
  • [21] K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, Int. J. Mod. Phys. C 11 (2000), 1157-1165.
  • [22] V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems, Phys. Chem. Chemical Physics 11 (2009), no. 6, 897-912.
  • [23] K. White, M. Lewis and J. Murray, A model for wolf-pack territory formation and maintenance, J. Theoret. Biolo. 178 (1996), 29-43.
  • [24] K. White, J. Murray and M. Lewis, Wolf-Deer interactions: A mathematical model, Proc. Biolog. Sci. 263 (1996), no. 1368, 299-305.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-4cc17dd3-483a-4173-b612-a3db72988995
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