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Positive Steady States of a Prey-predator Model with Diffusion and Non-monotone Conversion Rate

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Abstract

In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.

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References

  1. Blat, J., Brown, K. J.: Global bifurcation on positive solutions in some systems of elliptic equations. SIAM J. Math. Anal., 17, 1339–1353 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Casal, A., Eilbeck, J. C., López–Gómez, J.: Existence and uniqueness of coexistence states for a prey– predator model with diffusion. Diff. Integral Equns., 7(2), 411–439 (1994)

    MATH  MathSciNet  Google Scholar 

  3. Du, Y. H., Lou, Y.: Some uniqueness and exact multiplicity results for a predator–prey model. Trans. Amer. Math. Soc., 349(6), 2442–2475 (1997)

    Article  Google Scholar 

  4. Pang, P. Y. H., Wang, M. X.: Qualitative analysis of a ratio–dependent predator–prey system with diffusion. Proc. Roy. Soc. Edinburgh A, 133(4), 919–942 (2003)

    MATH  Google Scholar 

  5. Du, Y. H., Lou, Y.: S–shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator–prey model. J. Diff. Equns., 144, 390–440 (1998)

    Article  MATH  Google Scholar 

  6. Freedman, H. I.,Wolkowicz, G. S. K.: Predator–prey systems with group defense: The paradox of enrichment revisited. Bull. Math. Biol., 57, 493–508 (1986)

    Google Scholar 

  7. Mischaikow, K., Wolkowicz, G. S. K.: A predator–prey system involving group defence: A connection matrix approch. Nonlinear Analysis, 14, 955–969 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Pang, P. Y. H., Wang, M. X.: Non–constant positive steady states of a predator–prey system with nonmonotonic functional response and diffusion. Proc. London Math. Soc., 88(1), 135–157 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Wang, M. X.: Stationary patterns of strongly coupled prey–predator models. J. Math. Anal. Appl., 292(2), 484–505 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wang, M. X.: Stationary patterns for a prey–predator model with prey–dependent and ratio–dependent functional responses and diffusion. Physica D, 196, 172–192 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Wolkowicz, G. S. K.: Bifurcation analysis of a predator–prey system involving group defense. SIAM J. Appl. Math., 48, 592–606 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Murray, J. D.: Mathematical Biology, second edition, Springer–Verlag, Berlin, 1993

  13. Okubo, A.: Diffusion and Ecological Problem: Mathematical Models, Springer–Verlag, Berlin, New York, 1980

  14. Ruan, W., Feng, W.: On the fixed point index and multiple steady states of reaction–diffusion systems. Diff. and Integral Equns., 8, 371–391 (1995)

    MATH  Google Scholar 

  15. Mimura, M., Nishiura, Y.: Pattern formation in coupled reaction–diffusion systems. Japan J. Indust. Appl. Math., 12(3), 385–424 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ni, W. M.: Diffusion, cross–diffusion and their spike–layer steady states. Notices Amer. Math. Soc., 45(1), 9–18 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Wang, M. X.: Non–constant positive steady states of the Sel’kov model. J. Diff. Equns., 190(2), 600–620 (2003)

    Article  MATH  Google Scholar 

  18. Yamada, Y.: Stability of steady states for prey–predator diffusion equations with homogeneous Dirichlet conditions. SIAM J. Math. Anal., 21, 327–345 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Alves, C. O., De Figueiredo, D. G.: Nonvariational elliptic systems. Discrete Contin. Dynam. Systems, 8, 289–302 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 18, 620–709 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dancer, E. N.: On the indices of fixed ponits of mappings in cones and applications. J. Math. Anal. Appl., 91, 131–151 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  22. Dancer, E. N., Du, Y. L.: Effects of certain degeneracies in the predator–prey model. SIAM J. Math. Anal., 34, 292–314 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Delgado, M., López–Gómez, J.: On the symbiotic Lotka–Volterra model with diffusion and transport effects. J. Diff. Equns., 160, 175–262 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. De Figueiredo, D. G., Yang, J.: A priori bounds for positive solutions of a non–variational elliptic system. Comm. Partial Diff. Equns., 26, 2305–2321 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gui, C. F., Lou, Y.: Uniqueness and nonuniqueness of coexistence states in the Lotka–Volterra competition model. Comm. Pure Appl. Math., XLVII, 1571–1594 (1994)

    MathSciNet  Google Scholar 

  26. Li, L.: Coexistence theorems of steady–states for predator–prey interacting systems. Trans. Amer. Math. Soc., 305, 143–166 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lou, Y.: Necessary and sufficient condition for the existence of positive solutions of certain cooperative system. Nonlinear Analysis, 26, 1079–1095 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Wang, M. X.: Nonlinear Partial Differential Equations of Parabolic Type (in Chinese), Science Press, Beijing, 1993

  29. Hei, L. J., Wu, J. H.: Existence and Stability of Positive Solutions for an Elliptic Cooperative System. Acta Mathematica Sinica, English Series, 21(5), 1113–1120 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Crandall, M. G., Rabinowitz, P. H.: Bifurcation from simple eigenvalues. J. Funct. Anal., 8, 321–340 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Nirenberg, L.: Topics in nonlinear functional analysis, American Mathematical Society, Providence, RI, 2001

  33. Crandall, M. G., Rabinowitz, P. H.: Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rational Mech. Anal., 52, 161–180 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kato, K.: Perturbation Theory for Linear Operator, Springer–Verlag, New York, 1966

  35. López–Gómez, J., Pardo, R.: Existence and uniqueness of coexistence states for the predator–prey Lotka– Volterra model with diffusion on intervals. Diff. Integral Equns., 6, 1025–1031 (1993)

    MATH  Google Scholar 

  36. Protter, M. H., Weinberger, H. F.: Maximum Principle in Differential Equations, Englewoods Cliffs, Prentice–Hall, N. J., 1967

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Authors and Affiliations

  1. Institute of Nonlinear Complex System, College of Science, China Three Gorges University, Yichang 443002, P. R. China

    Rui Peng

  2. Department of Mathematics, Southeast University, Nanjing 210018, P. R. China

    Rui Peng

  3. Department of Mathematics, Southeast University, Nanjing 210018, P. R. China

    Ming Xin Wang

  4. Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, P. R. China

    Ming Xin Wang

  5. Department of Mathematics, Southeast University, Nanjing 210018, P. R. China

    Wen Yan CHEN

Authors
  1. Rui Peng
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  2. Ming Xin Wang
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  3. Wen Yan CHEN
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Corresponding author

Correspondence to Ming Xin Wang.

Additional information

This work is supported by the National Natural Science Foundation of China 10471022, and the Ministry of Education of China Science and Technology Major Projects Grant 104090. The work of R. Peng is also supported by the Foundation of Excellent Doctoral Dissertation of Southeast University YBJJ0405

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Peng, R., Wang, M.X. & CHEN, W.Y. Positive Steady States of a Prey-predator Model with Diffusion and Non-monotone Conversion Rate. Acta Math Sinica 23, 749–760 (2007). https://doi.org/10.1007/s10114-005-0789-9

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  • DOI: https://doi.org/10.1007/s10114-005-0789-9

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