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Persistence in reaction diffusion models with weak allee effect

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Abstract

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation.

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References

  1. Allee, W.C.: The social life of animals. W.W Norton, New York (1938)

  2. Bradford, E., Philip, J.P.: Stability of steady distributions of asocial populations dispersing in one dimension. J. Theor. Biol. 29 (1), 13–26 (1970)

    Article  Google Scholar 

  3. Bradford, E., Philip, J.P.: Note on asocial populations dispersing in two dimensions. J. Theor. Biol. 29 (1), 27–33 (1970)

    Article  Google Scholar 

  4. Cantrell, R.S., Cosner, C.: Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. Roy. Soc. Edinburgh Sect. A 112 (3–4), 293–318 (1989)

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Cantrell, R.S., Cosner, C.: Conditional persistence in logistic models via nonlinear diffusion. Proc. Roy. Soc. Edinburgh Sect. A 132 (2), 267–281 (2002)

    MathSciNet  Google Scholar 

  7. Cantrell, R.S., Cosner, C.: Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003)

  8. Cantrell, R.S., Cosner, C., Fagan, W.F.: Habitat edges and predator-prey interactions: effects on critical patch size. Math. Biosci. 175 (1), 31–55 (2002)

    Article  MathSciNet  Google Scholar 

  9. Clark, C.W.: Mathematical Bioeconomics, The Optimal Management of Renewable Resources. John Wiley & Sons, Inc. New York 1991

  10. Conway, E.D.: Diffusion and predator-prey interaction: Steady states with flux at the boundaries. Contemporary Mathematics, 17, 217–234 (1983)

    MathSciNet  Google Scholar 

  11. Conway, E. D.: Diffusion and predator-prey interaction: pattern in closed systems. In Partial differential equations and dynamical systems, 85–133, Res. Notes in Math., 101, Pitman, Boston-London 1984

  12. Crandall, M.G., Rabinowitz, P.H: Bifurcation from simple eigenvalues. J. Functional Analysis, 8, 321–340 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  13. de Figueiredo, D.G.: Positive solutions of semilinear elliptic problems. In Differential equations (Sao Paulo, 1981), 34–87, Lecture Notes in Math., 957, Springer, Berlin-New York 1982

  14. de Roos, A.M., McCawley, E., Wilson, W.G.: Pattern formation and the spatial scale of interaction between predators and their prey. Theo. Popu. Biol. 53, 108–130 (1998)

    Article  MATH  Google Scholar 

  15. Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction. Natur. Resource Modeling 3 (4), 481–538 (1989)

    MathSciNet  Google Scholar 

  16. Du, Y., Shi, J.: Allee Effect and Bistability in a Spatially Heterogeneous Predator-Prey Model. To appear in Trans. Amer. Math. Soc. (2006)

  17. Fife, P.C.: Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin (1979)

  18. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics, 7, 353–369 (1937)

    Google Scholar 

  19. Groom, M.J.: Allee effects limit population viability of an annual plant. Amer. Naturalist 151, 487–496 (1998)

    Article  Google Scholar 

  20. Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York (1981)

  21. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 45, 5–60 (1965)

    Google Scholar 

  22. Hopf, F.A., Hopf, F.W.: The role of the Allee effect in species packing. Theo. Popu. Biol. 27 (1), 27–50 (1985)

    Article  MathSciNet  Google Scholar 

  23. Jiang, J., Liang, X., Zhao, X.-Q.: Saddle-point behavior for monotone semiflows and reaction-diffusion models. J. Differential Equations 203 no. 2, 313–330 (2004)

    Google Scholar 

  24. Keitt, T.H., Lewis, M.A., Holt, R.D.: Allee effect, invasion Pinning, and species' borders. Amer. Naturalist 157, 203–216 (2001)

    Article  Google Scholar 

  25. Kolmogoroff, A., Petrovsky, I., Piscounoff, N.: Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. (French) Moscow Univ. Bull. Math. 1, 1–25 (1937)

    MATH  Google Scholar 

  26. Korman, P., Shi, J.: New exact multiplicity results with an application to a population model. Proc. Roy. Soc. Edinburgh Sect. A 131 (5), 1167–1182 (2001)

    MathSciNet  Google Scholar 

  27. Lee, Y.H., Sherbakov, L., Taber, J., Shi, J.: Bifurcation Diagrams of Population Models with Nonlinear Diffusion. To appear in Jour. Compu. Appl. Math., (2006)

  28. Lewis, M.A., Kareiva, P.: Allee dynamics and the spread of invading organisms. Theo. Popu. Biol. 43, 141–158 (1993)

    Article  MATH  Google Scholar 

  29. Logan, R.: Positive solutions to a system of differential equations modeling a competitive interactive system with nonlogistic growth rates. Differential Integral Equations 10 no. 5, 929–945 (1997)

    Google Scholar 

  30. Ludwig, D., Aronson, D.G., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8 (3), 217–258 (1979)

    MathSciNet  Google Scholar 

  31. Murray, J.D.: Mathematical biology. Third edition. I. An introduction. Interdisciplinary Applied Mathematics, 17; II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York 2003

  32. Okubo, A., Levin, S.: Diffusion and ecological problems: modern perspectives. Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York (2001)

  33. Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problem. J. Differential Equations 146 no. 1, 121–156 (1998)

    Google Scholar 

  34. Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problem:II. J. Differential Equations 158 no. 1, 94–151 (1999)

    Google Scholar 

  35. Owen, M.R., Lewis, M.A.: How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 63, 655–684 (2001)

    Article  Google Scholar 

  36. Philip, J.R.: Sociality and sparse populations. Ecology 38 107–111 (1957)

    Google Scholar 

  37. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7, 487–513 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  38. Shi, J.: Persistence and Bifurcation of Degenerate Solutions. Jour. Func. Anal., 169, no. 2, 494-531 (2000)

  39. Shi, J., Shivaji, R.: Global bifurcation for concave semipositon problems. Ed: G.R. Goldstein, R. Nagel, S. Romanelli, Advances in Evolution Equations: Proceedings in honor of J.A. Goldstein's 60th birthday, Marcel Dekker, Inc., New York, Basel, 385–398, (2003)

  40. Shi, J., Yao, M.: On a singular nonlinear semilinear elliptic problem. Proc. Roy. Soc. Edinburgh Sect. A 128 (6), 1389–1401 (1998)

    Google Scholar 

  41. Skellam, J.G.: Random dispersal in theoritical populations. Biometrika 38 196–218 (1951)

    Google Scholar 

  42. Smoller, J.: Shock waves and reaction-diffusion equations. Second edition. Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York (1994)

  43. Smoller, J., Wasserman, A.: Global bifurcation of steady-state solutions. J. Differential Equations 39 no. 2, 269–290 (1981)

  44. Struwe, M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Third edition. Springer-Verlag, Berlin (2000)

  45. Taira, K.: Diffusive logistic equations in population dynamics. Adv. Differential Equations 7 no. 2, 237–256 (2002)

  46. Thieme, H.R.: Mathematics in population biology. Princeton Series in Theoritical and Computational Biology. Princeton University Press, (2003)

  47. Turchin, P.: Population consequences of aggregative movement. Jour. Anim. Ecol. 58 (1), 75–100 (1989)

    Google Scholar 

  48. Veit, R.R., Lewis, M.A.: Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America. Amer. Naturalist 148, 255–274 (1996)

    Article  Google Scholar 

  49. Wang, M.-H., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171 (1), 83–97 (2001)

    Article  MathSciNet  Google Scholar 

  50. Wang, M.-H., Kot, M., Neubert, M.G.: Integrodifference equations, Allee effects, and invasions. J. Math. Biol. 44 (2), 150–168 (2002)

    Article  MathSciNet  Google Scholar 

  51. Wilson, W.G., Nisbet, R.M.: Cooperation and competition along smooth environment gradients. Ecology 78, 2004–2017 (1997)

    Article  Google Scholar 

  52. Yoshizawa, S.: Population growth process described by a semilinear parabolic equation. Math. Biosci. 7, 291–303 (1970)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23185, USA

    Junping Shi

  2. School of Mathematics, Harbin Normal University, Harbin, Heilongjiang, 150025, P.R.China

    Junping Shi

  3. Department of Mathematics, Mississippi State University, Mississippi State, MS, 39762, USA

    Ratnasingham Shivaji

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  1. Junping Shi
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  2. Ratnasingham Shivaji
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Correspondence to Junping Shi.

Additional information

J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.

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Shi, J., Shivaji, R. Persistence in reaction diffusion models with weak allee effect. J. Math. Biol. 52, 807–829 (2006). https://doi.org/10.1007/s00285-006-0373-7

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  • DOI: https://doi.org/10.1007/s00285-006-0373-7

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