On the steady-states of a two-species non-local cross-diffusion model
-
Nancy Rodríguez
and
Yi Hu
Abstract
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.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1516778
Funding statement: This work was partially supported by NSF DMS-1516778.
A Appendix
We provide the proof of the scaling lemma for
Lemma 7 (Scaling lemma).
Let (ii) in Theorem 2 hold. Then, for all
Proof.
Let
where the sequence
Taking into account that
one obtains for all
Therefore, if we choose a sequence such that
Therefore, there exists a sufficiently small λ such that
for R sufficiently large
References
[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. 10.1111/1365-2656.12267Search in Google Scholar PubMed
[2] J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett. 24 (2011), no. 11, 1927–1932. 10.1016/j.aml.2011.05.022Search in Google Scholar
[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. 10.1088/0951-7715/24/6/001Search in Google Scholar
[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. 10.3934/cpaa.2010.9.1617Search in Google Scholar
[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. 10.4310/CMS.2013.v11.n3.a3Search in Google Scholar
[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. 10.1007/BF00276056Search in Google Scholar PubMed
[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. 10.4208/cicp.160214.010814aSearch in Google Scholar
[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. 10.1093/acprof:oso/9780198508809.003.0013Search in Google Scholar
[9] J. M. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science, Santa Fe Inst. Stud. Sci. Complex. Lecture Notes IV, Addison-Wesley, Reading, 1997. Search in Google Scholar
[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. 10.1090/qam/736508Search in Google Scholar
[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. 10.1137/16M1072450Search in Google Scholar
[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. 10.1137/110823584Search in Google Scholar
[13] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. 10.1007/978-3-642-55925-9_43Search in Google Scholar
[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. 10.4171/RMI/12Search in Google Scholar
[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. 10.1890/0012-9658(1999)080[1656:HRAUAM]2.0.CO;2Search in Google Scholar
[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. 10.1007/s00285-004-0279-1Search in Google Scholar
[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). Search in Google Scholar
[18] M. Morisita, Measuring of habitat value by the “environmental density” method, Spatial Ecology 1 (1971), 379–401. Search in Google Scholar
[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. 10.4310/CMS.2016.v14.n2.a3Search in Google Scholar
[20] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83–99. 10.1016/0022-5193(79)90258-3Search in Google Scholar
[21] K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, Int. J. Mod. Phys. C 11 (2000), 1157–1165. 10.1142/S0129183100000936Search in Google Scholar
[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. 10.1039/B813825GSearch in Google Scholar
[23] K. White, M. Lewis and J. Murray, A model for wolf-pack territory formation and maintenance, J. Theoret. Biolo. 178 (1996), 29–43. 10.1006/jtbi.1996.0004Search in Google Scholar
[24] K. White, J. Murray and M. Lewis, Wolf–Deer interactions: A mathematical model, Proc. Biolog. Sci. 263 (1996), no. 1368, 299–305. 10.1098/rspb.1996.0046Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the steady-states of a two-species non-local cross-diffusion model
- Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations
-
Existence of positive solutions of a new class of nonlocal
- A function fitting method
-
Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in
- Some properties of modified Szász–Mirakyan operators in polynomial spaces via the power summability method
- Some sufficient conditions for univalence and close-to-convexity
- Approximation by Stancu–Chlodowsky type λ-Bernstein operators
- Radius problems for univalent functions
- Convergence theorem for a finite family of asymptotically demicontractive multi-valued mappings in CAT(0) spaces
- Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets
- Delaunay surfaces expressed in terms of a Cartan moving frame
Articles in the same Issue
- Frontmatter
- On the steady-states of a two-species non-local cross-diffusion model
- Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations
-
Existence of positive solutions of a new class of nonlocal
- A function fitting method
-
Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in
- Some properties of modified Szász–Mirakyan operators in polynomial spaces via the power summability method
- Some sufficient conditions for univalence and close-to-convexity
- Approximation by Stancu–Chlodowsky type λ-Bernstein operators
- Radius problems for univalent functions
- Convergence theorem for a finite family of asymptotically demicontractive multi-valued mappings in CAT(0) spaces
- Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets
- Delaunay surfaces expressed in terms of a Cartan moving frame
