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

Nonlocal elliptic system arising from the growth of cancer stem cells

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.

Citation: Manuel Delgado, Ítalo Bruno Mendes Duarte, Antonio Suárez Fernández. Nonlocal elliptic system arising from the growth of cancer stem cells. Discrete and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[9]

M. DelgadoJ. 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.

[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.

[11]

H. EnderlingP. 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.

[12]

A. FasanoA. 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.

[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.

[14]

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

[15]

T. HillenB. GreeseJ. 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.

[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.

[17]

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

[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. 

[19]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[9]

M. DelgadoJ. 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.

[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.

[11]

H. EnderlingP. 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.

[12]

A. FasanoA. 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.

[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.

[14]

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

[15]

T. HillenB. GreeseJ. 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.

[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.

[17]

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

[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. 

[19]

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

[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.

[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.

[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.

Figure 1.  Coexistence region of ($1$) for $\delta\neq 1$
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 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 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 5.  Coexistence regions of ($1$) for $\delta$ close to 0
Figure 6.  Coexistence regions of ($1$) for $\delta$ close to 1
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