Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY-PREDATOR MODEL WITH NON-MONOTONIC FUNCTIONAL RESPONSE AND CROSS-DIFFUSION

  • Xu, Shenghu (Department of Mathematics, Longdong University)
  • Received : 2010.03.17
  • Accepted : 2010.05.20
  • Published : 2011.01.30
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Abstract

In this paper, using the energy estimates and the bootstrap arguments, the global existence of classical solutions for a prey-predator model with non-monotonic functional response and cross-diffusion where the prey and predator both have linear density restriction is proved when the space dimension n < 10.

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References

  1. J. F. Andrews, A mathematical model for the continuous culture of microoganisms utilizing inhibitorysubstrates, Biotech. Bioeng10(1968), 707-723. https://doi.org/10.1002/bit.260100602
  2. S. H. Xu, Existence of global solutions for a predator-prey model with cross-diffusion, Electronic. J. Diff. Eqns 06(2008), 1-14.
  3. S. Ruan and D. Xiao, Global analysis in a predatorCprey system with nonmonotonic functional response, SIAM J. Appl. Math61(4)(2000), 1445C1472.
  4. H. Zhu, S. Campbell and G. Wolkowicz, Bifurcation analysis of a predatorCprey system with nonmonotonic functional response, SIAM J. Appl. Math63(2) (2002), 636C682.
  5. X. F. Chen, Y. W. Qi and M. X. Wang, A Strongly coupled predator-prey system with non-monotonic functional response, Nonlinear Analysis: RWA 67(2007), 1966-1979. https://doi.org/10.1016/j.na.2006.08.022
  6. C. Holing, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent, Soc. Can 45(1965), 1-65.
  7. W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Analysis: RWA 10(4)(2009), 2558-2573. https://doi.org/10.1016/j.nonrwa.2008.05.012
  8. W. Ko and K. Ryu, coexistence states of a predator-prey system with non-monotonic functional response, Nonlinear Analysis: RWA8(2007), 769-786. https://doi.org/10.1016/j.nonrwa.2006.03.003
  9. H. Amann, Dynamic theory of quasilinear parabolic equations: Abstract evolution equations, Nonlinear Analysis12(1988), 859-919.
  10. H. Amann, Dynamic theory of quasilinear parabolic equations: Reaction-diffusion, Diff. Int. Eqs3(1990), 13-75.
  11. H. Amann, Dynamic theory of quasilinear parabolic equations: Global existence, Math. Z 202(1989), 219-250. https://doi.org/10.1007/BF01215256
  12. A. Shim, $W\frac{1}{2}$-estimates on the prey-predator systems with cross-diffusions and functional responses. Commun. Keorean. Math. Soc23(2008), 211-227. https://doi.org/10.4134/CKMS.2008.23.2.211
  13. N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species. J. Theor. Biology79(1979), 83-99. https://doi.org/10.1016/0022-5193(79)90258-3
  14. Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete Contin. Dynam. Systems9(2003), 1193-1200.
  15. Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dynam. Systems10(2004), 719-730.
  16. P. V. Tuoc, On global existence of solutions to a cross-diffusion system. J. Math. Anal. Appl343(2008), 826-834. https://doi.org/10.1016/j.jmaa.2008.01.089
  17. O. A. Ladyzenskaja and V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, AMS, 1968.