Taiwanese Journal of Mathematics

POSITIVE SOLUTIONS FOR A PREDATOR-PREY INTERACTION MODEL WITH HOLLING-TYPE FUNCTIONAL RESPONSE AND DIFFUSION

Yunfeng Jia, Jianhua Wu, and Hong-Kun Xu

Full-text: Open access

Abstract

We deal with a predator-prey interaction model with Holling-type monotonic functional response and diffusion and which is endowed with a second homogeneous boundary condition. Via spectrum analysis and bifurcation theory, we investigate the local and global bifurcation solutions of the model which emanate from a positive constant solution by taking the growth rate as a bifurcation parameter. Basing on the fixed point index theory, we prove the existence of positive steady-state solutions of the model.

Article information

Source
Taiwanese J. Math., Volume 15, Number 5 (2011), 2013-2034.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406420

Digital Object Identifier
doi:10.11650/twjm/1500406420

Mathematical Reviews number (MathSciNet)
MR2880390

Zentralblatt MATH identifier
1230.92051

Subjects
Primary: 92D25: Population dynamics (general)
Secondary: 93C20: Systems governed by partial differential equations 35K57: Reaction-diffusion equations

Keywords
predator-prey model positive solution functional response bifurcation theory fixed point index theory stability Crandall-Rabinowitz's bifurcation theorem

Citation

Jia, Yunfeng; Wu, Jianhua; Xu, Hong-Kun. POSITIVE SOLUTIONS FOR A PREDATOR-PREY INTERACTION MODEL WITH HOLLING-TYPE FUNCTIONAL RESPONSE AND DIFFUSION. Taiwanese J. Math. 15 (2011), no. 5, 2013--2034. doi:10.11650/twjm/1500406420. https://projecteuclid.org/euclid.twjm/1500406420


Export citation

References

  • R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973.
  • R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
  • D. J. Wollkind and J. A. Logan, Tempreture-dependent predator-prey mite ecosystem on apple tree foliage, J. Math. Biol., 6 (1978), 265-283.
  • D. J. Wollkind, J. B. Collings and J. A. Logan, Metastability in a tempreture-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. Math. Biol., 50 (1988), 379-409.
  • J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168.
  • M. W. Sabelis, Predation on spider mites, in: Spide Mites: Their Biology, Natural Enemies and Control (World Crop Pests Vol. 1B, 103-129), Elsevier, Amsterdam, 1985.
  • R. J. Taylor, Predation, Chapman and Hall, New York, 1984.
  • C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Canad. Ent., 91 (1959), 293-320.
  • C. S. Holling, Principles of insect predation, Ann. Rev. Entomol., 6 (1961), 163-182.
  • S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey sestems, SIAM J. Appl. Math., 55 (1995), 763-783.
  • J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.
  • Y. Jia, J. Wu and H. Nie, The coexistence states of a predator-prey model with nonmonotonic functional response and diffusion, Acta Appl. Math., 108 (2009), 413-428.
  • E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.
  • Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345.
  • P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
  • M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-181.
  • K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Diff. Eqns., 197 (2004), 315-348.
  • I. J. Maddox, Elements of Functional Analysis, Cambridge Univ. Press, Cambridge, 1970.
  • D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Univ. Press, Oxford, 1990.
  • P. Y. H. Pang and M. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh A, 133 (2003), 919-942.
  • R. Peng and M. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh A, 135 (2005), 149-164.
  • H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
  • P. A. Braza, The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing, SIAM J. Appl. Math., 63 (2003), 889-904.
  • H. I. Freedman and R. M. Mathsen, Persistence in predator-prey systems with ratio-dependence predation influence, Bull. Math. Biol., 55 (1993), 817-827.