Bulletin of Mathematical Biology

, Volume 78, Issue 2, pp 254–279 | Cite as

Evolution of Dispersal with Starvation Measure and Coexistence

  • Yong-Jung Kim
  • Ohsang KwonEmail author
Original Article


Many biological species increase their dispersal rate if starvation starts. To model such a behavior, we need to understand how organisms measure starvation and response to it. In this paper, we compare three different ways of measuring starvation by applying them to starvation-driven diffusion. The evolutional selection and coexistence of such starvation measures are studied within the context of Lotka–Volterra-type competition model of two species. We will see that, if species have different starvation measures and different motility functions, both the coexistence and selection are possible.


Coexistence Evolutional selection Global asymptotic stability Linear stability Starvation driven diffusion 



This work was started when the second author was working at the Center for Partial Differential Equation in East China Normal University, Shanghai, China. He would like to thank all the hospitality and supports provided by the center during his stay.


  1. Beck M, Wayne CE (2011) Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity [reprint of mr 2551255]. SIAM Rev 53(1):129–153CrossRefMathSciNetzbMATHGoogle Scholar
  2. Cantrell RS, Cosner C (2003) Spatial ecology via reaction-diffusion equations, Wiley series in mathematical and computational biology. Wiley, ChichesterGoogle Scholar
  3. Cho E, Kim YJ (2013) Starvation driven diffusion as a survival strategy of biological organisms. Bull Math Biol 75(5):845–870CrossRefMathSciNetzbMATHGoogle Scholar
  4. Cohen D, Levin SA (1991) Dispersal in patchy environments: the effects of temporal and spatial structure. Theor Popul Biol 39(1):63–99CrossRefMathSciNetzbMATHGoogle Scholar
  5. Desvillettes L, Lepoutre T, Moussa A, Trescases A (2015) On the entropic structure of reaction-cross diffusion systems. Comm Partial Differ Equ 40(9):1705–1747CrossRefMathSciNetGoogle Scholar
  6. Dieckman U, OHara B, Weisser W (1999) The evolutionary ecology of dispersal. Trends Ecol Evol 14(3):88–90CrossRefGoogle Scholar
  7. Dockery J, Hutson V, Mischaikow K, Pernarowski M (1998) The evolution of slow dispersal rates: a reaction diffusion model. J Math Biol 37(1):61–83CrossRefMathSciNetzbMATHGoogle Scholar
  8. Hess P (1991) Periodic-parabolic boundary value problems and positivity, Pitman research notes in mathematics series, vol 247. Longman Scientific & Technical, HarlowGoogle Scholar
  9. Holt R, McPeek M (1996) Chaotic population dynamics favors the evolution of dispersal. Am Nat 148:709–718CrossRefGoogle Scholar
  10. Hsu S, Smith H, Waltman P (1996) Competitive exclusion and coexistence for competitive systems on ordered Banach spaces. Trans Am Math Soc 348(10):4083–4094CrossRefMathSciNetzbMATHGoogle Scholar
  11. Huisman G, Kolter R (2013) Sensing starvation: a homoserine lactone—dependent signaling pathway in Escherichia coli. Science 341:1236566CrossRefGoogle Scholar
  12. Hutson V, Mischaikow K, Poláčik P (2001) The evolution of dispersal rates in a heterogeneous time-periodic environment. J Math Biol 43(6):501–533CrossRefMathSciNetzbMATHGoogle Scholar
  13. Johnson M, Gaines M (1990) Evolution of dispersal: theoretical models and empirical tests using birds and mammals. Ann Rev Ecol Syst 21:449–480CrossRefGoogle Scholar
  14. Kang S, Pacold M, Cervantes C, Lim D, Lou H, Ottina K, Gray N, Turk B, Yaffe M, Sabatini D (2013) mTORC1 phosphorylation sites encode their sensitivity to starvation and rapamycin. Science 341:1236566CrossRefGoogle Scholar
  15. Keeling M (1999) Spatial models of interacting populations, advanced ecological theory: principles and applications. J McGlade, (ed.) Blackwell Science, OxfordGoogle Scholar
  16. Kim YJ, Kwon O, Li F (2013) Evolution of dispersal toward fitness. Bull Math Biol 75(12):2474–2498CrossRefMathSciNetzbMATHGoogle Scholar
  17. Kim YJ, Kwon O, Li F (2014) Global asymptotic stability and the ideal free distribution in a starvation driven diffusion. J Math Biol 68(6):1341–1370CrossRefMathSciNetzbMATHGoogle Scholar
  18. Kim YJ, Tzavaras AE (2001) Diffusive \(N\)-waves and metastability in the Burgers equation. SIAM J Math Anal 33(3):607–633 (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  19. Lam KY, Lou Y (2014) Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal. Bull Math Biol 76(2):261–291CrossRefMathSciNetzbMATHGoogle Scholar
  20. Lam KY, Ni WM (2010) Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discret Contin Dyn Syst 28(3):1051–1067CrossRefMathSciNetzbMATHGoogle Scholar
  21. McPeek M, Holt R (1992) The evolution of dispersal in spatially and temporally varying environments. Am Nat 140:1010–1027CrossRefGoogle Scholar
  22. Nagylaki T (1992) Introduction to theoretical population genetics, biomathematics, vol 21. Springer, BerlinCrossRefGoogle Scholar
  23. Ni WM (2011) The mathematics of diffusion, CBMS-NSF regional conference series in applied mathematics, vol 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PAGoogle Scholar
  24. Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives, In: Interdisciplinary applied mathematics, (2nd ed.), vol 14. Springer, New YorkGoogle Scholar
  25. Pao CV (1992) Nonlinear parabolic and elliptic equations. Plenum Press, New YorkzbMATHGoogle Scholar
  26. Seo HW (2013) Optimal selection under satisfaction dependent dispersal strategy, Master’s Thesis, KAISTGoogle Scholar
  27. Skellam JG (1972) Some philosophical aspects of mathematical modelling in empirical science with special reference to ecology, mathematical models in ecology. Blackwell Sci. Publ, LondonGoogle Scholar
  28. Skellam JG (1973) The formulation and interpretation of mathematical models of diffusionary processes in population biology, the mathematical theory of the dynamics of biological populations. Academic Press, New YorkGoogle Scholar
  29. Travis JMJ, Dytham C (1999) Habitat persistence, habitat availability and the evolution of dispersal. Proc R Soc Lond B 266:723–728CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  1. 1.National Institute for Mathematical SciencesDaejeonKorea
  2. 2.Department of Mathematical SciencesKAISTDaejeonKorea
  3. 3.Department of MathematicsChungbuk National UniversityChungbukKorea

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