American Institute of Mathematical Sciences

Advanced
September  2020, 25(9): 3609-3630. doi: 10.3934/dcdsb.2020075

Influence of mutations in phenotypically-structured populations in time periodic environment

Cécile Carrère 1,, and  Grégoire Nadin 2,

1. 

Instituto Gulbenkian de Ciência, Rua da Quinta Grande, 6, 2780-156 Oeiras, Portugal
2. 

Laboratoire Jacques-Louis Lions, 5 place Jussieu, 75005 Paris, France

* Corresponding author: cmcarrere@igc.gulbenkian.pt

Received  August 2019 Revised  October 2019 Published  January 2020

Fund Project: This work was partially supported by grants from Région Ile-de-France and Inria team Mamba

We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solution to the unique positive and periodic solution of the problem, we study the influence of different factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the influence of the diffusion rate, the period length and the time variance of the environment fluctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the influence of a periodic treatment. In this framework, we show that the population might benefit from a intermediate rate of mutation.

Keywords: Parabolic integro-differential equations, time-periodic coefficients, evolution of mutation rates, adaptive evolution, principal eigenvalue of parabolic operators.
Mathematics Subject Classification: 35B10, 35K57, 35R09, 92D15.
Citation: Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075
References:
[1]

M. Alfaro and M. Veruete, Evolutionary branching via replicator mutator equations, Journal of Dynamics and Differential Equations, 31 (2019), 2029-2052.  doi: 10.1007/s10884-018-9692-9.  Google Scholar

[2]

A. Ardaševa, R. A. Gatenby, A. R. A. Anderson, H. M. Byrne, P. K. Maini and T. Lorenzi, Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments, 2019, arXiv: 1905.11712. Google Scholar

[3]

D. Basanta, H. Hatzikirou and A. Deutsch, Studying the emergence of invasiveness in tumours using game theory, The European Physical Journal B, 63 (2008), 393-397.  doi: 10.1140/epjb/e2008-00249-y.  Google Scholar

[4]

P. L. Bedard, A. R. Hansen, M. J. Ratain and L. L. Siu, Tumour heterogeneity in the clinic, Nature, 501 (2013), 355-364.  doi: 10.1038/nature12627.  Google Scholar

[5]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ-species persistence, Journal of Mathematical Biology, 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[6]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅱ-biological invasions and pulsating travelling fronts, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[7]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, Journal of Mathematical Biology, 48 (2004), 135-159.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM Journal on Mathematical Analysis, 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

[9]

N. Champagnat, R. Ferrière and S. Méléard, Individual-based probabilistic models of adaptive evolution and various scaling approximations, In Seminar on Stochastic Analysis, Random Fields and Applications V, Springer, 59 (2008), 75–113. doi: 10.1007/978-3-7643-8458-6_6.  Google Scholar

[10]

N. Champagnat and A. Lambert, Evolution of discrete populations and the canonical diffusion of adaptive dynamics, The Annals of Applied Probability, 17 (2007), 102-155.  doi: 10.1214/105051606000000628.  Google Scholar

[11]

J. Coville., Convergence to equilibrium for positive solutions of some mutation-selection model., arXiv: 1308.6471 [math], 2013, arXiv: 1308.6471. Google Scholar

[12]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

[13]

T. Epstein, R. A. Gatenby and J. S. Brown, The Warburg effect as an adaptation of cancer cells to rapid fluctuations in energy demand, PLOS ONE, 12 (2017), e0185085. doi: 10.1371/journal.pone.0185085.  Google Scholar

[14]

S. Figueroa Iglesias and S. Mirrahimi, Long time evolutionary dynamics of phenotypically structured populations in time-periodic environments, SIAM Journal on Mathematical Analysis, 50 (2018), 5537-5568.  doi: 10.1137/18M1175185.  Google Scholar

[15]

S. Geritz, Evolution and the theory of games, Evolution and the Theory of Games, 2011, page 6. Google Scholar

[16]

C. A. Gravenmier, M. Siddique and R. A. Gatenby, Adaptation to stochastic temporal variations in intratumoral blood flow: The warburg effect as a bet hedging strategy, Bulletin of Mathematical Biology, 80 (2018), 954-970.  doi: 10.1007/s11538-017-0261-x.  Google Scholar

[17]

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

[18]

D. Horvath and B. Brutovsky, Study of selected phenotype switching strategies in time varying environment, Physics Letters A, 380 (2016), 1267-1278.  doi: 10.1016/j.physleta.2016.01.037.  Google Scholar

[19]

J. Hùska, Harnack inequality and exponential separation for oblique derivative problems on lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557.  doi: 10.1016/j.jde.2006.02.008.  Google Scholar

[20]

V. Hutson, K. Mischaikow and P. Polik, The evolution of dispersal rates in a heterogeneous time-periodic environment, Journal of Mathematical Biology, 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[21]

V. Hutson, W. Shen and G. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proceedings of the American Mathematical Society, 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[22]

E. Kisdi, Dispersal: Risk spreading versus local adaptation, The American Naturalist, 159 (2002), 579-596.  doi: 10.1086/339989.  Google Scholar

[23]

E. Kisdi, Conditional dispersal under kin competition: Extension of the hamiltonmay model to brood size-dependent dispersal, Theoretical Population Biology, 66 (2004), 369-380.  doi: 10.1016/j.tpb.2004.06.009.  Google Scholar

[24]

U. Ledzewicz, H. Schttler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering, 10 (2013), 803-819.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[25]

S. Liu, Y. Lou, R. Peng and M. Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc., 147 (2019), 5291–5302, arXiv: 1903.11757. doi: 10.1090/proc/14653.  Google Scholar

[26]

T. Lorenzi, R. H. Chisholm, L. Desvillettes and B. D. Hughes, Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments, Journal of Theoretical Biology, 386 (2015), 166-176.  doi: 10.1016/j.jtbi.2015.08.031.  Google Scholar

[27]

G. Nadin, The principal eigenvalue of a spacetime periodic parabolic operator, Annali di Matematica Pura ed Applicata, 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[28]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the spacetime periodic FisherKPP equation, European Journal of Applied Mathematics, 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.  Google Scholar

[29]

R. Peng and X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications, Calculus of Variations and Partial Differential Equations, 54 (2015), 1611-1642.  doi: 10.1007/s00526-015-0838-x.  Google Scholar

[30]

B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2007.  Google Scholar

[31]

A. O. Pisco, A. Brock, J. Zhou, A. Moor, M. Mojtahedi, D. Jackson and S. Huang, Non-Darwinian dynamics in therapy-induced cancer drug resistance, Nature Communications, 4 (2013), Article number: 2467. doi: 10.1038/ncomms3467.  Google Scholar

[32]

C. Pouchol, J. Clairambault, A. Lorz and E. Trlat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy, Journal de Mathmatiques Pures et Appliques, 116 (2018), 268-308.  doi: 10.1016/j.matpur.2017.10.007.  Google Scholar

show all references

References:
[1]

M. Alfaro and M. Veruete, Evolutionary branching via replicator mutator equations, Journal of Dynamics and Differential Equations, 31 (2019), 2029-2052.  doi: 10.1007/s10884-018-9692-9.  Google Scholar

[2]

A. Ardaševa, R. A. Gatenby, A. R. A. Anderson, H. M. Byrne, P. K. Maini and T. Lorenzi, Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments, 2019, arXiv: 1905.11712. Google Scholar

[3]

D. Basanta, H. Hatzikirou and A. Deutsch, Studying the emergence of invasiveness in tumours using game theory, The European Physical Journal B, 63 (2008), 393-397.  doi: 10.1140/epjb/e2008-00249-y.  Google Scholar

[4]

P. L. Bedard, A. R. Hansen, M. J. Ratain and L. L. Siu, Tumour heterogeneity in the clinic, Nature, 501 (2013), 355-364.  doi: 10.1038/nature12627.  Google Scholar

[5]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ-species persistence, Journal of Mathematical Biology, 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[6]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅱ-biological invasions and pulsating travelling fronts, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[7]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, Journal of Mathematical Biology, 48 (2004), 135-159.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments Ⅱ, SIAM Journal on Mathematical Analysis, 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

[9]

N. Champagnat, R. Ferrière and S. Méléard, Individual-based probabilistic models of adaptive evolution and various scaling approximations, In Seminar on Stochastic Analysis, Random Fields and Applications V, Springer, 59 (2008), 75–113. doi: 10.1007/978-3-7643-8458-6_6.  Google Scholar

[10]

N. Champagnat and A. Lambert, Evolution of discrete populations and the canonical diffusion of adaptive dynamics, The Annals of Applied Probability, 17 (2007), 102-155.  doi: 10.1214/105051606000000628.  Google Scholar

[11]

J. Coville., Convergence to equilibrium for positive solutions of some mutation-selection model., arXiv: 1308.6471 [math], 2013, arXiv: 1308.6471. Google Scholar

[12]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

[13]

T. Epstein, R. A. Gatenby and J. S. Brown, The Warburg effect as an adaptation of cancer cells to rapid fluctuations in energy demand, PLOS ONE, 12 (2017), e0185085. doi: 10.1371/journal.pone.0185085.  Google Scholar

[14]

S. Figueroa Iglesias and S. Mirrahimi, Long time evolutionary dynamics of phenotypically structured populations in time-periodic environments, SIAM Journal on Mathematical Analysis, 50 (2018), 5537-5568.  doi: 10.1137/18M1175185.  Google Scholar

[15]

S. Geritz, Evolution and the theory of games, Evolution and the Theory of Games, 2011, page 6. Google Scholar

[16]

C. A. Gravenmier, M. Siddique and R. A. Gatenby, Adaptation to stochastic temporal variations in intratumoral blood flow: The warburg effect as a bet hedging strategy, Bulletin of Mathematical Biology, 80 (2018), 954-970.  doi: 10.1007/s11538-017-0261-x.  Google Scholar

[17]

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

[18]

D. Horvath and B. Brutovsky, Study of selected phenotype switching strategies in time varying environment, Physics Letters A, 380 (2016), 1267-1278.  doi: 10.1016/j.physleta.2016.01.037.  Google Scholar

[19]

J. Hùska, Harnack inequality and exponential separation for oblique derivative problems on lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557.  doi: 10.1016/j.jde.2006.02.008.  Google Scholar

[20]

V. Hutson, K. Mischaikow and P. Polik, The evolution of dispersal rates in a heterogeneous time-periodic environment, Journal of Mathematical Biology, 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[21]

V. Hutson, W. Shen and G. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proceedings of the American Mathematical Society, 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[22]

E. Kisdi, Dispersal: Risk spreading versus local adaptation, The American Naturalist, 159 (2002), 579-596.  doi: 10.1086/339989.  Google Scholar

[23]

E. Kisdi, Conditional dispersal under kin competition: Extension of the hamiltonmay model to brood size-dependent dispersal, Theoretical Population Biology, 66 (2004), 369-380.  doi: 10.1016/j.tpb.2004.06.009.  Google Scholar

[24]

U. Ledzewicz, H. Schttler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering, 10 (2013), 803-819.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[25]

S. Liu, Y. Lou, R. Peng and M. Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc., 147 (2019), 5291–5302, arXiv: 1903.11757. doi: 10.1090/proc/14653.  Google Scholar

[26]

T. Lorenzi, R. H. Chisholm, L. Desvillettes and B. D. Hughes, Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments, Journal of Theoretical Biology, 386 (2015), 166-176.  doi: 10.1016/j.jtbi.2015.08.031.  Google Scholar

[27]

G. Nadin, The principal eigenvalue of a spacetime periodic parabolic operator, Annali di Matematica Pura ed Applicata, 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[28]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the spacetime periodic FisherKPP equation, European Journal of Applied Mathematics, 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.  Google Scholar

[29]

R. Peng and X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications, Calculus of Variations and Partial Differential Equations, 54 (2015), 1611-1642.  doi: 10.1007/s00526-015-0838-x.  Google Scholar

[30]

B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2007.  Google Scholar

[31]

A. O. Pisco, A. Brock, J. Zhou, A. Moor, M. Mojtahedi, D. Jackson and S. Huang, Non-Darwinian dynamics in therapy-induced cancer drug resistance, Nature Communications, 4 (2013), Article number: 2467. doi: 10.1038/ncomms3467.  Google Scholar

[32]

C. Pouchol, J. Clairambault, A. Lorz and E. Trlat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy, Journal de Mathmatiques Pures et Appliques, 116 (2018), 268-308.  doi: 10.1016/j.matpur.2017.10.007.  Google Scholar

Figure 1.  Proliferation and sensitivity functions chosen for the simulations
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Final mean populations $ \bar{\rho}_N $ represented as functions of the mutation rate $ D $ for various treatment schedules. The treatments are of the form $ C(t) = \mathbf{1}_{0\leq t \leq \tau}M/\tau $, with $ M = 2 $ and $ \tau $ varying between $ 1/4 $ and $ T = 2 $. Populations $ A $, $ B $ and $ C $ are detailed in figure 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Populations $ A $, $ B $ and $ C $ of figure 2 are detailed for $ \tau = 1/4 $. We represent the population repartition in phenotypes just before treatment, just after treatment and the mean population
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Illustration of the conjecture that $ \bar{\rho}_N $ is a decreasing function of the time of drug administration $ \tau $. These numerical simulations were performed for $ T = 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Illustration of Proposition 4 on the influence of $ T $ on the population size. These numerical simulations were performed for $ \tau = T/2 $ and a fixed diffusion coefficient $ D = 0.1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Mean limit population $ \bar{\rho}_N $ as a function of both diffusion and drug dosage per period $ M $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Populations for random changing of environment with same mean value
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[2]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[3]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517
[4]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069
[5]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17
[6]

Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047
[7]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[8]

Pierre-A. Vuillermot. On the time evolution of Bernstein processes associated with a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1073-1090. doi: 10.3934/dcdsb.2018142
[9]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
[10]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039
[11]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[12]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[13]

Jin Liang, James H. Liu, Ti-Jun Xiao. Condensing operators and periodic solutions of infinite delay impulsive evolution equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023
[14]

Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277
[15]

Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5
[16]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160
[17]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[18]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[19]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[20]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (65)
  • HTML views (270)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences