Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A Dynamic Game Approach to Uninvadable Strategies for Biotrophic Pathogens

  • 42 Accesses

Abstract

This paper studies a zero-sum state-feedback game for a system of nonlinear ordinary differential equations describing one-seasonal dynamics of two biotrophic fungal cohorts within a common host plant. From the perspective of adaptive dynamics, the cohorts can be interpreted as resident and mutant populations. The invasion functional takes the form of the difference between the two marginal fitness criteria and represents the cost in the definition of the value of the differential game. The presence of a specific competition term in both equations and marginal fitnesses substantially complicates the reduction in the game to a two-step problem that can be solved by using optimal control theory. Therefore, a general game-theoretic formulation involving uninvadable strategies has to be considered. First, the related Cauchy problem for the Hamilton–Jacobi–Isaacs equation is investigated analytically by the method of characteristics. A number of important properties are rigorously derived. However, the complete theoretical analysis still remains an open challenging problem due to the high complexity of the differential game. That is why an ad hoc conjecture is additionally proposed. An informal but rather convincing and practical justification for the latter relies on numerical simulation results. We also establish some asymptotic properties and provide biological interpretations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. 1.

    Akhmetzhanov AR, Grognard F, Mailleret L, Bernhard P (2012) Join forces or cheat: evolutionary analysis of a consumer-resource system. In: Advances in dynamic games, volume 12 of the series Annals of the International Society of Dynamic Games, Springer, New York, pp 73–95

  2. 2.

    Akhmetzhanov AR, Grognard F, Mailleret L (2011) Optimal life-history strategies in seasonal consumer-resource dynamics. Evolution 65(11):3113–3125

  3. 3.

    Bancal MO, Hansart A, Sache I, Bancal P (2012) Modelling fungal sink competitiveness with grains for assimilates in wheat infected by a biotrophic pathogen. Ann Bot 110(1):113–123

  4. 4.

    Berkovitz LD (1985) The existence of value and saddle point in games of fixed duration. SIAM J Control Optim 23(2):172–196

  5. 5.

    Bernhard P (2014) Pursuit-evasion games and zero-sum two-person differential games. Encycl Syst Control. https://doi.org/10.1007/978-1-4471-5102-9_270-1

  6. 6.

    Bernhard P (1977) Singular surfaces in differential games: an introduction. In: Hagedorn P, Knobloch HW, Olsder GJ (eds) Differential games and applications, volume 3 of the series Lecture Notes in Control and Information Sciences. Springer, Berlin, pp 1–33

  7. 7.

    Bernhard P (1987) Differential games: closed loop. In: Singh MG (ed) Systems & control encyclopedia: Theory, technology, applications. Pergamon Press, Oxford, New York, pp 1004–1009

  8. 8.

    Bernhard P (1987) Differential games: Isaacs equation. In: Singh MG (ed) Systems & control encyclopedia: theory, technology, applications. Pergamon Press, Oxford, New York, pp 1010–1016

  9. 9.

    Bernhard P (2015) Evolutionary dynamics of the handicap principle: an example. Dyn Games Appl 5:214–227

  10. 10.

    Bernhard P, Grognard F, Mailleret L, Akhmetzhanov A (2010) ESS for life-history traits of cooperating consumers facing cheating mutants. [Research Report] RR–7314, INRIA. https://hal.inria.fr/inria-00491489v2

  11. 11.

    Bokanowski O, Desilles A, Zidani H, Zhao J (2017) User’s guide for the ROC-HJ solver. May 10. Version 2.3. https://uma.ensta-paristech.fr/soft/ROC-HJ

  12. 12.

    Botkin ND, Hoffmann K-H, Turova VL (2011) Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J Sci Comput 33(2):992–1007

  13. 13.

    Boyle B, Hamelin RC, Séguin A (2005) In vivo monitoring of obligate biotrophic pathogen growth by kinetic PCR. Appl Environ Microbiol 71(3):1546–1552

  14. 14.

    Clarke FH, Ledyaev YuS, Stern RJ, Wolenski PR (1998) Nonsmooth analysis and control theory. Springer, New York

  15. 15.

    Crandall MG, Lions P-L (1984) Two approximations of solutions of Hamilton–Jacobi equations. Math Comput 43:1–19

  16. 16.

    Day T (2001) Parasite transmission modes and the evolution of virulence. Evolution 55:2389–2400

  17. 17.

    Day T (2003) Virulence evolution and the timing of disease life-history events. Trends Ecol Evol 18:113–118

  18. 18.

    Deacon JW (1997) Modern mycology. Blackwell Scientific, Oxford

  19. 19.

    Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton University Press, Princeton

  20. 20.

    Dieckmann U, Heino M, Parvinen K (2006) The adaptive dynamics of function-valued traits. J Theor Biol 241:370–389

  21. 21.

    Elliott RJ, Kalton NJ (1972) The existence of value in differential games. Mem Am Math Soc 126:1–67

  22. 22.

    Eshel I, Motro U (1981) Kin selection and strong stability of mutual help. Theor Popul Biol 19:420–433

  23. 23.

    Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions. Springer, New York

  24. 24.

    Friedman A (1971) Differential games. Wiley, New York

  25. 25.

    Geritz SAH, Kisdi É, Meszéna G, Metz JAJ (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12(1):35–57

  26. 26.

    Gilchrist MA, Sulsky DL, Pringle A (2006) Identifying fitness and optimal life-history strategies for an asexual filamentous fungus. Evolution 60:970–979

  27. 27.

    Hahn M (2000) The rust fungi: cytology, physiology and molecular biology of infection. In: Kronstad JW (ed) Fungal pathology. Springer, Dordrecht

  28. 28.

    Isaacs R (1965) Differential games. Wiley, New York

  29. 29.

    Ivanov GE (1997) Saddle point for differential games with strongly convex-concave integrand. Math Notes 62(5):607–622

  30. 30.

    Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow In Russian

  31. 31.

    Mailleret L, Lemesle V (2009) A note on semi-discrete modelling in life sciences. Philos Trans R Soc Lond A 367:4779–4799

  32. 32.

    Maynard Smith J (1974) The theory of games and the evolution of animal conflicts. J Theor Biol 47:209–221

  33. 33.

    Maynard Smith J, Price GR (1973) The logic of animal conflicts. Nature 246:15–18

  34. 34.

    Melikyan AA (1998) Generalized characteristics of first order PDEs: application in optimal control and differential games. Birkhauser, Boston

  35. 35.

    Metz JAJ (2008) Fitness. In: Jørgensen SE, Fath BD (eds) Encyclopedia of ecology. Academic Press, Oxford, pp 1599–1612

  36. 36.

    Metz JAJ, Stankova K, Johansson J (2016) The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin’s maximum principle. J Math Biol 72:1125–1152

  37. 37.

    Murray JD (2002) Mathematical biology. I. An introduction. Interdisciplinary applied mathematics, volume 17. Springer, Berlin, Heidelberg (2002)

  38. 38.

    Nowak MA (1990) An evolutionary stable strategy may be inaccessible. J Theor Biol 142:237–241

  39. 39.

    Osher S, Shu C-W (1991) High order essentially non-oscillatory schemes for Hamilton–Jacobi equations. SIAM J Numer Anal 28(4):907–922

  40. 40.

    Parthasarathy T, Raghavan TES (1975) Existence of saddle points and Nash equilibrium points for differential games. SIAM J Control 13(5):977–980

  41. 41.

    Parvinen K, Heino M, Dieckmann U (2013) Function-valued adaptive dynamics and optimal control theory. J Math Biol 67:509–533

  42. 42.

    Pontryagin LS, Boltyansky VG, Gamkrelidze RV, Mishchenko EF (1964) The mathematical theory of optimal processes. Macmillan, New York

  43. 43.

    Précigout P-A, Claessen D, Robert C (2017) Crop fertilization impacts epidemics and optimal latent period of biotrophic fungal pathogens. Phytopathology 107:1256–1267

  44. 44.

    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing. Cambridge University Press, New York

  45. 45.

    Roxin E (1969) Axiomatic approach in differential games. J Optim Theory Appl 3(3):153–163

  46. 46.

    Sasaki A, Iwasa Y (1991) Optimal-growth schedule of pathogens within a host: switching between lytic and latent cycles. Theor Popul Biol 39:201–239

  47. 47.

    Schmitendorf WE (1970) Differential games with open-loop saddle point conditions. IEEE Trans Autom Control 15:320–325

  48. 48.

    Schmitendorf WE (1970) Existence of optimal open-loop strategies for a class of differential games. J Optim Theory Appl 5:363–375

  49. 49.

    Schmitendorf WE (1976) Differential games without pure strategy saddle-point solutions. J Optim Theory Appl 18:81–92

  50. 50.

    Shaiju A, Bernhard P (2009) Evolutionarily robust strategies: two nontrivial examples and a theorem. In: Pourtallier O, Gaitsgory V, Bernhard P (eds) Advances in dynamic games and their applications, vol 10. Annals of the International Society of Dynamic Games. Birkhäuser, Boston

  51. 51.

    Silvani VA, Bidondo LF, Bompadre MJ, Colombo RP, Pérgola M, Bompadre A, Fracchia S, Godeas A (2014) Growth dynamics of geographically different arbuscular mycorrhizal fungal isolates belonging to the ‘Rhizophagus clade’ under monoxenic conditions. Mycologia 106(5):963–975

  52. 52.

    Subbotin AI (1995) Generalized solutions of first-order PDEs: the dynamical optimization perspective. Birkhauser, Boston

  53. 53.

    Subbotin AI, Chentsov AG (1981) Optimization of guaranteed result in control problems. Nauka, Moscow In Russian

  54. 54.

    Varaiya PP (1967) On the existence for solution to a differential game. SIAM J Control 5(1):153–162

  55. 55.

    Vincent TL, Brown JS (2005) Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge

  56. 56.

    Yegorov I, Grognard F, Mailleret L, Halkett F (2017) Optimal resource allocation for biotrophic plant pathogens. IFAC-PapersOnline 50(1):3154–3159

  57. 57.

    Yong J (2015) Differential games: a concise introduction. World Scientific, Singapore

Download references

Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche 664 (ANR) under Grant ANR-13-BSV7-0011 (project FunFit).

Author information

Correspondence to Ivan Yegorov.

Additional information

Publisher's Note

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

Ivan Yegorov: Also known as I. Egorov.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 428 KB)

Supplementary material 2 (pdf 340 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yegorov, I., Grognard, F., Mailleret, L. et al. A Dynamic Game Approach to Uninvadable Strategies for Biotrophic Pathogens. Dyn Games Appl 10, 257–296 (2020). https://doi.org/10.1007/s13235-019-00307-1

Download citation

Keywords

  • Uninvadable strategy
  • Zero-sum differential game
  • Biotrophic pathogens
  • Resource allocation
  • State-feedback control
  • Hamilton–Jacobi–Isaacs equation
  • Method of characteristics
  • Finite-difference approximation