Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

References

• S. Baigent, Lotka-Volterra Dynamics: an introduction, preprint (2010).
   Google Scholar
• G. Barles, S. Mirrahimi, and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result, Methods Appl. Anal. 16(3) (2009), pp. 321–340.
   Google Scholar
• O. Bonnefon, J. Coville, and G. Legendre, Concentration phenomenon in some non-local equation, preprint (2015). Available at arXiv:1510.01971.
   Google Scholar
• B. Brutovsky and D. Horvath, Structure of intratumor heterogeneity: Is cancer hedging its bets? (2013). arxiv, page 1307.0607.
   Google Scholar
• J.-E. Busse, P. Gwiazda, A. Marciniak-Czochra, Mass concentration in a nonlocal model of clonal selection. J. Math. Biol. 73 (2016), pp. 1–33. doi: 10.1007/s00285-016-0979-3
   Google Scholar
• J.A. Cañizo, J.A. Carrillo, and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math. 123(1) (2013), pp. 141–156. doi: 10.1007/s10440-012-9758-3
   Google Scholar
• N. Champagnat, R. Ferrière, and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models 24(S1) (2008), pp. 2–44. doi: 10.1080/15326340802437710
   Google Scholar
• N. Champagnat, P.-E. Jabin, and G. Raoul, Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems, Comptes Rendus Math. 348(23) (2010), pp. 1267–1272. doi: 10.1016/j.crma.2010.11.001
   Google Scholar
• R.H. Chisholm, T. Lorenzi, and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimization, Biochimic. Biophys. Acta (BBA) - General Subjects 1860(11) (Nov 2016), pp. 2627–2645. doi: 10.1016/j.bbagen.2016.06.009
   Google Scholar
• R.H. Chisholm, T. Lorenzi, and A. Lorz, Effects of an advection term in nonlocal lotka-volterra equations, Commun. Math. Sci. 14 (2016), pp. 1181–8. doi: 10.4310/CMS.2016.v14.n4.a16
   Google Scholar
• R.H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L.N. de Almeida, A. Escargueil, and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation, Cancer Res. 75(6) (2015), pp. 930–939. doi: 10.1158/0008-5472.CAN-14-2103
   Google Scholar
• J. Coville, Convergence to equilibrium for positive solutions of some mutation-selection model, preprint (2013). Available at arXiv:1308.6471.
   Google Scholar
• J. Coville and F. Fabre, Convergence to the equilibrium in a Lotka-Volterra ODE competition system with mutations, preprint (2013). Available at arXiv:1301.6237.
   Google Scholar
• L. Desvillettes, P.E. Jabin, S. Mischler, and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci. 6(3) (2008), pp. 729–747. doi: 10.4310/CMS.2008.v6.n3.a10
   Google Scholar
• O. Diekmann, A beginner's guide to adaptive dynamics, Banach Center Publications 63 (2004), pp. 47–86.
   Google Scholar
• O. Diekmann, M. Gyllenberg, and J. Metz, Steady-state analysis of structured population models, Theor. Popul. Biol. 63(4) (2003), pp. 309–338. doi: 10.1016/S0040-5809(02)00058-8
   Google Scholar
• L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, Boca Raton, 2015.
   Google Scholar
• B. Goh, Stability in models of mutualism. Am. Nat. 113 (1979), pp. 261–275. doi: 10.1086/283384
   Google Scholar
• B.S. Goh, Global stability in many-species systems. Am. Nat. 111 (1977), pp. 135–143. doi: 10.1086/283144
   Google Scholar
• J. Greene, O. Lavi, M.M. Gottesman, and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol. 76(3) (2014), pp. 627–653. doi: 10.1007/s11538-014-9936-8
   Google Scholar
• J.M. Greene, D. Levy, K.L. Fung, P.S. Souza, M.M. Gottesman, and O. Lavi, Modeling intrinsic heterogeneity and growth of cancer cells, J. Theor. Biol. 367 (2015), pp. 262–277. doi: 10.1016/j.jtbi.2014.11.017
   Google Scholar
• M. Gyllenberg and G. Meszéna, On the impossibility of coexistence of infinitely many strategies, J. Math. Biol. 50(2) (2005), pp. 133–160. doi: 10.1007/s00285-004-0283-5
   Google Scholar
• F. Hamel and L. Ryzhik, On the nonlocal fisher–kpp equation: Steady states, spreading speed and global bounds, Nonlinearity 27(11) (2014), pp. 2735. doi: 10.1088/0951-7715/27/11/2735
   Google Scholar
• P.-E. Jabin and G. Raoul, On selection dynamics for competitive interactions, J. Math. Biol. 63(3) (2011), pp. 493–517. doi: 10.1007/s00285-010-0370-8
   Google Scholar
• H. Leman, S. Méléard, and S. Mirrahimi, Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system, Discrete Continuous Dyn. Syst. Ser. B. J. Bridging Math. Sci. 20(2) (2015), pp. 469–493. doi: 10.3934/dcdsb.2015.20.469
   Google Scholar
• A. Lorz, T. Lorenzi, M.E. Hochberg, J. Clairambault, and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Math. Model. Numer. Anal. 47(2) (2013), pp. 377–399. doi: 10.1051/m2an/2012031
   Google Scholar
• A. Lorz, S. Mirrahimi, and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Commun. Part. Diff. Equ. 36(6) (2011), pp. 1071–1098. doi: 10.1080/03605302.2010.538784
   Google Scholar
• A.J. Lotka, Elements of physical biology, (1924). Reprinted 1956 as elements of mathematical biology.
   Google Scholar
• J.A. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations, Vol. 68, Springer, Berlin, 2014.
   Google Scholar
• S. Mirrahimi and B. Perthame, Asymptotic analysis of a selection model with space, J. Math. Pures Appl. 104(6) (2015), pp. 1108–1118. doi: 10.1016/j.matpur.2015.07.006
   Google Scholar
• J.D. Murray, Mathematical Biology I: An Introduction, Vol. 17 of Interdisciplinary Applied Mathematics, 2002.
   Google Scholar
• B. Perthame, Transport Equations in Biology, Springer Science & Business Media, Basel, 2006.
   Google Scholar
• R.J. Plemmons, M-matrix characterizations. I-nonsingular M-matrices, Linear Algebra Appl. 18(2) (1977), pp. 175–188. doi: 10.1016/0024-3795(77)90073-8
   Google Scholar
• C. Pouchol, J. Clairambault, A. Lorz, E. Trélat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy. J. Math. Pures Appl. 116 (2017), pp. 268–308. doi: 10.1016/j.matpur.2017.10.007
   Google Scholar
• R. Vance, Predation and resource partitioning in one predator – two prey model communities. Am. Nat. 112 (1978), pp. 797–813. doi: 10.1086/283324
   Google Scholar
• V. Volterra and M. Brelot, Leçons sur la théorie mathématique de la lutte pour la vie, Vol. 1, Gauthier-Villars, Paris, 1931.
   Google Scholar
• J.X. Zhou, A.O. Pisco, H. Qian, and S. Huang, Nonequilibrium population dynamics of phenotype conversion of cancer cells, PloS one 9(12) (2014), pp. e110714. doi: 10.1371/journal.pone.0110714
   Google Scholar