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Bifurcation structure of stationary solutions for a chemotaxis system with bistable growth

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Abstract

From the viewpoint of pattern formation, Keller–Segel systems with growth terms are studied. These models exhibit various stationary and spatio-temporal patterns which are caused by a combination of three effects: chemotaxis, diffusion and growth. In this paper, we consider Keller–Segel system with the cubic growth term known as the Allee effect in ecology and its shadow system in the limiting case that the mobility of biological population tends to infinity. We show the existence and stability of stationary solutions of the shadow system in one space dimension. Our proof is based on the bifurcation theory, a singular perturbation method and a level set analysis. We also show some numerical results on global structures of stationary solutions in the systems by using AUTO package. Moreover, we mention the difference in dynamics between Keller–Segel system with the cubic growth term and that with the logistic growth term with the aid of a computer.

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References

  1. Aida, M., Tsujikawa, T., Efendiev, M., Yagi, A., Mimura, M.: Lower estimate of the attractor dimension for a chemotaxis growth system. J. Lond. Math. Soc. 74, 453–474 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aida, M., Yagi, A.: Target pattern solutions for chemotaxis–growth system. Sci. Math. Jpn. 59, 577–590 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Alt, W., Lauffenburger, D.A.: Transient behavior of a chemotaxis system modelling certain types of tissue inflammation. J. Math. Biol. 24, 691–722 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aotani, A., Mimura, M., Mollee, T.: A model aided understanding of spot pattern formation in chemotactic \(E. coli\) colonies. Jpn. J. Ind. Appl. Math. 27, 5–22 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chafee, N., Infante, E.F.: A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4, 17–37 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52, 161–180 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Doedel, E.J., Oldeman, B.E., Champneys, A.R., Dercole, F., Fairgrieve, T., Kuznetsov, Y.A., Paffenroth, R.C., Sandstede, B., Wang, X., Zhang, C.: AUTO-07p: continuation and bifurcation software for ordinary differential equations 2012. https://sourceforge.net/projects/auto-07p/files/auto07p/

  10. Ei, S.-I., Izuhara, H., Mimura, M.: Infinite dimensional relaxation oscillation in aggregation–growth systems. Discrete Contin. Dyn. Syst. Ser. B 17, 1859–1887 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ei, S.-I., Izuhara, H., Mimura, M.: Spatio-temporal oscillations in the Keller–Segel system with logistic growth. Phys. D 277, 1–21 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greenberg, J.M.: Stability of equilibrium solutions for the Fisher equation. Q. Appl. Math. 39, 239–247 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hai, D.D., Yagi, A.: Numerical computations and pattern formation for chemotaxis–growth model. Sci. Math. Jpn. 70, 205–211 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Izuhara, H., Tsujikawa, T.: Pattern dynamics for some shadow system of the Keller–Segel model with bistable growth (in preparation)

  16. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    Article  MATH  Google Scholar 

  17. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    Article  MATH  Google Scholar 

  18. Kurata, N., Kuto, K., Osaki, K., Tsujikawa, T., Sakurai, T.: Bifurcation phenomena of pattern solution to Mimura–Tsujikawa model in one dimension. Math. Sci. Appl. 29, 265–278 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Kuto, K., Osaki, K., Sakurai, T., Tsujikawa, T.: Spatial pattern formation in a chemotaxis–diffusion–growth model. Phys. D 241, 1629–1639 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kuto, K., Tsujikawa, T.: Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system. Nonlinearity 26, 1313–1343 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kuto, K., Tsujikawa, T.: Limiting structure of steady-states to the Lotka–Volterra competition model with large diffusion and advection. J. Differ. Equ. 258, 1801–1858 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lauffenburger, D.A., Kennedy, C.R.: Localized bacterial infection in a distributed model for tissue inflammation. J. Math. Biol. 16, 141–163 (1983)

    Article  MATH  Google Scholar 

  23. Maini, P.K., Myerscough, M.R., Winters, K.H., Murray, J.D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53, 701–719 (1991)

    Article  MATH  Google Scholar 

  24. Mimura, M., Tsujikawa, T.: Aggregation pattern dynamics in a chemotaxis model including growth. Phys. A 230, 499–543 (1996)

    Article  Google Scholar 

  25. Murray, J.D.: Mathematical Biology I, II. Springer, New York (2003)

    Google Scholar 

  26. Murray, J.D., Myerscough, M.R.: Pigmentation pattern formation on snakes. J. Theor. Biol. 149, 339–360 (1991)

    Article  Google Scholar 

  27. Myerscough, M.R., Maini, P.K., Painter, K.J.: Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60, 1–26 (1998)

    Article  MATH  Google Scholar 

  28. Nishiura, Y., Fujii, H.: Stability of singularly perturbed solutions to systems of reaction–diffusion equations. SIAM J. Math. Anal. 18, 1726–1770 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nishiura, Y., Tsujikawa, T.: Instability of singularly perturbed Neumann layer solutions in reaction–diffusion systems. Hiroshima Math. J. 20, 297–329 (1990)

    MathSciNet  MATH  Google Scholar 

  30. Okuda, T., Osaki, K.: Bifurcation of hexagonal patterns in a chemotaxis–diffusion–growth system. Nonlinear Anal. Real World Appl. 12, 3294–3305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M.: Exponential attractor for a chemotaxis–growth system of equations. Nonlinear Anal. 51, 119–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Painter, K.J., Hillen, T.: Spatio-temporal chaos in a chemotaxis model. Phys. D 240, 363–375 (2011)

    Article  MATH  Google Scholar 

  33. Schaaf, R.: Global behaviour of solution branches for some Neumann problems depending on one or several parameters. J. Reine Angew Math. 364, 1–31 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Schaaf, R.: Global solution branches of two-point boundary value problems. In: Lecture Notes in Mathematics, vol. 1458. Springer, Berlin (1990)

  35. Shi, J.: Semilinear Neumann boundary value problems on a rectangle. Trans. Am. Math. Soc. 354, 3117–3154 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Smaller, J., Wasserman, A.: Global bifurcation of steady-state solutions. J. Differ. Equ. 39, 269–290 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tsujikawa, T.: Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth. Methods Appl. Anal. 3, 401–431 (1996)

    MathSciNet  MATH  Google Scholar 

  38. Tsujikawa, T.: Stationary problem of a simple chemotaxis–growth model. RIMS Kokyuroku 1924, 55–63 (2014)

    Google Scholar 

  39. Tsujikawa, T., Kuto, K., Miyamoto, Y., Izuhara, H.: Stationary solutions for some shadow system of the Keller–Segel model with logistic growth. Discrete Contin. Dyn. Syst. Ser. S 8, 1023–1034 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, Q., Yan, J., Gai, C.: Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth. Z. Angew Math. Phys. 67, 51 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Woodward, D.E., Tyson, R., Myerscough, M.R., Murray, J.D., Budrene, E.O., Berg, H.C.: Spatio-temporal patterns generated by \(Salmonella\, typhimurium\). Biophys. J. 68, 2181–2189 (1995)

    Article  Google Scholar 

  42. Winkler, M.: Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors were partially supported by JSPS KAKENHI Grants (HI: 17K14237 and 16KT0135, KK: 15K04948, TT: 17K05334).

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Authors and Affiliations

  1. Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki, 889-2192, Japan

    Hirofumi Izuhara & Tohru Tsujikawa

  2. Department of Communication Engineering and Informatics, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo, 182-8585, Japan

    Kousuke Kuto

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  1. Hirofumi Izuhara
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Correspondence to Hirofumi Izuhara.

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Dedicated to the 75th birthday of Professor Masayasu Mimura.

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Izuhara, H., Kuto, K. & Tsujikawa, T. Bifurcation structure of stationary solutions for a chemotaxis system with bistable growth. Japan J. Indust. Appl. Math. 35, 441–475 (2018). https://doi.org/10.1007/s13160-017-0298-0

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  • DOI: https://doi.org/10.1007/s13160-017-0298-0

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