Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF NON-CONSTANT POSITIVE SOLUTIONS FOR A RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DISEASE IN THE PREY

  • Ryu, Kimun (Department of Mathematics Education Cheongju University)
  • Received : 2018.01.05
  • Accepted : 2018.01.19
  • Published : 2018.02.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider ratio-dependent predator-prey models with disease in the prey under Neumann boundary condition. We investigate sufficient conditions for the existence and non-existence of non-constant positive steady-state solutions by the effects of the induced diffusion rates.

Keywords

Acknowledgement

Supported by : Cheongju University

References

  1. I. Ahn, W. Ko, and K. Ryu, Asymptotic behavior of a ratio-dependent predatorprey system with disease in the prey, Discrete Contin. Dyn. Syst. Special (2013), 11-19.
  2. R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol. 139 (1989), 311-326.
  3. R. Arditi, L. R. Ginzburg, and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent models, American Naturalist. 138 (1991), 1287-1296.
  4. R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.
  5. R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 257 (2001), no. 1, 206-222. https://doi.org/10.1006/jmaa.2000.7343
  6. C. Cosner, D. L. DeAngelis, J. S. Ault, and D. B. Olson, CEffects of spatial grouping on the functional response of predators, Theoret. Population Biol. 56 (1999), 65-75.
  7. A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson's blow ies as an example, Ecology. 73 (1992), 1552-1563.
  8. D. Henry, Geometric Theory of Semilinear Parabolic Equations: Lecture Notes in Mathematics, Springer, Berlin, New York, 840, 1993.
  9. S. B. Hsu, T. W. Hwang, and Y. Kuang, Global analysis of the Michaelis-Mententype ratio-dependent predator-prey system, J. Math. Biol. 42 (2001), no. 6, 489-506. https://doi.org/10.1007/s002850100079
  10. S. B. Hsu, T. W. Hwang, and Y. Kuang, Rich dynamics of a ratio-dependent one-prey two-predators model, J. Math. Biol. 43 (2001), no. 5, 377-396. https://doi.org/10.1007/s002850100100
  11. S. B. Hsu, T. W. Hwang, and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosci. 181 (2003), no. 1, 55-83. https://doi.org/10.1016/S0025-5564(02)00127-X
  12. Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36 (1998), no. 4, 389-406. https://doi.org/10.1007/s002850050105
  13. Z. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal. 57 (2004), 421-433.
  14. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1973.
  15. P. Y. H. Pang and M. X. Wang, Stragey and stationary pattern in a three-species predator-prey model, J. Differential Equations. 200 (2004), 245-273.
  16. P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A. 133 (2003), no. 4, 919-942.