Advertisement

Journal of Mathematical Biology

, Volume 51, Issue 1, pp 75–113 | Cite as

Analysis of the periodically fragmented environment model : I – Species persistence

  • Henri Berestycki
  • François Hamel
  • Lionel Roques
Article

Abstract.

This paper is concerned with the study of the stationary solutions of the equation

Open image in new window

where the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.

Keywords

Evolution Equation Stationary Solution Stationary Equation Large Time Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agmon, S.: Differential equations (Birmingham, Ala., 1983). North-Holland Math. Stud. 92, 7–17 (1984)Google Scholar
  2. 2.
    Ammerman, A.J., Cavalli-Sforza, L.L.: The Neolithic Transition and the Genetics of Populations in Europe. Princeton Univ. Press, Princeton, NJ, 1984Google Scholar
  3. 3.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978)CrossRefGoogle Scholar
  4. 4.
    Berestycki, H.: Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Func. Anal. 40, 1–29 (1981)Google Scholar
  5. 5.
    Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002)Google Scholar
  6. 6.
    Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. I - Periodic framework. J. Europ. Math. Soc. 2005. J. Eur. Math. Soc. 7, 173–213 (2005)Google Scholar
  7. 7.
    Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model : II - Biological invasions and pulsating travelling fronts. J. Math. Pures Appl. 2005. To appearGoogle Scholar
  8. 8.
    Berestycki, H., Hamel, F., Rossi, L.: Liouville theorems for semilinear elliptic equations in unbounded domains. PreprintGoogle Scholar
  9. 9.
    Berestycki, H., Lachand-Robert, T.: Some properties of monotone rearrangement with applications to elliptic equations in cylinders. Math. Nachr. 266, 3–19 (2004)Google Scholar
  10. 10.
    Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré, Anal. Non Linéaire 9, 497–572 (1992)Google Scholar
  11. 11.
    Berestycki, H., Nirenberg, L., Varadhan, S.R.S.: The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Commun. Pure Appl. Math. 47, 47–92 (1994)Google Scholar
  12. 12.
    Brown, K., Lin, S.S.: On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function. J. Math. Anal. Appl. 75, 112–120 (1980)Google Scholar
  13. 13.
    Cantrell, R.S., Cosner, C.: Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. Roy. Soc. Edinburgh 112, 293–318 (1989)Google Scholar
  14. 14.
    Cantrell, R.S., Cosner, C.: Diffusive logistic equations with indefinite weights: population models in disrupted environments II. SIAM J. Math. Anal. 22 (4), 1043–1064 (1991)Google Scholar
  15. 15.
    Cantrell, R.S., Cosner, C.: The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315–338 (1991)Google Scholar
  16. 16.
    Cantrell, R.S., Cosner, C.: On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol. 37, 103–145 (1998)Google Scholar
  17. 17.
    Capdeboscq, Y.: Homogenization of a neutronic critical diffusion problem with drift. Proc. Royal Soc. Edinburgh 132 A, 567–594 (2002)Google Scholar
  18. 18.
    Cano-Casanova, S., López-Gómez, J.: Permanence under strong aggressions is possible. Ann. Inst. H. Poincaré, Anal. Non Linéaire 20, 999–1041 (2003)Google Scholar
  19. 19.
    Cavalli-Sforza, L.L., Feldman, M.W.: Cultural Transmission and Evolution : A Quantitative Approach. Princeton Univ. Press, Princeton, NJ, 1981Google Scholar
  20. 20.
    Engländer, J., Kyprianou, A.E.: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32, 78–99 (2004)MathSciNetGoogle Scholar
  21. 21.
    Engländer, J., Pinsky, R.G.: On the construction and support properties of measure-valued diffusions on Open image in new window with spatially dependent branching. Ann. Probab. 27, 684–730 (1999)Google Scholar
  22. 22.
    Engländer, J., Pinsky, R.G.: Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form ut=Lu+Vu–γ up in Rn. J. Diff. Equations 192, 396–428 (2003)Google Scholar
  23. 23.
    Fife, P.C., McLeod, J.B.: The approach of solutions of non-linear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)Google Scholar
  24. 24.
    Fisher, R.A.: The advance of advantageous genes. Ann. Eugenics 7, 335–369 (1937)Google Scholar
  25. 25.
    Freidlin, M.: On wavefront propagation in periodic media. Stochastic analysis and applications, Adv. Probab. (Related Topics, 7, Dekker, New York, 1984) 147–166Google Scholar
  26. 26.
    Freidlin, M.: Semi-linear PDEs and limit theorem for large deviations. Springer Lecture Notes in Mathematics, 1527, 1992Google Scholar
  27. 27.
    Freidlin, M.: Markov Processes and Differential Equations: Asymptotic Problems. Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1996Google Scholar
  28. 28.
    Gärtner, J., Freidlin, M.: On the propagation of concentration waves in periodic and random media. Sov. Math. Dokl. 20, 1282–1286 (1979)Google Scholar
  29. 29.
    Hardy, G.H., Littlewood, J.E., P’olya, G.: Inequalities. Cambridge University Press, Cambridge, 1952Google Scholar
  30. 30.
    Harrell, E.M., Kröger, P., Kurata, K.: On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal. 33 (1), 240–259 (2001)Google Scholar
  31. 31.
    Hess, P.: Periodic-parabolic boundary value problems and positivity. Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, UK, 1991Google Scholar
  32. 32.
    Hiebeler, D.E.: Populations on Fragmented Landscapes with Spatially Structured Heterogeneities: Landscape Generation and Local Dispersal. Ecology 81 (6), 1629–1641 (2000)Google Scholar
  33. 33.
    Hudson, W., Zinner, B.: Existence of travelling waves for reaction-diffusion equations of Fisher type in periodic media. In: Boundary Value Problems for Functional-Differential Equations. J. Henderson (ed.), World Scientific, 1995, pp. 187–199Google Scholar
  34. 34.
    Hughes, J.: Modeling the effect of landscape pattern on mountain pine beetles. Thesis, Simon Fraser University, 2002Google Scholar
  35. 35.
    Hutson, V., Mischaikow, K., Polacik, P.: The Evolution of dispersal rates in a heterogeneous time-periodic environment. J. Math. Biol. 43, 501–533 (2001)PubMedGoogle Scholar
  36. 36.
    Kaipio, J.P., Tervo, J., Vauhkonen, M.: Simulations of the heterogeneity of environments by finite element methods. Math. Comput. Simul. 39, 155–172 (1995)Google Scholar
  37. 37.
    Kawohl, B.: On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems. Arch. Ration. Mech. Anal. 94, 227–243 (1986)Google Scholar
  38. 38.
    Kinezaki, N., Kawasaki, K., Takasu, F., Shigesada, N.: Modeling biological invasion into periodically fragmented environments. Theor. Population Biol. 64, 291–302 (2003)Google Scholar
  39. 39.
    Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d’État à Moscou (Bjul. Moskowskogo Gos. Univ.), Série internationale A 1, 1–26 (1937)Google Scholar
  40. 40.
    Ludwig, D., Aronson, D.G., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979)Google Scholar
  41. 41.
    Murray, J.D., Sperb, R.P.: Minimum domains for spatial patterns in a class of reaction-diffusion equations. J. Math. Biol. 18, 169–184 (1983)PubMedGoogle Scholar
  42. 42.
    Pinsky, R.G.: Positive harmonic functions and diffusion. Cambridge University Press, 1995Google Scholar
  43. 43.
    Pinsky, R.G.: Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions. J. Funct. Anal. 129, 80–107 (1995)Google Scholar
  44. 44.
    Pinsky, R.G.: Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24, 237–267 (1996)Google Scholar
  45. 45.
    Senn, S.: On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics. Commun. Part. Diff. Equations 8, 1199–1228 (1983)Google Scholar
  46. 46.
    Senn, S., Hess, P.: On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions. Math. Ann. 258, 459–470 (1981/82)Google Scholar
  47. 47.
    Shigesada, N., Kawasaki, K.: Biological invasions: theory and practice. Oxford Series in Ecology and Evolution, Oxford : Oxford University Press, 1997Google Scholar
  48. 48.
    Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Population Biol. 30, 143–160 (1986)Google Scholar
  49. 49.
    Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)PubMedGoogle Scholar
  50. 50.
    Weinberger, H.: On spreading speed and travelling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)PubMedGoogle Scholar
  51. 51.
    Xin, X.: Existence of planar flame fronts in convective-diffusive periodic media. Arch. Ration. Mech. Anal. 121, 205–233 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Henri Berestycki
    • 1
  • François Hamel
    • 2
  • Lionel Roques
    • 2
  1. 1.EHESS, CAMSParisFrance
  2. 2.Faculté des Sciences et TechniquesUniversité Aix-Marseille III, LATPMarseille Cedex 20France

Personalised recommendations