×

zbMATH — the first resource for mathematics

A non-local cross-diffusion model of population dynamics II: exact, approximate, and numerical traveling waves in single- and multi-species populations. (English) Zbl 1448.92212
Summary: We study traveling waves in a non-local cross-diffusion-type model, where organisms move along gradients in population densities. Such models are valuable for understanding waves of migration and invasion and how directed motion can impact such scenarios. In this paper, we demonstrate the emergence of traveling wave solutions, studying properties of both planar and radial wave fronts in one- and two-species variants of the model. We compute exact traveling wave solutions in the purely diffusive case and then perturb these solutions to analytically capture the influence directed motion has on these exact solutions. Using linear stability analysis, we find that the minimum wavespeeds correspond to the purely diffusive case, but numerical simulations suggest that advection can in general increase or decrease the observed wavespeed substantially, which allows a single species to more rapidly move into unoccupied resource-rich spatial regions or modify the speed of an invasion for two populations. We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally, we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wavespeed but also on the existence and structure of emergent patterns, as predicted in the first part of our study [N. P. Taylor et al., Bull. Math. Biol. 82, No. 8, Paper No. 112, 40 p. (2020; Zbl 1448.92253)].
MSC:
92D25 Population dynamics (general)
35C07 Traveling wave solutions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ablowitz, MJ; Zeppetella, A., Explicit solutions of Fisher’s equation for a special wave speed, Bull Math Biol, 41, 6, 835-840 (1979) · Zbl 0423.35079
[2] Alhasanat, A.; Ou, C., Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J Diff Equ, 266, 11, 7357-7378 (2019) · Zbl 1408.35067
[3] Al-Kiffai, A.; Crooks, E., Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems, Tamkang J Math, 47, 1, 51-70 (2016) · Zbl 1355.35102
[4] Aronson, DG; Weinberger, HF, Multidimensional nonlinear diffusion arising in population genetics, Adv Math, 30, 1, 33-76 (1978) · Zbl 0407.92014
[5] Ben-Jacob, E.; Brand, H.; Dee, G.; Kramer, L.; Langer, JS, Pattern propagation in nonlinear dissipative systems, Phys D Nonlinear Phenom, 14, 3, 348-364 (1985) · Zbl 0622.76051
[6] Berestycki H (2002) The influence of advection on the propagation of fronts in reaction-diffusion equations. In: Nonlinear PDE’s in condensed matter and reactive flows, Springer, pp 11-48 · Zbl 1073.35113
[7] Berestycki, H.; Hamel, F., Generalized travelling waves for reaction-diffusion equations, Contemp Math, 446, 101-124 (2007) · Zbl 1200.35169
[8] Bertsch, M.; Gurtin, ME; Hilhorst, D.; Peletier, L., On interacting populations that disperse to avoid crowding: preservation of segregation, J Math Biol, 23, 1, 1-13 (1985) · Zbl 0596.35074
[9] Castillo-Chavez, C.; Li, B.; Wang, H., Some recent developments on linear determinacy, Math Biosci Eng, 10, 5-6, 1419-1436 (2013) · Zbl 1273.92045
[10] del Castillo-Negrete, D.; Carreras, B.; Lynch, V., Front propagation and segregation in a reaction-diffusion model with cross-diffusion, Phys D Nonlinear Phenom, 168, 45-60 (2002) · Zbl 1022.35017
[11] Chen, X.; Hambrock, R.; Lou, Y., Evolution of conditional dispersal: a reaction-diffusion-advection model, J Math Biol, 57, 3, 361-386 (2008) · Zbl 1141.92040
[12] Dee, G.; Langer, JS, Propagating pattern selection, Phys Rev Lett, 50, 6, 383 (1983)
[13] Dee, GT; van Saarloos, W., Bistable systems with propagating fronts leading to pattern formation, Phys Rev Lett, 60, 25, 2641 (1988)
[14] Dormand, JR; Prince, PJ, A family of embedded Runge-Kutta formulae, J Comput Appl Math, 6, 1, 19-26 (1980) · Zbl 0448.65045
[15] Dunbar, SR, Travelling wave solutions of diffusive Lotka-Volterra equations, J Math Biol, 17, 1, 11-32 (1983) · Zbl 0509.92024
[16] Dunbar, SR, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in \({\mathbb{R}}^4\), Trans Am Math Soc, 286, 2, 557-594 (1984) · Zbl 0556.35078
[17] Fisher, RA, The wave of advance of advantageous genes, Ann Eugenics, 7, 4, 355-369 (1937)
[18] Gambino, G.; Lombardo, MC; Sammartino, M., Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion, Math Comput Simul, 82, 6, 1112-1132 (2012) · Zbl 1320.35170
[19] Girardin, L.; Nadin, G., Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed, Eur J Appl Math, 26, 4, 521-534 (2015) · Zbl 1375.92049
[20] Girardin, L., Non-cooperative Fisher-KPP systems: asymptotic behavior of traveling waves, Math Models Methods Appl Sci, 28, 6, 1067-1104 (2018) · Zbl 06894968
[21] Girardin, L.; Lam, KY, Invasion of open space by two competitors: spreading properties of monostable two-species competition-diffusion systems, Proc Lond Math Soc, 119, 5, 1279-1335 (2019) · Zbl 1428.35158
[22] Goriely, A., Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J Math Phys, 37, 4, 1871-1893 (1996) · Zbl 0867.58036
[23] Grindrod, P., Models of individual aggregation or clustering in single and multi-species communities, J Math Biol, 26, 6, 651-660 (1988) · Zbl 0714.92024
[24] Grindrod, P., Patterns and waves: the theory and applications of reaction-diffusion equations (1991), USA: Oxford University Press, USA · Zbl 0743.35032
[25] Hambrock, R.; Lou, Y., The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull Math Biol, 71, 8, 1793 (2009) · Zbl 1179.92060
[26] Hearns, J.; Van Gorder, RA; Choudhury, SR, Painlevé test, integrability, and exact solutions for density-dependent reaction-diffusion equations with polynomial reaction functions, Appl Math Comput, 219, 6, 3055-3064 (2012) · Zbl 1309.35040
[27] Hillen, T.; Painter, KJ, A user’s guide to PDE models for chemotaxis, J Math Biol, 58, 1-2, 183 (2009) · Zbl 1161.92003
[28] Horstmann, D., Remarks on some Lotka-Volterra type cross-diffusion models, Nonlinear Anal Real World Appl, 8, 1, 90-117 (2007) · Zbl 1134.35058
[29] Hosono, Y., The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bull Math Biol, 60, 3, 435-448 (1998) · Zbl 1053.92519
[30] Huang, W.; Han, M., Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J Diff Equ, 251, 6, 1549-1561 (2011) · Zbl 1263.92047
[31] Gurtin, ME; MacCamy, RC, On the diffusion of biological populations, Math Biosci, 33, 1-2, 35-49 (1977) · Zbl 0362.92007
[32] Ibrahim, H.; Nasreddine, E., Traveling waves for a model of individual clustering with logistic growth rate, J Math Phys, 58, 8, 081505 (2017) · Zbl 1370.92136
[33] Jensen, O.; Pannbacker, VO; Mosekilde, E.; Dewel, G.; Borckmans, P., Localized structures and front propagation in the Lengyel-Epstein model, Phys Rev E, 50, 2, 736 (1994)
[34] Jones, CK, Spherically symmetric solutions of a reaction-diffusion equation, J Diff Equ, 49, 1, 142-169 (1983) · Zbl 0523.35059
[35] Kareiva, P.; Odell, G., Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Am Nat, 130, 2, 233-270 (1987)
[36] Keener, JP, A geometrical theory for spiral waves in excitable media, SIAM J Appl Math, 46, 6, 1039-1056 (1986) · Zbl 0655.35046
[37] Keener, JP, An eikonal-curvature equation for action potential propagation in myocardium, J Math Biol, 29, 7, 629-651 (1991) · Zbl 0744.92015
[38] Kiselev A, Ryzhik L (2001) Enhancement of the traveling front speeds in reaction-diffusion equations with advection. Annales de l’Institut Henri Poincaré (C) Non Linear Anal 18(3):309-358 · Zbl 1002.35069
[39] Kurowski, L.; Krause, AL; Mizuguchi, H.; Grindrod, P.; Van Gorder, RA, Two-species migration and clustering in two-dimensional domains, Bull Math Biol, 79, 10, 2302-2333 (2017) · Zbl 1378.92058
[40] Kuznetsov, YA; Antonovsky, MY; Biktashev, V.; Aponina, E., A cross-diffusion model of forest boundary dynamics, J Math Biol, 32, 3, 219-232 (1994) · Zbl 0790.92028
[41] Lewis, MA; Petrovskii, SV; Potts, JR, The mathematics behind biological invasions (2016), Berlin: Springer, Berlin · Zbl 1338.92001
[42] Lewis, MA; Li, B.; Weinberger, HF, Spreading speed and linear determinacy for two-species competition models, J Math Biol, 45, 3, 219-233 (2002) · Zbl 1032.92031
[43] Li, B.; Weinberger, HF; Lewis, MA, Spreading speeds as slowest wave speeds for cooperative systems, Math Biosci, 196, 1, 82-98 (2005) · Zbl 1075.92043
[44] Liebhold, AM; Tobin, PC, Population ecology of insect invasions and their management, Annu Rev Entomol, 53, 387-408 (2008)
[45] Lockwood, JL; Hoopes, MF; Marchetti, MP, Invasion ecology (2013), Hoboken: Wiley, Hoboken
[46] Miller, PD, Nonmonotone waves in a three species reaction-diffusion model, Methods Appl Anal, 4, 3, 261-282 (1997) · Zbl 0908.35060
[47] Murray, JD, Mathematical biology II: spatial models and biomedical applications (2003), New York: Springer, New York
[48] Myerscough, MR; Murray, JD, Analysis of propagating pattern in a chemotaxis system, Bull Math Biol, 54, 1, 77-94 (1992) · Zbl 0733.92002
[49] Okubo, A.; Maini, PK; Williamson, MH; Murray, JD, On the spatial spread of the grey squirrel in Britain, Proc R Soc Lond B Biol Sci, 238, 1291, 113-125 (1989)
[50] Pettet, G.; McElwain, D.; Norbury, J., Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves, Math Med Biol A J IMA, 17, 4, 395-413 (2000) · Zbl 0969.92020
[51] Potts, JR; Lewis, MA, Spatial memory and taxis-driven pattern formation in model ecosystems, Bull Math Biol, 81, 2725-2747 (2019) · Zbl 1417.92217
[52] Ramani, A.; Grammaticos, B.; Bountis, T., The Painlevé property and singularity analysis of integrable and non-integrable systems, Phys Rep, 180, 3, 159-245 (1989)
[53] Roques, L.; Garnier, J.; Hamel, F.; Klein, EK, Allee effect promotes diversity in traveling waves of colonization, Proc Natl Acad Sci, 109, 23, 8828-8833 (2012)
[54] Roussier, V., Stability of radially symmetric travelling waves in reaction-diffusion equations, Annales de l’IHP Analyse non linéaire, 21, 3, 341-379 (2004) · Zbl 1066.35018
[55] Russo, M.; Van Gorder, RA; Choudhury, SR, Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities, Commun Nonlinear Sci Numer Simul, 18, 7, 1623-1634 (2013) · Zbl 1277.35102
[56] Satnoianu, RA, Coexistence of stationary and traveling waves in reaction-diffusion-advection systems, Phys Rev E, 68, 3, 032101 (2003)
[57] Sherratt, JA, Cellular growth control and travelling waves of cancer, SIAM J Appl Math, 53, 6, 1713-1730 (1993) · Zbl 0811.92018
[58] Shigesada, N.; Kawasaki, K.; Takeda, Y., Modeling stratified diffusion in biological invasions, Am Nat, 146, 2, 229-251 (1995)
[59] Strobl MAR, Krause AL, Damaghi M, Gillies R, Anderson ARA, Maini PK (2020) Mix & match: phenotypic coexistence as a key facilitator of solid tumour invasion. Bull Math Biol 82:15 · Zbl 1432.92031
[60] Taylor NP, Kim H, Krause AL, Van Gorder RA (2020) A non-local cross-diffusion model of population dynamics I: emergent spatial and spatiotemporal patterns. Bull Math Biol. 10.1007/s11538-020-00786-z
[61] Turing, AM, The chemical basis of morphogenesis, Philos Trans R Soc Lond Ser B Biol Sci, 237, 641, 37-72 (1952) · Zbl 1403.92034
[62] Volpert, V.; Petrovskii, S., Reaction-diffusion waves in biology, Phys Life Rev, 6, 4, 267-310 (2009)
[63] Wang, MH; Kot, M., Speeds of invasion in a model with strong or weak Allee effects, Math Biosci, 171, 1, 83-97 (2001) · Zbl 0978.92033
[64] Wang, ZA, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin Dyn Syst-B, 18, 3, 601-641 (2013) · Zbl 1277.35006
[65] Weiss J (1983) The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J Math Phys 24(6):1405-1413 · Zbl 0531.35069
[66] Weiss, J.; Tabor, M.; Carnevale, G., The Painlevé property for partial differential equations, J Math Phys, 24, 3, 522-526 (1983) · Zbl 0514.35083
[67] White A, Lurz PW, Jones HE, Boots M, Bryce J, Tonkin M, Ramoo K, Bamforth L, Jarrott A (2015) The use of mathematical models in red squirrel conservation: assessing the threat from grey invasion and disease to the Fleet basin stronghold. Red Squirrels Ecology, Conservation Management in Europe; Shuttleworth C, Lurz PWW, Hayward MW, Eds, pp 265-279
[68] Wu, YP, Traveling waves for a class of cross-diffusion systems with small parameters, J Diff Equ, 123, 1, 1-34 (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.