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POSITIVE COEXISTENCE FOR A SIMPLE FOOD CHAIN MODEL WITH RATIO-DEPENDENT FUNCTIONAL RESPONSE AND CROSS-DIFFUSION

  • Published : 2006.10.31

Abstract

The positive coexistence of a simple food chain model with ratio-dependent functional response and cross-diffusion is discussed. Especially, when a cross-diffusion is small enough, the existence of positive solutions of the system concerned can be expected. The extinction conditions for all three interacting species and for one or two of three species are studied. Moreover, when a cross-diffusion is sufficiently large, the extinction of prey species with cross-diffusion interaction to predator occurs. The method employed is the comparison argument for elliptic problem and fixed point theory in a positive cone on a Banach space.

References

  1. R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol. 139 (1989), 311-326 https://doi.org/10.1016/S0022-5193(89)80211-5
  2. 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 https://doi.org/10.1086/285286
  3. R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratiodependent consumption, Ecology 73 (1992), 1544-1551 https://doi.org/10.2307/1940007
  4. C. Cosner, D. L. DeAngelis, J. S. Ault, and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Population Biol, 56 (1999), 65-75 https://doi.org/10.1006/tpbi.1999.1414
  5. E. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1976), 131-151 https://doi.org/10.1016/0022-247X(83)90098-7
  6. W. Feng and X. Lu, Permanence effect in a three-species food chain model, Appl. Anal. 54 (1994), no. 3-4, 195-209 https://doi.org/10.1080/00036819408840277
  7. W. Feng and X. Lu, Some coexistence and extinction results for a 3-species ecological system, Differential Integral Equations 8 (1995), no. 3, 617-626
  8. H. I. Freedman and P. Waltman, Mathematical analysis of some three-species food-chain models, Math. Biosci. 33 (1977), no. 3-4, 257-276 https://doi.org/10.1016/0025-5564(77)90142-0
  9. A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson's blowflies as an example, Ecology 73 (1992), 1552-1563 https://doi.org/10.2307/1940008
  10. L. Hsiao and P. De Mottoni, Persistence in reaction-diffusion systems: interaction of two predators and one prey, Nonlinear Analysis 11 (1987), 877-891 https://doi.org/10.1016/0362-546X(87)90058-7
  11. 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
  12. 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
  13. 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
  14. Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. BioI. 36 (1998), no. 4, 389-406 https://doi.org/10.1007/s002850050105
  15. K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Diff. Eqs. 197 (2004), 315-348 https://doi.org/10.1016/j.jde.2003.08.003
  16. N. Lakos, Existence of steady-state soludions for a one-predator two-prey system, SIAM J. Math. Analysis 21 (1990), 647-659 https://doi.org/10.1137/0521034
  17. A. Leung, A study of 3-species prey-predator reaction-diffusions by monotone schemes, J. Math. Analysis Applic. 100 (1984), 583-604 https://doi.org/10.1016/0022-247X(84)90103-3
  18. L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1998), 143-166 https://doi.org/10.2307/2001045
  19. C. S. Lin, W. M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), 1-27 https://doi.org/10.1016/0022-0396(88)90147-7
  20. J. Lopez-Gomez and R. Pardo San Gil, Coexistence in a simple food chain with diffusion, J. Math. Biol. 30 (1992), no. 7, 655-668
  21. Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996), 79-131 https://doi.org/10.1006/jdeq.1996.0157
  22. Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations 154 (1999), no. 1, 157-190 https://doi.org/10.1006/jdeq.1998.3559
  23. W. Ruan and W. Feng, On the fixed point index and mutiple steady-state solutions of reaction-diffusion coefficients, Differ. Integral Equ. 8 (1995), no.2, 371-391
  24. K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 1049-1061 https://doi.org/10.3934/dcds.2003.9.1049
  25. K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl. 283 (2003), no. 1, 46-65 https://doi.org/10.1016/S0022-247X(03)00162-8
  26. K. Ryu and I. Ahn, Positive solutions to ratio-dependent predator-prey interacting systems, J. Differ. Equations 218 (2005), 117-135 https://doi.org/10.1016/j.jde.2005.06.020
  27. N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83-99 https://doi.org/10.1016/0022-5193(79)90258-3
  28. J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983
  29. M. Wang, Z. Y. Li, and Q. X. Ye, Existence of positive solutions for semilinear elliptic system, in 'School on qualitative aspects and applications of nonlinear evolution equations (Trieste, 1990)', 256-259, World Sci. Publishing, River Edge, NJ, 1991

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