
-
Previous Article
Approximating exit times of continuous Markov processes
- DCDS-B Home
- This Issue
-
Next Article
Assignability of dichotomy spectra for discrete time-varying linear control systems
Influence of mutations in phenotypically-structured populations in time periodic environment
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. |
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. 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. |
[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. 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. |
[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. 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. |
[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. |






[1] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
[2] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440 |
[3] |
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 |
[4] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[5] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282 |
[6] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020109 |
[7] |
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398 |
[8] |
Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021006 |
[9] |
Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 |
[10] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[11] |
Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 |
[12] |
Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020383 |
[13] |
Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171 |
[14] |
Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274 |
[15] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[16] |
Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 |
[17] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174 |
[18] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[19] |
Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021026 |
[20] |
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]