Skip to main content
SpringerLink
Log in

Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii), and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimisation of free energy functionals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Akin, E.: The Geometry of Population Genetics. Lecture Notes in Biomathematics, vol. 31. Springer, Berlin (1979)

    MATH  Google Scholar 

  2. Akin, E.: The differential geometry of population genetics and evolutionary games. In: Lessard, S. (ed.) Mathematical and Statistical Developments of Evolutionary Theory. Proc. Semin., Montréal, Canada, 1987. NATO ASI Ser., Ser. C, vol. 299, pp. 1–93. Kluwer Academic, Dordrecht (1990)

    Google Scholar 

  3. Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Modelling and Optimisation of Flows on Networks, Cetraro, Italy, 2009 Lecture Notes in Mathematics, vol. 2062, pp. 1–155. Springer, Berlin (2013)

    Chapter  MATH  Google Scholar 

  4. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon, Oxford (2000)

    MATH  Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Birkhäuser, Basel (2008)

    MATH  Google Scholar 

  6. Antonelli, P.L., Strobeck, C.: The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Probab. 9(2), 238–249 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Athreya, K.B., Ney, P.E.: Branching Processes, vol. 196. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  8. Behera, N.: Variational principles in evolution. Bull. Math. Biol. 58(1), 175–202 (1996)

    Article  MATH  Google Scholar 

  9. Bellomo, N., Delitala, M.: From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells. Phys. Life Rev. 5(4), 183–206 (2008)

    Article  Google Scholar 

  10. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Blanchet, A., Laurencot, P.: The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in \(\mathbb{R}^{d}\), \(d\ge 3\). Commun. Partial Differ. Equ. 38(4), 658–686 (2013)

    Article  MATH  Google Scholar 

  12. Bürger, R.: The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester (2000)

    MATH  Google Scholar 

  13. Carlier, G., Duval, V., Peyré, G., Schmitzer, B.: Convergence of entropic schemes for optimal transport and gradient flows. SIAM J. Math. Anal. 49(2), 1385–1418 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cattiaux, P., Méléard, S.: Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60(6), 797–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., San Martín, J., et al.: Quasi-stationary distributions and diffusion models in population dynamics. Ann. Appl. Probab. 37(5), 1926–1969 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Cavalli-Sforza, L.L., Bodmer, W.F.: The Genetics of Human Populations. Freeman, New York (1971)

    Google Scholar 

  17. Chalub, F.A.C.C., Souza, M.O.: From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics. Theor. Popul. Biol. 76(4), 268–277 (2009)

    Article  MATH  Google Scholar 

  18. Chalub, F.A.C.C., Souza, M.O.: A non-standard evolution problem arising in population genetics. Commun. Math. Sci. 7(2), 489–502 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chalub, F.A.C.C., Souza, M.O.: The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations. J. Math. Biol. 68(5), 1089–1133 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chalub, F.A.C.C., Souza, M.O.: Fixation in large populations: a continuous view of a discrete problem. J. Math. Biol. 72(1–2), 283–330 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chalub, F.A.C.C., Souza, M.O.: On the stochastic evolution of finite populations. J. Math. Biol. 75(6–7), 1735–1774 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chalub, F.A.C.C., Souza, M.O.: Fitness potentials and qualitative properties of the Wright-Fisher dynamics. J. Theor. Biol. 457, 57–65 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Champagnat, N., Villemonais, D.: Exponential convergence to quasi-stationary distribution and \(Q\)-process. Probab. Theory Relat. 164(1–2), 243–283 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Champagnat, N., Ferrière, R., Méléard, S.: Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69(3), 297–321 (2006)

    Article  MATH  Google Scholar 

  25. Chow, S.-N., Huang, W., Li, Y., Zhou, H.: Fokker–Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203(3), 969–1008 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chui, C.K.: Concerning rates of convergence of Riemann sums. J. Approx. Theory 4(3), 279–287 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  27. Collet, P., Martínez, S., Martín, J.S.: Quasi-Stationary Distributions. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  28. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Proceedings of the 26th International Conference on Neural Information Processing Systems, USA, vol. 2, NIPS’13, pp. 2292–2300 (2013). Curran Associates Inc.

    Google Scholar 

  29. Dal Maso, G.: An Introduction to \(\varGamma \)-Convergence, vol. 8. Birkhäuser, Basel (1993)

    Book  MATH  Google Scholar 

  30. Danilkina, O., Souza, M.O., Chalub, F.A.C.C.: Conservative parabolic problems: nondegenerated theory and degenerated examples from population dynamics. Math. Methods Appl. Sci. 41(12), 4391–4406 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. De Giorgi, E.: New problems on minimizing movements. In: Boundary Value Problems for Partial Differential Equations and Applications. Dedicated to Enrico Magenes on the Occasion of his 70th Birthday, pp. 81–98. Masson, Paris (1993)

    Google Scholar 

  32. De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei 8(68), 180–187 (1980)

    MathSciNet  MATH  Google Scholar 

  33. DeAngelis, D.L., Mooij, W.M.: Individual-based modeling of ecological and evolutionary processes. Annu. Rev. Ecol. Evol. Syst. 36, 147–168 (2005)

    Article  Google Scholar 

  34. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, vol. 38, 2nd edn. Springer, Berlin (2010), corrected 2nd printing edition

    Book  MATH  Google Scholar 

  35. Di Francesco, M., Fagioli, S.: Measure solutions for non-local interaction PDEs with two species. Nonlinearity 26(10), 2777–2808 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Disser, K., Liero, M.: On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Netw. Heterog. Media 10(2), 233–253 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Edwards, A.W.F.: Likelihood. Cambridge University Press, Cambridge (1972)

    MATH  Google Scholar 

  38. Erbar, M., Fathi, M., Laschos, V., Schlichting, A.: Gradient flow structure for McKean-Vlaslov equations on discrete spaces. Discrete Contin. Dyn. Syst. 3612, 6799–6833 (2016)

    MATH  Google Scholar 

  39. Etheridge, A.: Some Mathematical Models from Population Genetics: École D’Été de Probabilités de Saint-Flour XXXIX-2009, vol. 2012. Springer, Berlin (2011)

    MATH  Google Scholar 

  40. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley, Hoboken (1986)

    Book  MATH  Google Scholar 

  41. Ewens, W.: An optimizing principle of natural selection in evolutionary population genetics. Theor. Popul. Biol. 42(3), 333–346 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ewens, W.J.: Mathematical Population Genetics. I: Theoretical Introduction, 2nd edn. Springer, New York (2004)

    Book  MATH  Google Scholar 

  43. Ewens, W.J.: What is the gene trying to do? Br. J. Philos. Sci. 62(1), 155–176 (2011)

    Article  MathSciNet  Google Scholar 

  44. Ewens, W.J., Lessard, S.: On the interpretation and relevance of the fundamental theorem of natural selection. Theor. Popul. Biol. 104, 59–67 (2015)

    Article  MATH  Google Scholar 

  45. Figalli, A., Gigli, N.: A new transportation distance between non-negative measures with applications to gradient flows with Dirichlet boundary conditions. J. Math. Pures Appl. 94(2), 107–130 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–341 (1922)

    Article  Google Scholar 

  47. Fisher, R.A.: The Genetical Theory of Natural Selection. Clarendon, Oxford (1930)

    Book  MATH  Google Scholar 

  48. Furuichi, S., Yanagi, K., Kuriyama, K.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45(12), 4868–4877 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gigli, N., Maas, J.: Gromov–Hausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal. 45(2), 879–899 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. Gladbach, P., Kopfer, E., Maas, J.: Scaling limits of discrete optimal transport, arXiv preprint (2018). arXiv:1809.01092

  51. Gladbach, P., Kopfer, E., Maas, J., Portinale, L.: Homogenisation of one-dimensional discrete optimal transport, arXiv preprint (2019). arXiv:1905.05757

  52. Grafen, W.J.E.: The Price equations, fitness maximization, optimisation and the fundamental theorem of natural selection. Biol. Philos. 29(2), 197–205 (2014)

    Article  MathSciNet  Google Scholar 

  53. Gyllenberg, M., Parvinen, K.: Necessary and sufficient conditions for evolutionary suicide. Bull. Math. Biol. 63(5), 981–993 (2001)

    Article  MATH  Google Scholar 

  54. Gyllenberg, M., Service, R.: Necessary and sufficient conditions for the existence of an optimisation principle in evolution. J. Math. Biol. 62(3), 359–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Gyllenberg, M., et al.: Necessary and sufficient conditions for the existence of an optimisation principle in evolution. J. Math. Biol. 62(3), 359–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. Gyllenberg, M., Metz, J.A.J.H., Service, R.: When do optimisation arguments make evolutionary sense? In: Chalub, F.A.C.C., Rodrigues, J.F. (eds.) The Mathematics of Darwin’s Legacy, Mathematics and Biosciences in Interaction, pp. 233–268. Birkhäuser, Basel (2011)

    Google Scholar 

  57. Harper, M.: Escort evolutionary game theory. Physica D 240(18), 1411–1415 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  58. Hauert, C., Doebeli, M.: Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428(6983), 643 (2004)

    Article  Google Scholar 

  59. Hofbauer, J.: The selection mutation equation. J. Math. Biol. 23(1), 41–53 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  60. Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  61. Hofbauer, J., Sigmund, K.: Evolutionary game dynamics. Bull. Am. Math. Soc. 40(4), 479–519 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  62. Hofrichter, J., Jost, J., Tran, T.D.: Information geometry and population genetics. The mathematical structure of the Wright-Fisher model. Springer, Cham (2017)

    Book  MATH  Google Scholar 

  63. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  64. Jones, W.: Variational principles for entropy production and predictive statistical mechanics. J. Phys. A, Math. Gen. 16(15), 3629 (1983)

    Article  MATH  Google Scholar 

  65. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  66. Karev, G.P., Koonin, E.V.: Parabolic replicator dynamics and the principle of minimum Tsallis information gain. Biol. Direct 8, 19 (2013)

    Article  Google Scholar 

  67. Karlin, S., Taylor, H.E.: A Second Course in Stochastic Processes. Elsevier, Amsterdam (1981)

    MATH  Google Scholar 

  68. Kelly, F.P.: Reversibility and Stochastic Networks. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  69. Kimura, M.: On the change of population fitness by natural selection. Heredity 12, 145–167 (1958)

    Article  Google Scholar 

  70. Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47, 713–719 (1962)

    Article  Google Scholar 

  71. Kinderlehrer, D., Monsaingeon, L., Xu, X.: A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations. ESAIM Control Optim. Calc. Var. 23(1), 137–164 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  72. Kisdi, É.: Frequency dependence versus optimization. Trends Ecol. Evol. 13(12), 508 (1998)

    Article  Google Scholar 

  73. Laguzet, L.: High order variational numerical schemes with application to Nash-MFG vaccination games. Ric. Mat. 67(1, SI), 247–269 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  74. Lamperti, J., Ney, P.: Conditioned branching processes and their limiting diffusions. Theory Probab. Appl. 13(1), 128–139 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  75. Laurençot, P., Matioc, B.-V.: A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differ. Equ. 47(1–2), 319–341 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  76. Léonard, C.: From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. 262(4), 1879–1920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  77. Lisini, S.: Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM Control Optim. Calc. Var. 15(3), 712–740 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  78. Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  79. Manturov, O.V.: The product integral. J. Sov. Math. 55(5), 2042–2076 (1991)

    Article  MATH  Google Scholar 

  80. Mariani, M.: A \(\varGamma \)-convergence approach to large deviations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18(3), 951–976 (2018)

    MathSciNet  MATH  Google Scholar 

  81. Martyushev, L.M., Seleznev, V.D.: Maximum entropy production principle in physics, chemistry and biology. Phys. Rep. 426(1), 1–45 (2006)

    Article  MathSciNet  Google Scholar 

  82. Meszena, G., Kisdi, É., Dieckmann, U., Geritz, S.A., Metz, J.A.: Evolutionary optimisation models and matrix games in the unified perspective of adaptive dynamics. Selection 2(1–2), 193–220 (2002)

    Article  Google Scholar 

  83. Metz, J.A.J., Mylius, S.D., Diekmann, O.: When does evolution optimize? On the relation between types of density dependence and evolutionarily stable life history parameters. In: IIASA Working Paper WP-96-004, IIASA, Laxenburg, Austria (1996)

    Google Scholar 

  84. Metz, J., Mylius, S., Diekmann, O.: When does evolution optimize? Evol. Ecol. Res. 10, 629–654 (2008)

    Google Scholar 

  85. Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differ. Equ. 48(1–2), 1–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  86. Mielke, A.: On evolutionary \(\varGamma \)-convergence for gradient systems. In: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, pp. 187–249. Springer, Berlin (2016)

    Chapter  Google Scholar 

  87. Mielke, A., Montefusco, A., Peletier, M.A.: Exploring families of energy-dissipation landscapes via tilting–three types of EDP convergence, arXiv preprint (2020). arXiv:2001.01455

  88. Moran, P.: The Statistical Processes of Evolutionary Theory. Clarendon, Oxford (1962)

    MATH  Google Scholar 

  89. Mylius, S.D., Metz, J.A.J.: When does evolution optimize? On the relationship between evolutionary stability, optimization and density dependence. In: Elements of Adaptive Dynamics. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  90. Norris, J.R.: Markov Chains. Reprint, vol. 2. Cambridge University Press, Cambridge (1998), reprint edition

    MATH  Google Scholar 

  91. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  92. Reda, F.A., Maury, B.: Interpretation of finite volume discretization schemes for the Fokker-Planck equation as gradient flows for the discrete Wasserstein distance. In: Topological Optimization and Optimal Transport in the Applied Sciences, pp. 400–416. de Gruyter, Berlin (2017)

    Chapter  MATH  Google Scholar 

  93. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Anal. 57(12), 1627–1672 (2004)

    MathSciNet  MATH  Google Scholar 

  94. Santambrogio, F.: Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, vol. 87. Springer, Berlin (2015)

    MATH  Google Scholar 

  95. Santambrogio, F.: {Euclidean, metric, and Wasserstein} gradient flows: an overview. Bull. Math. Sci. 7(1), 87–154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  96. Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100(3), 533–538 (1983)

    Article  MathSciNet  Google Scholar 

  97. Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31(4), 1427–1451 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  98. Shahshahani, S.: A new mathematical framework for the study of linkage and selection. Mem. Am. Math. Soc. 211, 34 (1979)

    MathSciNet  MATH  Google Scholar 

  99. Sigmund, K.: Game dynamics, mixed strategies, and gradient systems. Theor. Popul. Biol. 32(1), 114–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  100. Sigmund, K.: A maximum principle for frequency dependent selection. Math. Biosci. 84(2), 189–195 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  101. Sigmund, K., Nowak, M.A.: Evolutionary game theory. Curr. Biol. 9(14), R503–R505 (1999)

    Article  Google Scholar 

  102. Smith, J.M.: Optimization theory in evolution. Annu. Rev. Ecol. Syst. 9(1), 31–56 (1978)

    Article  Google Scholar 

  103. Sober, E., Steel, M.: Entropy increase and information loss in Markov models of evolution. Biol. Philos. 26(2), 223–250 (2011)

    Article  Google Scholar 

  104. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Text in Applied Mathematics, vol. 12. Springer, New York (2002)

    Book  MATH  Google Scholar 

  105. Taylor, P.D., Jonker, L.B.: Evolutionarily stable strategies and game dynamics. Math. Biosci. 40(1–2), 145–156 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  106. Veloz, T., Razeto-Barry, P., Dittrich, P., Fajardo, A.: Reaction networks and evolutionary game theory. J. Math. Biol. 68(1–2), 181–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  107. Villani, C.: Topics in Optimal Transportation, vol. 58. Am. Math. Soc. Providence (2003)

    MATH  Google Scholar 

  108. Villani, C.: Optimal Transport. Old and New, vol. 338. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  109. Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge (1997)

    MATH  Google Scholar 

  110. Wright, S.: Evolution in Mendelian populations. Genetics 16(2), 0097 (1931)

    Article  Google Scholar 

  111. Xuan, L.V., Lan, N.T., Viet, N.A.: On application of non-extensive statistical mechanics to studying ecological diversity. J. Phys. Conf. Ser. 726, 012024 (2016)

    Article  Google Scholar 

  112. Zettl, A.: Sturm-Liouville Theory. Am. Math. Soc., Providence (2005)

    MATH  Google Scholar 

  113. Ziman, J.: The general variational principle of transport theory. Can. J. Phys. 34(12A), 1256–1273 (1956)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

FACCC and AMR were partially supported by FCT/Portugal Strategic Project UID/MAT/00297/2019 (Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa). FACCC also benefited from an “Investigador FCT” grant. LM was supported by FCT/Portugal projects PTDC/MAT-STA/0975/2014 and PTDC/MAT-STA/28812/2017. MOS was partially supported by CNPq under grants # 309079/2015-2, 310293/2018-9 and by CAPES — Finance Code 001. We also thank an anonymous reviewer for many comments that helped us to improve the paper.

Author information

Authors and Affiliations

  1. Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Fabio A. C. C. Chalub & Ana Margarida Ribeiro

  2. IECL Université de Lorraine, 54506, Vandœuvre-lès-Nancy Cedex, France

    Léonard Monsaingeon

  3. Grupo de Física Matemática, GFMUL, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Léonard Monsaingeon

  4. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, Campus do Gragoatá, Niterói, RJ, 24210-201, Brazil

    Max O. Souza

Authors
  1. Fabio A. C. C. Chalub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Léonard Monsaingeon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ana Margarida Ribeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Max O. Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio A. C. C. Chalub.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chalub, F.A.C.C., Monsaingeon, L., Ribeiro, A.M. et al. Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective. Acta Appl Math 171, 24 (2021). https://doi.org/10.1007/s10440-021-00391-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-021-00391-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation