August 2018, 11(4): 891-909. doi: 10.3934/krm.2018035

Traveling wave and aggregation in a flux-limited Keller-Segel model

1. 

Institut Camille Jordan, UMR 5208 CNRS/Université Claude Bernard Lyon 1, and Project-team Inria NUMED, Lyon, France

2. 

Sorbonne Universités, UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions UMR CNRS 7598, Université Paris Diderot, Inria de Paris, F75005 Paris, France

3. 

Graduate School of Simulation Studies, University of Hyogo, Kobe 650-0047, Japan

Received  September 2017 Revised  January 2018 Published  April 2018

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term.

Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response.

In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counter-balancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.

Citation: Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035
References:
[1]

J. Adler, Chemotaxis in Bacteria, Science, 153 (1966), 708-716.

[2]

D. G. Aronson and H. F. Weinberger, Lecture Notes in Mathematics, Springer, 1975.

[3]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.

[4]

E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. doi: 10.1038/376049a0.

[5]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Model Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[6]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and abasence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473. doi: 10.1080/03605302.2016.1277237.

[7]

S. M. BlockJ. E. Segall and H. C. Berg, Adaptation kinetics in bacterial chemotaxis, J. Bacteriol., 154 (1983), 312-323.

[8]

V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429.

[9]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and related models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[10]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6.

[11]

S. EiH. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Physica D, 277 (2014), 1-21. doi: 10.1016/j.physd.2014.03.002.

[12]

C. Emako, C. Gayrard, A. Buguin, L. Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. doi: 10. 1371/journal. pcbi. 1004843.

[13]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 65 (1937), 335-369.

[14]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[15]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Differ. Equ. Appl., 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4.

[16]

Y. V. KalininL. JiangY. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[18]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de et son application à un problème biologique, Moskow. Univ. Math. Bull., 1 (1937), 1–25.

[20]

C. LiuX. FuL. LiuX. RenC. K. L. ChauS. LiL. XiangH. ZengG. ChenL. TangP. LenzX. CuiW. HuangT. Hwa and J.-.D. Huang, Sequential Establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. doi: 10.1126/science.1209042.

[21]

P. K. MainiM. R. MyerscoughK. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.

[22]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9.

[23]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interface free bound., 10 (2008), 517-538. doi: 10.4171/IFB/200.

[24]

G. NadinB. Perthame and M. Tang, Can a traveling wave connect two unstable state? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008.

[25]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[26]

B. Perthame, Parabolic Equations in Biology, Springer, 2015. doi: 10.1007/978-3-319-19500-1.

[27]

B. Perthame, N. Vauchelet and Z. Wang, Modulation of stiff response in E. coli bacterial populations, (in preparation).

[28]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, preprint, arXiv: 1703.08386.

[29]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.

[30]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240. doi: 10.1073/pnas.1101996108.

[31]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607. doi: 10.1007/s11538-008-9322-5.

show all references

References:
[1]

J. Adler, Chemotaxis in Bacteria, Science, 153 (1966), 708-716.

[2]

D. G. Aronson and H. F. Weinberger, Lecture Notes in Mathematics, Springer, 1975.

[3]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.

[4]

E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. doi: 10.1038/376049a0.

[5]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Model Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[6]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and abasence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473. doi: 10.1080/03605302.2016.1277237.

[7]

S. M. BlockJ. E. Segall and H. C. Berg, Adaptation kinetics in bacterial chemotaxis, J. Bacteriol., 154 (1983), 312-323.

[8]

V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429.

[9]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and related models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[10]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6.

[11]

S. EiH. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Physica D, 277 (2014), 1-21. doi: 10.1016/j.physd.2014.03.002.

[12]

C. Emako, C. Gayrard, A. Buguin, L. Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. doi: 10. 1371/journal. pcbi. 1004843.

[13]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 65 (1937), 335-369.

[14]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[15]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Differ. Equ. Appl., 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4.

[16]

Y. V. KalininL. JiangY. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[18]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de et son application à un problème biologique, Moskow. Univ. Math. Bull., 1 (1937), 1–25.

[20]

C. LiuX. FuL. LiuX. RenC. K. L. ChauS. LiL. XiangH. ZengG. ChenL. TangP. LenzX. CuiW. HuangT. Hwa and J.-.D. Huang, Sequential Establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. doi: 10.1126/science.1209042.

[21]

P. K. MainiM. R. MyerscoughK. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.

[22]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9.

[23]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interface free bound., 10 (2008), 517-538. doi: 10.4171/IFB/200.

[24]

G. NadinB. Perthame and M. Tang, Can a traveling wave connect two unstable state? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008.

[25]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[26]

B. Perthame, Parabolic Equations in Biology, Springer, 2015. doi: 10.1007/978-3-319-19500-1.

[27]

B. Perthame, N. Vauchelet and Z. Wang, Modulation of stiff response in E. coli bacterial populations, (in preparation).

[28]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, preprint, arXiv: 1703.08386.

[29]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.

[30]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240. doi: 10.1073/pnas.1101996108.

[31]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607. doi: 10.1007/s11538-008-9322-5.

Figure 1.  The schematic of domain decomposition in the relative coordinate system $\xi$ introduced in Sec. 2.3
Figure 2.  Figure (a) shows the solution curves of Eq. (44) in $\hat \xi_c$-$\hat \chi$ plane with variation in the diffusion constant $d$, while the proliferation rate $p = 0.5$ is fixed. Figure (b) shows the parameter regime which satisfies the constraints Eqs. (39) and (40). Here the modulation amplitude $\hat \chi = 3.0$ is fixed. The contour shows the peak value of population wave, i.e., $\alpha$ defined in Eq. (34). The symbols "$\times$" in figure (b) show the solutions of Eq. (44) with $\hat \chi$ = 3.0 and $d = $1, 5, 10, 25, and 50, respectively, from left to right
Figure 3.  The diagram of different types of numerical solutions with variation in the modulation $\chi$ and stiffness $\delta^{-1}$. The circles (Type Ⅰ and Ⅱ) refer to Fig. 5, the squares (Type Ⅲ) refer to Fig. 6, the triangles (Type Ⅳ) refer to Fig. 7, and the diamonds (Type Ⅴ) refer to Fig. 8. The colors of each symbol show the maximum value of population density in the spatial profile. The diffusion coefficient $d$ and proliferation rate $p$ are fixed as $d = 4$ and $p = 0.5$. The dotted horizontal line shows the critical value determined by the instability condition
Figure 4.  The speed of propagating front with variation in the modulation $\chi$ and stiffness $\delta^{-1}$. The values of parameters $d$ and $p$ are same as in Fig. 3. The vertical line on the horizontal axis indicates the critical value of the instability condition. The dotted lines show the minimum speeds obtained by Eq. (16). The numbers (Ⅰ)-(Ⅴ) illustrate the solution types shown in Fig. 3
Figure 5.  The snapshots of monotonic and non-monotonic traveling waves for $2\chi/(\pi\delta) = 0.01$ (a) and $2\chi/(\pi\delta) = 5.0$ (b), respectively. The other parameters are set as $\hat \chi = 1.5$, $d = 4.0$, and $p = 0.5$
Figure 6.  The snapshots of backward traveling wave for $\tilde \chi = 3.0$, $2\chi/(\pi\delta) = 9$, $d = 4.0$, and $p = 0.5$
Figure 7.  The snapshots of periodic pattern formation with a moving front in forward (a) and backward (b) directions. The modulation parameter is set as $\hat\chi = 1.0$ (a) and $\hat\chi = 3.0$ (b). The other parameters are set as $2\chi/(\pi\delta) = 10.0$, $d = 4.0$, and $p = 0.5$ in both (a) and (b)
Figure 8.  The snapshots of pattern formation of localized spikes. The parameters are set as $\hat \chi = 3.0$, $2\chi/(\pi\delta) = 20.0$, $d = 4.0$, and $p = 0.5$. Note that the results at $t = $200 and 500 are almost overlapped only except the region around $\hat x = 350$
Figure 9.  The comparison of the traveling speed measured from numerical solution $c^*$ to the minimum traveling speed $c_\mathrm{min}$ obtained by Eq. (16). The modulation amplitude $\hat\chi = 1.5$ and diffusion constant $d = 1.0$ are fixed while the proliferation rate $p$ varies. The dispersion relation Eq. (14) between the traveling speed and exponential decay of population density far ahead the front, i.e., $c(\lambda^*)$ is also plotted. The way to measure $c^*$ and $\lambda^*$ is described in the last paragraph of Sec. 3.1. Note that $c^*$ and $c(\lambda^*)$ are almost overlapped in the figure
Figure 10.  The snapshot of chemotactic drift speed $U_\delta$ for Fig. 6 (i.e., Type Ⅲ in Fig. 4) at time $\hat t = 500$
Figure 11.  Numerical solutions for large-stiffness parameters are compared to the analytical solution for the stiff flux Eq. (18), i.e., $2\chi/(\pi\delta)\rightarrow \infty$. The modulation amplitude $\chi$ and proliferation rate $p$ are fixed as $\hat\chi = 2.5$ and $p = 0.5$, respectively. The diffusion coefficient $d$ is set as $d = 4$ in figure (a) and $d = 16$ in figure (b)
Table 1.  The accuracy tests performed with different numbers of mesh interval $I$, i.e., $I$ = 5000, 10000, 20000. The subscripts $f$ and $c$ represent the finer and coarser mesh systems, respectively. The traveling speeds $c^*$ and exponential decay $\lambda^*$ are directly measured from the numerical solutions
$I_f$ -$I_c$ $|\frac{c^*_f-c^*_c}{c^*_f} |$ $|\frac{\lambda^*_f-\lambda^*_c}{\lambda^*_f}|$ $\frac{c^*_f-c(\lambda_f^*)}{c(\lambda_f^*)}$
10000 -5000$1.7\times 10^{-3}$ $2.8\times 10^{-3}$$1.1\times 10^{-3}$
20000 -10000 $3.6\times 10^{-4}$ $5.6\times 10^{-4}$$8.1\times 10^{-4}$
$I_f$ -$I_c$ $|\frac{c^*_f-c^*_c}{c^*_f} |$ $|\frac{\lambda^*_f-\lambda^*_c}{\lambda^*_f}|$ $\frac{c^*_f-c(\lambda_f^*)}{c(\lambda_f^*)}$
10000 -5000$1.7\times 10^{-3}$ $2.8\times 10^{-3}$$1.1\times 10^{-3}$
20000 -10000 $3.6\times 10^{-4}$ $5.6\times 10^{-4}$$8.1\times 10^{-4}$
Table 2.  The decay rate $\lambda$ defined in Eq. (12) and the distance from the peak of chemoattractant to the position where the population density equals to $\rho_c$, i.e., $\xi_c = x_c-x_S$ where $\rho(x_c) = \rho_c$ and $\partial_x S(x_S) = 0$, with variation in the stiffness. The modulation amplitude $\chi$ and proliferation rate $p$ are fixed as $\hat \chi = 2.5$ and $p = 0.5$, respectively
$d=4.0$$d=16.0$
$\frac{2\chi}{\pi\delta}$$\lambda/\sqrt{p}$$\hat \xi_c$$\frac{2\chi}{\pi\delta}$$\lambda/\sqrt{p}$$\hat \xi_c$
7.01.303.0821.01.0655.57
8.01.262.9623.01.0625.44
9.01.232.7925.01.0595.28
10.01.202.6120.01.0585.16
$\infty$1.003.09 $\infty$1.006.95
$d=4.0$$d=16.0$
$\frac{2\chi}{\pi\delta}$$\lambda/\sqrt{p}$$\hat \xi_c$$\frac{2\chi}{\pi\delta}$$\lambda/\sqrt{p}$$\hat \xi_c$
7.01.303.0821.01.0655.57
8.01.262.9623.01.0625.44
9.01.232.7925.01.0595.28
10.01.202.6120.01.0585.16
$\infty$1.003.09 $\infty$1.006.95
[1]

Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717

[2]

Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017

[3]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046

[4]

Qi Wang, Jingyue Yang, Lu Zhang. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179

[5]

Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012

[6]

Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023

[7]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231

[8]

Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3463-3478. doi: 10.3934/dcdsb.2016107

[9]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

[10]

Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066

[11]

Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339

[12]

Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79

[13]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[14]

Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911

[15]

Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109

[16]

Shen Bian, Jian-Guo Liu, Chen Zou. Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$. Kinetic & Related Models, 2014, 7 (1) : 9-28. doi: 10.3934/krm.2014.7.9

[17]

Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015

[18]

Xinru Cao. Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3369-3378. doi: 10.3934/dcdsb.2017141

[19]

José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319

[20]

Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (43)
  • HTML views (193)
  • Cited by (0)

Other articles
by authors

[Back to Top]