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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  September 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075
References:
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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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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.

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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.

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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.

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[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

Figure 1.  Proliferation and sensitivity functions chosen for the simulations
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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
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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
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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 $
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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 $
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Figure 6.  Mean limit population $ \bar{\rho}_N $ as a function of both diffusion and drug dosage per period $ M $
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Figure 7.  Populations for random changing of environment with same mean value
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