Abstract
We consider nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis-growth system
in \(\Omega :=(0,L)\subset \mathbb {R}\) with \(L>0, \varepsilon >0, r\ge 0\) and \(\mu >0\), along with the corresponding limit problem formally obtained upon taking \(\varepsilon \searrow 0\). For the latter hyperbolic–elliptic problem, we establish results on local existence and uniqueness within an appropriate generalized solution concept. In this context we shall moreover derive an extensibility criterion involving the norm of \(u(\cdot ,t)\) in \(L^\infty (\Omega )\). This will enable us to conclude that in this case \(\varepsilon =0\),
-
if \(\mu \ge 1\), then all solutions emanating from sufficiently regular initial data are global in time, whereas
-
if \(\mu <1\), then some solutions blow-up in finite time.
The latter will reveal that the original parabolic–elliptic problem (\(\star \)), though known to possess no such exploding solutions, exhibits the following property of dynamical structure generation: given any \(\mu \in (0,1)\), one can find smooth bounded initial data with the property that for each prescribed number \(M>0\) the solution of (\(\star \)) will attain values above \(M\) at some time, provided that \(\varepsilon \) is sufficiently small. In particular, this means that the associated carrying capacity given by \(\frac{r}{\mu }\) can be exceeded during evolution to an arbitrary extent. We finally present some numerical simulations that illustrate this type of solution behavior and that, moreover, inter alia, indicate that achieving large population densities is a transient dynamical phenomenon occurring on intermediate time scales only.
Similar content being viewed by others
References
Aida, M., Tsujikawa, T., Efendiev, M., Yagi, A., Mimura, M.: Lower estimate of the attractor dimension for a chemotaxis growth system. J. London Math. Soc. 74(2), 453–474 (2006)
Biler, P.: Local and global solvability of come parabolic systems modeling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)
Bournaveas, N., Calvez, V., Gutiérrez, S., Perthame, B.: Global existence for a Kinetic model of chemotaxis via dispersion and strichartz estimates. Commun. Part. Differ. Equ. 33(1), 79–95 (2008)
Carrillo, J.A., Hittmeir, S., Jüngel, A.: Cross diffusion and nonlinear diffusion preventing blow up in the Keller–Segel model. Math. Models Methods Appl. Sci. 22, 1250041 (2012)
Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Mod. Methods Appl. Sci. 15, 1685–1734 (2005)
Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)
Cieślak, T., Laurençot, P.H.: Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27(1), 437–446 (2010)
Cieślak, T., Laurençot, Ph.: Global existence vs. blowup in a one-dimensional Smoluchowski-Poisson system. Escher, Joachim (ed.) et al., Parabolic problems. The Herbert Amann Festschrift. Based on the conference on nonlinear parabolic problems held in celebration of Herbert Amann’s 70th birthday at the Banach Center in Bedlewo, Poland, May 1016, 2009. Birkhäuser, Basel. Progress in Nonlinear Differential Equations and Their Applications 80, 95–109 (2011).
Cieślak, T., Stinner, C.: Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. J. Differ. Equ. 252(10), 5832–5851 (2012)
Cieślak, T., Winkler, M.: Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)
Eberl, H.J., Parker, D.F., van Loosdrecht, M.C.M.: A new deterministic spatio-temporal continuum model for biofilm development. J. Theor. Med. 3(3), 161175 (2001)
Funaki, M., Mimura, M., Tsujikawa, T.: Travelling front solutions arising in the chemotaxis-growth model. Interfaces Free Bound. 8(2), 223–245 (2006)
Hašcovec, J., Schmeiser, C.: Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. J. Stat. Phys. 135, 133–151 (2009)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1981)
Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scu. Norm. Super. Pisa Cl. Sci. 24, 663–683 (1997)
Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1), 183–217 (2009)
Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 (2003)
Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819824 (1992)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Kuto, K., Osaki, K., Sakurai, T., Tsujikawa, T.: Spatial pattern formation in a chemotaxis-diffusion-growth model. Physica D 241, 1629–1639 (2012)
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(5), 701719 (1991)
Meral, G., Stinner, C., Surulescu, C.: On a multiscale model involving cell contractivity and its effects on tumor invasion (2014)
Mizoguchi, N., Winkler, M.: Is finite-time blow-up a generic phenomenon in the two-dimensional Keller–Segel system? (2014)
Nadin, G., Perthame, B., Ryzhik, L.: Traveling waves for the Keller–Segel system with Fisher birth terms. Interfaces Free Bound. 10(4), 517–538 (2008)
Nagai, T.: Blowup of nonradial solutions to parabolicelliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)
Nakaguchi, E., Osaki, K.: Global existence of solutions to a parabolic–parabolic system for chemotaxis with weak degradation. Nonlinear Anal. TMA 74(1), 286–297 (2011)
Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002)
Osaki, K., Yagi, A.: Global existence for a chemotaxis-growth system in \({\mathbb{R}}^2\). Adv. Math. Sci. Appl. 12(2), 587–606 (2002)
Othmer, H.G., Stevens, A.: Aggregation, blowup and collapse: the ABCs of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 10441081 (1997)
Painter, K.J., Hillen, T.: Volume-filling and quroum-sensing in models for chemosensitive movement. Can. Appl. Am. Quart. 10(4), 501–543 (2002)
Painter, K.J., Hillen, T.: Spatio-temporal chaos in a chemotaxis model. Physica D 240, 363–375 (2011)
Painter, K.J., Maini, P.K., Othmer, H.G.: Complex spatial patterns in a hybrid chemotaxis reaction-diffusion model. J. Math. Biol. 41(4), 285314 (2000)
Perthame, B.: Transport Equations in Biology. Birkhäuser-Verlag, Basel (2007)
Poupaud, F.: Diagonal defect measures, adhesion dynamics and Euler equation. Methods Appl. Anal. 9(4), 533–561 (2002)
Szymańska, Z., Morales, Rodrigo C., Lachowicz, M., Chaplain, M.A.: Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Mod. Methods Appl. Sci. 19, 257–281 (2009)
Tello, J.I., Winkler, M.: A chemotaxis system with logistic source. Commun. Partial Differ. Equ. 32(6), 849–877 (2007)
Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and its Applications, vol. 2. North-Holland, Amsterdam (1977)
Winkler, M.: Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010)
Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
Winkler, M.: Does a volume-filling effect always prevent chemotactic collapse? Math. Methods Appl. Sci. 33, 12–24 (2010)
Winkler, M.: Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384, 261–272 (2011)
Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Winkler, M., Djie, K.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. TMA 72(2), 1044–1064 (2010)
Woodward, D.E., Tyson, R., Myerscough, M.R., Murray, J.D., Budrene, E.O., Berg, H.C.: Spatiotemporal patterns generated by Salmonella typhimurium. Biophys. J. 68(5), 21812189 (1995)
Wrzosek, D.: Volume filling effect in modelling chemotaxis. Math. Model. Nat. Phenom. 5, 123–147 (2010)
Yagi, A.: Norm behavior of solutions to a parabolic system of chemotaxis. Math. Jpn. 45, 241–265 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. K. Maini.
Rights and permissions
About this article
Cite this article
Winkler, M. How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities?. J Nonlinear Sci 24, 809–855 (2014). https://doi.org/10.1007/s00332-014-9205-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-014-9205-x