American Institute of Mathematical Sciences

Advanced
June  2018, 23(4): 1767-1795. doi: 10.3934/dcdsb.2018083

Nonlocal elliptic system arising from the growth of cancer stem cells

Manuel Delgado 1, , Ítalo Bruno Mendes Duarte 2, and  Antonio Suárez Fernández 1,

1. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, Sevilla, C/. Tarfia s/n, 41012, Spain
2. 

Universidade Federal do Pará, Faculdade de Matemática, Belém, PA 66075-110, Brazil

Received  October 2016 Revised  October 2017 Published  June 2018 Early access  March 2018

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

Keywords: Nonlocal elliptic system, coexistence states, bifurcation method, fixed point index, cancer stem cells.
Mathematics Subject Classification: Primary: 35J60, 35P05; Secondary: 92B05.
Citation: Manuel Delgado, Ítalo Bruno Mendes Duarte, Antonio Suárez Fernández. Nonlocal elliptic system arising from the growth of cancer stem cells. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1767-1795. doi: 10.3934/dcdsb.2018083
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[2]

J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh, 97A (1984), 21-34.  doi: 10.1017/S0308210500031802.  Google Scholar

[3]

I. Borsi, A. Fasano, M. Primicerio and T. Hillen, A non-local model for cancer stem cells and the tumor growth paradox, Math. Med. Biol., 34 (2017), 59-75.   Google Scholar

[4]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[7]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes in Math. Ser., 279 Longman, New York, 1992.  Google Scholar

[8]

M. Delgado, I. B. M. Duarte and A. Suárez, Nonlocal problem arising from the birth-jump processes, Proc. Roy. Soc. Edinburgh, to appear. Google Scholar

[9]

M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262.  doi: 10.1006/jdeq.1999.3655.  Google Scholar

[10]

M. Delgado and A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations, 244 (2008), 3119-3150.  doi: 10.1016/j.jde.2007.12.007.  Google Scholar

[11]

H. Enderling, P. Hahnfeldt and T. Hillen, The tumor growth paradox and immune system-mediated selection for cancer stem cells, Bull. Math. Biology, 75 (2013), 161-184.  doi: 10.1007/s11538-012-9798-x.  Google Scholar

[12]

A. Fasano, A. Mancini and M. Primicerio, Tumours with cancer stem cells: A PDE model, Math. Biosci., 272 (2016), 76-80.  doi: 10.1016/j.mbs.2015.12.003.  Google Scholar

[13]

J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh, 127 (1997), 281-336.  doi: 10.1017/S0308210500023659.  Google Scholar

[14]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, 1991.  Google Scholar

[15]

T. Hillen, B. Greese, J. Martin and G. de Vries, Birth-jump processes and application to forest fire spotting, J. of Biological Dynamics, 9 (2015), 104-127.  doi: 10.1080/17513758.2014.950184.  Google Scholar

[16]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

[17]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2013.  Google Scholar

[18]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.   Google Scholar

[19]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Ma-thematics, 426 CRC Press, Boca Raton, Florida, 2001.  Google Scholar

[20]

J. López-Gómez and J. Sabina de Lis, Coexistence states and global attractivity for some convective diffusive competing species models, Trans. Amer. Math. Soc., 347 (1995), 3797-3833.  doi: 10.1090/S0002-9947-1995-1311910-8.  Google Scholar

[21]

L. Maddalena, Analysis of an integro-differential system modeling tumor growth, Appl. Math. Comput., 245 (2014), 152-157.  doi: 10.1016/j.amc.2014.07.081.  Google Scholar

[22]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[2]

J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh, 97A (1984), 21-34.  doi: 10.1017/S0308210500031802.  Google Scholar

[3]

I. Borsi, A. Fasano, M. Primicerio and T. Hillen, A non-local model for cancer stem cells and the tumor growth paradox, Math. Med. Biol., 34 (2017), 59-75.   Google Scholar

[4]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[7]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes in Math. Ser., 279 Longman, New York, 1992.  Google Scholar

[8]

M. Delgado, I. B. M. Duarte and A. Suárez, Nonlocal problem arising from the birth-jump processes, Proc. Roy. Soc. Edinburgh, to appear. Google Scholar

[9]

M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262.  doi: 10.1006/jdeq.1999.3655.  Google Scholar

[10]

M. Delgado and A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations, 244 (2008), 3119-3150.  doi: 10.1016/j.jde.2007.12.007.  Google Scholar

[11]

H. Enderling, P. Hahnfeldt and T. Hillen, The tumor growth paradox and immune system-mediated selection for cancer stem cells, Bull. Math. Biology, 75 (2013), 161-184.  doi: 10.1007/s11538-012-9798-x.  Google Scholar

[12]

A. Fasano, A. Mancini and M. Primicerio, Tumours with cancer stem cells: A PDE model, Math. Biosci., 272 (2016), 76-80.  doi: 10.1016/j.mbs.2015.12.003.  Google Scholar

[13]

J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh, 127 (1997), 281-336.  doi: 10.1017/S0308210500023659.  Google Scholar

[14]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, 1991.  Google Scholar

[15]

T. Hillen, B. Greese, J. Martin and G. de Vries, Birth-jump processes and application to forest fire spotting, J. of Biological Dynamics, 9 (2015), 104-127.  doi: 10.1080/17513758.2014.950184.  Google Scholar

[16]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

[17]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2013.  Google Scholar

[18]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.   Google Scholar

[19]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Ma-thematics, 426 CRC Press, Boca Raton, Florida, 2001.  Google Scholar

[20]

J. López-Gómez and J. Sabina de Lis, Coexistence states and global attractivity for some convective diffusive competing species models, Trans. Amer. Math. Soc., 347 (1995), 3797-3833.  doi: 10.1090/S0002-9947-1995-1311910-8.  Google Scholar

[21]

L. Maddalena, Analysis of an integro-differential system modeling tumor growth, Appl. Math. Comput., 245 (2014), 152-157.  doi: 10.1016/j.amc.2014.07.081.  Google Scholar

[22]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

Figure 1.  Coexistence region of ($1$) for $\delta\neq 1$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Possible coexistence region of ($1$) for $\delta = 1$. In this case the sum of the index of the coexistence states of ($1$) is 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Possible coexistence region of ($1$) for $\delta = 1$. In this case the sum of the index of the coexistence states of ($1$) is -1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Possible coexistence region of ($1$) for $\delta = 1$. In this case, there are regions where the sum of the index of the coexistence states of ($1$) is 1 (when $\mathcal{F}_1$ is above $\mathcal{G}$) and others where the sum is -1 (when $\mathcal{G}$ is above $\mathcal{F}_1$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Coexistence regions of ($1$) for $\delta$ close to 0
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Coexistence regions of ($1$) for $\delta$ close to 1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013
[2]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263
[3]

Chengjun Guo, Chengxian Guo, Sameed Ahmed, Xinfeng Liu. Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2473-2489. doi: 10.3934/dcdsb.2016056
[4]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[5]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[6]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[7]

Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467
[8]

Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021211
[9]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[10]

Pablo Álvarez-Caudevilla, Julián López-Gómez. Characterizing the existence of coexistence states in a class of cooperative systems. Conference Publications, 2009, 2009 (Special) : 24-33. doi: 10.3934/proc.2009.2009.24
[11]

Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333
[12]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129
[13]

Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021210
[14]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[15]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178
[16]

Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092
[17]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021046
[18]

Evans K. Afenya, Calixto P. Calderón. Growth kinetics of cancer cells prior to detection and treatment: An alternative view. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 25-28. doi: 10.3934/dcdsb.2004.4.25
[19]

Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335
[20]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (193)
  • HTML views (287)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy