Advertisement

Nonlinear Dynamics

, Volume 98, Issue 2, pp 861–872 | Cite as

First-passage time study of a stochastic growth model

  • Claudio FlorisEmail author
Original paper
First Online:
  • 147 Downloads

Abstract

In this paper, a stochastic nonlinear growth model is proposed, which can be considered a generalization of the stochastic logistic model. It can be applied to the growth of a generic population as well as to the propagation of the fracture in engineering materials. The excitation is assumed to be a stationary Gaussian white noise stochastic process, which affects the system parametrically. A preliminary study of the stochastic differential equation governing the model reveals that there is a phase transition when the nonlinearity parameter c reaches one: when \(c<1\), the system tends to the stationary state, while it is never stationary when \(c\ge 1\). Then, attention is focused on the first-passage time problem, which is of crucial importance for dynamical systems. The first-order stochastic differential equation (SDE) that describes the model is transformed into an Itô’s SDE by adding the Wong–Zakai–Stratonovich corrective term. For the last equation, the backward Kolmogorov equation is formulated. By solving it with appropriate initial and boundary conditions, the probability of survival is obtained, that is, the probability of not exceeding a given threshold. The solution is looked for three cases \(c<1,c=1,c>1\). In any case, the numerical analyses show that the survival probability decays fast.

Keywords

Stochastic growth model Stochastic generalization of the logistic model Itô’s stochastic differential equations First-passage time Survival probability 
This is a preview of subscription content, log in to check access.

Notes

References

  1. 1.
    Malthus, T.R.: An Essay on the Principle of Population. J. Johnson, London (1798)Google Scholar
  2. 2.
    Verhulst, P.F.: Notice sur la loi que la population poursuit dans son accroissement. Correspondence Mathématique et Physique 10, 113–121 (1838)Google Scholar
  3. 3.
    Verhulst, P.F.: Recherches mathématiques sur la loi de l’accroissement de la population. Nouveaux Mémoires de l’Académie Royale de Sciences et Belles Lettres de Bruxelles 18, 1–42 (1845)Google Scholar
  4. 4.
    Ray, S., Chandrakishen, J.M.: Fatigue crack growth propagation model for plain concrete—an analogy with population growth. Eng. Fract. Mech. 77, 3418–3433 (2010)Google Scholar
  5. 5.
    Kendall, D.G.: Stochastic processes and population growth. J. R. Stat. Soc. Ser. B 11, 230–282 (1949)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Feller, W.: Diffusion processes in genetics. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 227–246 (1951)Google Scholar
  7. 7.
    Lewontin, R.C., Cohen, D.: On population growth in a randomly varying environment. Proc. Natl. Acad. Sci. 62, 1056–1060 (1969)MathSciNetGoogle Scholar
  8. 8.
    Levins, R.: The effects of random variations of different types on population growth. Proc. Natl. Acad. Sci. 62, 1061–1065 (1969)MathSciNetGoogle Scholar
  9. 9.
    Capocelli, R.M., Ricciardi, L.M.: A diffusion model for population growth in a random environment. Theor. Popul. Biol. 5, 28–41 (1974)zbMATHGoogle Scholar
  10. 10.
    Tuckwell, H.C.: A study of some diffusion models of population growth. Theor. Popul. Biol. 5, 345–357 (1974)zbMATHGoogle Scholar
  11. 11.
    Feldman, M.W., Roughgarden, J.: A population’s stationary distribution and chance of extinction in a stochastic environment with remarks on the theory of species packing. Theor. Popul. Biol. 7, 197–207 (1975)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Turelli, M.: Random environments and stochastic calculus. Theor. Popul. Biol. 12, 140–178 (1977)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Tuckwell, H.C., Koziol, J.A.: Logistic population growth under random dispersal. Bull. Math. Biol. 49, 495–506 (1987)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Artzrouni, M., Reneke, J.: Stochastic differential equations in mathematical demography: a review. Appl. Math. Comput. 39, 139–153 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ricciardi, L.M., Di Crescenzo, A., Giorno, V., Nobile, A.G.: An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological models. Math. Jpn. 50, 247–322 (1999)zbMATHGoogle Scholar
  16. 16.
    Allen, L.J.S., Allen, E.J.: A comparison of three different stochastic population models with regard to persistence time. Theor. Popul. Biol. 64, 439–449 (2003)zbMATHGoogle Scholar
  17. 17.
    Braumann, C.A.: Itô versus Stratonovich calculus in random population growth. Math. Biosci. 206, 81–107 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Braumann, C.A.: Growth and extinction of populations in randomly varying environments. Comput. Math. Appl. 56, 631–644 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Grigoriu, M.: Noise-induced transitions for random versions of Verhulst model. Probab. Eng. Mech. 38, 136–142 (2014)Google Scholar
  20. 20.
    Spencer, B.F., Tang, J.: Markov process model for fatigue crack growth. ASCE J. Eng. Mech. 114, 2134–2157 (1988)Google Scholar
  21. 21.
    Sobczyk, K.: Stochastic models for fatigue damage of materials. Adv. Appl. Probab. 19, 652–673 (1987)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Trębicki, J., Sobczyk, K.: Maximum entropy principle and non-stationary distributions of stochastic systems. Probab. Eng. Mech. 11, 169–178 (1996)zbMATHGoogle Scholar
  23. 23.
    Itô, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 289–302 (1951)Google Scholar
  24. 24.
    Itô, K.: On a formula concerning stochastic differentials. Nagoya Math. J. 3, 55–65 (1951)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Di Paola, M.: Stochastic differential calculus. In: Casciati, F. (ed.) Dynamic Motion: Chaotic and Stochastic Behaviour, pp. 29–92. Springer, Wien (1993)Google Scholar
  26. 26.
    Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw Hill, New York (1995)Google Scholar
  27. 27.
    Grigoriu, M.: Stochastic Calculus. Birkhauser, Boston (2002)zbMATHGoogle Scholar
  28. 28.
    Stratonovich, R.L.: A new representation for stochastic integrals and equations. SIAM J. Control 4, 362–371 (1966)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Stratonovich, R.L.: Topics in Theory of Random Noise. Gordon and Breach, New York (1967)zbMATHGoogle Scholar
  30. 30.
    Gray, A.H., Caughey, T.K.: A controversy in problems involving random parametric excitation. J. Math. Phys. 44, 288–296 (1965)zbMATHGoogle Scholar
  31. 31.
    Mortensen, R.E.: Mathematical problems of modeling stochastic nonlinear dynamic systems. J. Stat. Phys. 1, 271–296 (1969)Google Scholar
  32. 32.
    Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Feller, W.: Diffusion processes in one dimension. Trans. Am. Math. Soc. 77, 1–31 (1954)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Academic Press, New York (1965)zbMATHGoogle Scholar
  35. 35.
    Bharucha-Reid, A.T.: Elements of the Theory of Stochastic Processes and Their Applications. McGraw-Hill, New York (1960)zbMATHGoogle Scholar
  36. 36.
    Roberts, J.B.: First passage probabilities for randomly excited systems: diffusion methods. Probab. Eng. Mech. 1, 66–80 (1986)Google Scholar
  37. 37.
    Lennox, W.C., Fraser, D.A.: On the first-passage distribution for the envelope of a non-stationary narrow-band stochastic process. ASME J. Appl. Mech. 41, 793–797 (1974)zbMATHGoogle Scholar
  38. 38.
    Spanos, P.D.: Numerics for common first passage problem. ASCE J. Eng. Mech. 108, 864–882 (1982)Google Scholar
  39. 39.
    Spanos, P.D.: Survival probability of non-linear oscillator subjected to broad-band random disturbances. Int. J. Non-Linear Mech. 17, 303–317 (1982)zbMATHGoogle Scholar
  40. 40.
    Ronveaux, A.: Heun’s Differential Equations. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  41. 41.
    Ray, S., Chandra Kishen, J.M.: Fatigue crack propagation model for plain concrete—an analogy with population growth. Eng. Fract. Mech. 77, 3418–3433 (2010)Google Scholar
  42. 42.
    Zhu, W.Q., Lin, Y.K., Lei, Y.: On fatigue crack growth under random loading. Eng. Fract. Mech. 43, 1–12 (1992)Google Scholar
  43. 43.
    Rocha, M.M., Schuëller, G.I.: A probabilistic criterion for evaluating the goodness of fatigue crack growth models. Eng. Fract. Mech. 53, 707–731 (1996)Google Scholar
  44. 44.
    Sobczyk, K.: Modelling of random fatigue crack growth. Eng. Fract. Mech. 24, 609–623 (1986)Google Scholar
  45. 45.
    Oh, K.P.: A weakest link model for the prediction of fatigue crack growth rate. J. Eng. Mater. Technol. 100, 170–174 (1978)Google Scholar
  46. 46.
    Carpinteri, A., Paggi, M.: Self-similarity and crack growth instability in the correlation between the Paris’ constants. Eng. Fract. Mech. 74, 1041–1053 (2007)Google Scholar
  47. 47.
    Lindenberg, K., Shuler, K.E., Freeman, J., Lie, T.J.: First passage time and extremum properties of Markov and independent processes. J. Stat. Phys. 12, 217–251 (1975)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Giraudo, M.T.: A similarity solution for the Ornstein–Uhlenbeck diffusion process constrained by a reflecting and an absorbing boundary. Ricerche di Matematica XLIX, 47–63 (2000)MathSciNetGoogle Scholar
  49. 49.
    West, B.J., Bulsara, A.R., Lindenberg, K., Seshadri, V., Shuler, K.E.: Stochastic processes with non-additive fluctuations. Phys. A 97, 211–233 (1979)MathSciNetGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Civil and Enviromental Engineering (DICA)Politecnico di MilanoMilanItaly

Personalised recommendations

Cite article

Buy options