Spatial segregation limit of traveling wave solutions for a fully nonlinear strongly coupled competitive system

Léo Girardin; Danielle Hilhorst; Léo Girardin; Danielle Hilhorst

doi:10.3934/era.2022088

Electronic Research Archive

2022, Volume 30, Issue 5: 1748-1773. doi: 10.3934/era.2022088

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Spatial segregation limit of traveling wave solutions for a fully nonlinear strongly coupled competitive system

Léo Girardin ^{1
,
,},
Danielle Hilhorst ²

1.
Institut Camille Jordan, Université Claude Bernard Lyon-1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
2.
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay Cedex, France

Received: 10 September 2021 Revised: 14 December 2021 Accepted: 22 December 2021 Published: 29 March 2022

The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.
- Lotka–Volterra,
- traveling wave,
- bistability,
- nonlinear diffusion,
- nonlinear advection
Citation: Léo Girardin, Danielle Hilhorst. Spatial segregation limit of traveling wave solutions for a fully nonlinear strongly coupled competitive system[J]. Electronic Research Archive, 2022, 30(5): 1748-1773. doi: 10.3934/era.2022088

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Abstract

The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.

References

[1]	J. R. Potts, S. V. Petrovskii, Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations, J. Theor. Biol., 420 (2017), 190–199. https://doi.org/10.1016/j.jtbi.2017.03.011 doi: 10.1016/j.jtbi.2017.03.011
[2]	N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83–99. https://doi.org/10.1016/0022-5193(79)90258-3 doi: 10.1016/0022-5193(79)90258-3
[3]	L. Desvillettes, T. Lepoutre, A. Moussa, A. Trescases, On the entropic structure of reaction-cross diffusion systems, Comm. Partial Differ. Equ., 40 (2015), 1705–1747. https://doi.org/10.1080/03605302.2014.998837 doi: 10.1080/03605302.2014.998837
[4]	L. Girardin, G. Nadin, Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed, Eur. J. Appl. Math., 26 (2015), 521–534. https://doi.org/10.1017/S0956792515000170 doi: 10.1017/S0956792515000170
[5]	L. Malaguti, C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms, Math. Nachr., 242 (2002), 148–164. https://doi.org/10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J
[6]	R. A. Gardner, Existence and stability of travelling wave solutions of competition models: a degree theoretic approach, J. Differ. Equ., 44 (1982), 343–364. https://doi.org/10.1016/0022-0396(82)90001-8 doi: 10.1016/0022-0396(82)90001-8
[7]	Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340–363. https://doi.org/10.1137/S0036141093244556 doi: 10.1137/S0036141093244556
[8]	L. Girardin, G. Nadin, Competition in periodic media: II – Segregative limit of pulsating fronts and "Unity is not Strength"-type result, J. Differ. Equ., 265 (2018), 98–156. https://doi.org/10.1016/j.jde.2018.02.026 doi: 10.1016/j.jde.2018.02.026
[9]	D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, 2001.
[10]	L. Girardin, The effect of random dispersal on competitive exclusion – A review, Math. Biosci., 318 (2019), 108271. https://doi.org/10.1016/j.mbs.2019.108271 doi: 10.1016/j.mbs.2019.108271
[11]	M. Conti, S. Terracini, G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524–560. https://doi.org/10.1016/j.aim.2004.08.006 doi: 10.1016/j.aim.2004.08.006
[12]	E. C. M. Crooks, E. N. Dancer, D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1–36.
[13]	E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura, H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Anal. Real World Appl., 5 (2004), 645–665. https://doi.org/10.1016/j.nonrwa.2004.01.004 doi: 10.1016/j.nonrwa.2004.01.004
[14]	E. N. Dancer, Y. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differ. Equ., 114 1994,434–475.
[15]	E. N. Dancer, D. Hilhorst, M. Mimura, L. A. Peletier, Spatial segregation limit of a competition-diffusion system, Eur. J. Appl. Math., 10 (1999), 97–115. https://doi.org/10.1017/S0956792598003660 doi: 10.1017/S0956792598003660
[16]	E. N. Dancer, K. Wang, Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961–1005. https://doi.org/10.1090/S0002-9947-2011-05488-7 doi: 10.1090/S0002-9947-2011-05488-7
[17]	N. Soave, A. Zilio, Uniform bounds for strongly competing systems: the optimal Lipschitz case, Arch. Ration. Mech. Anal., 218 (2015), 647–697. https://doi.org/10.1007/s00205-015-0867-9 doi: 10.1007/s00205-015-0867-9
[18]	S. Zhang, L. Zhou, Z. Liu, Uniqueness and least energy property for solutions to a strongly coupled elliptic system, Acta Math. Sin. (Engl. Ser.), 33 (2017), 419–438. https://doi.org/10.1007/s10114-016-5686-x doi: 10.1007/s10114-016-5686-x
[19]	L. Zhou, S. Zhang, Z. Liu, Uniform Hölder bounds for a strongly coupled elliptic system with strong competition, Nonlinear Anal., 75 (2012), 6120–6129. https://doi.org/10.1016/j.na.2012.06.017 doi: 10.1016/j.na.2012.06.017
[20]	L. Zhou, S. Zhang, Z. Liu, Z. Lin, The spatial behavior of a strongly coupled non-autonomous elliptic system, Nonlinear Anal., 75 (2012), 3099–3106. https://doi.org/10.1016/j.na.2011.12.008 doi: 10.1016/j.na.2011.12.008
[21]	M. Dreher, Analysis of a population model with strong cross-diffusion in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 13875 (2008), 769–786. https://doi.org/10.1017/S0308210506001259 doi: 10.1017/S0308210506001259
[22]	Y. Wu, Y. Zhao, The existence and stability of traveling waves with transition layers for the {S}-{K}-{T} competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161–1184. https://doi.org/10.1007/s11425-010-0141-4 doi: 10.1007/s11425-010-0141-4
[23]	Q. Wang, C. Gai, J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239–1284. https://doi.org/10.3934/dcds.2015.35.1239 doi: 10.3934/dcds.2015.35.1239
[24]	Q. Wang, J. Yang, F. Yu, Global well-posedness of advective lotka–volterra competition systems with nonlinear diffusion, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, (2019), page 1–27. https://doi.org/10.1017/prm.2019.10
[25]	Q. Wang, L. Zhang, On the multi-dimensional advective Lotka-Volterra competition systems, Nonlinear Anal. Real World Appl., 37 (2017), 329–349. https://doi.org/10.1016/j.nonrwa.2017.02.011 doi: 10.1016/j.nonrwa.2017.02.011
[26]	Y. Zhang, Global solutions and uniform boundedness of attractive/repulsive LV competition systems, Adv. Differ. Equ., 8 (2018), pages Paper No. 52. https://doi.org/10.1186/s13662-018-1513-2
[27]	A. Kubo, J. I. Tello, Mathematical analysis of a model of chemotaxis with competition terms, Differ. Integral Equ., 29 (2016), 441–454.
[28]	A. L. Krause, R. A. Van Gorder, A non-local cross-diffusion model of population dynamics II: Exact, approximate, and numerical traveling waves in single- and multi-species populations, Bull. Math. Biol., 82 (2020), 113.
[29]	L. Bao, Z. Zhou, Traveling wave in backward and forward parabolic equations from population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1507–1522. https://doi.org/10.3934/dcdsb.2014.19.1507 doi: 10.3934/dcdsb.2014.19.1507
[30]	L. Ferracuti, C. Marcelli, F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Syst. Appl., 4 (2009), 19–33.
[31]	M. Kuzmin, S. Ruggerini, Front propagation in diffusion-aggregation models with bi-stable reaction, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 819–833. https://doi.org/10.3934/dcdsb.2011.16.819 doi: 10.3934/dcdsb.2011.16.819
[32]	P. K. Maini, L. Malaguti, C. Marcelli, S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1175–1189.
[33]	L. Malaguti, C. Marcelli, S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differ. Equ., 9 (2004), 1143–1166.
[34]	L. Malaguti, C. Marcelli, S. Matucci, Continuous dependence in front propagation of convective reaction-diffusion equations, Commun. Pure Appl. Anal., 9 (2010), 1083–1098. https://doi.org/10.3934/cpaa.2010.9.1083 doi: 10.3934/cpaa.2010.9.1083
[35]	C. Marcelli, F. Papalini, A new estimate of the minimal wave speed for travelling fronts in reaction-diffusion-convection equations, Electron. J. Qual. Theor. Differ. Equ., 19 (2018), pages Paper No. 10. https://doi.org/10.14232/ejqtde.2018.1.10
[36]	F. Sánchez-Garduño, P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163–192.
[37]	F. Sánchez-Garduño, P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differ. Equ., 117 (1995), 281–319
[38]	A. Ducrot, T. Giletti, H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), 5541–5566. https://doi.org/10.1090/S0002-9947-2014-06105-9 doi: 10.1090/S0002-9947-2014-06105-9
[39]	P. C. Fife, J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. https://doi.org/10.1007/BF00250432 doi: 10.1007/BF00250432
[40]	B. Sandstede, Stability of travelling waves, In Handbook of dynamical systems, Vol. 2, pages 983–1055. North-Holland, Amsterdam, 2002. https://doi.org/10.1016/S1874-575X(02)80039-X
[41]	V. Volpert, Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion Equations, volume 104. Springer, 2014.
[42]	L. Girardin, A. Zilio, Competition in periodic media: III–Existence and stability of segregated periodic coexistence states, J. Dyn. Differ. Equ., 32 (2020), 257–279. https://doi.org/10.1007/s10884-019-09732-7 doi: 10.1007/s10884-019-09732-7
[43]	J. Fang, X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc. (JEMS), 17 (2015), 2243–2288. https://doi.org/10.4171/JEMS/556 doi: 10.4171/JEMS/556
[44]	S. Cantrell, C. Cosner, S. Ruan, Spatial ecology. CRC Press, 2009.
[45]	J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61–83. https://doi.org/10.1007/s002850050120 doi: 10.1007/s002850050120

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