DOI QR코드

DOI QR Code

LONG-TIME PROPERTIES OF PREY-PREDATOR SYSTEM WITH CROSS-DIFFUSION

  • Shim Seong-A
  • Published : 2006.04.01

Abstract

Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.

Keywords

prey-predator system;cross-diffusion;self-diffusion;calculus inequalities;uniform bound;Liapunov functional;convergence

References

  1. E. Ahmed, A. S. Hegazi and A. S. Elgazzar, On persistence and stability of some biological systems with cross-diffusion, Advances in Complex Systems 7 (2004), no. 1, 65-76 https://doi.org/10.1142/S0219525904000056
  2. H. Amann, Dynamic theory of quasilinear parabolic equations, III. Global Existence, Math Z. 202 (1989), 219-250 https://doi.org/10.1007/BF01215256
  3. H. Amann, Non-homogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), 9-126, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993
  4. N. Boudiba and M. Pierre, Global existence for coupled reaction-diffusion systems, J. Math. Anal. Appl. 250 (2000), 1-12 https://doi.org/10.1006/jmaa.2000.6895
  5. P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z. 194 (1987), 375-396 https://doi.org/10.1007/BF01162244
  6. A. Friedman, Partial differential equations, Holt, Rinehart and Winston, New York, 1969
  7. J. U. Kim, Smooth solutions to a quasi-linear system of diffusion equations for a certain population model, Nonlinear Analysis, Theory, Methods & Applications 8 (1984), No. 10, 1121-1144
  8. K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314 https://doi.org/10.1016/j.jde.2003.10.016
  9. K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, in press
  10. Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains, Discrete Contin. Dynam. Systems 12 (2005), no. 2, 185-192
  11. Y. Lou and W. -M. Ni, Diffusion, Self-Diffusion and Cross-Diffusion, Journal of Differential Equations 131 (1996), 79-131 https://doi.org/10.1006/jdeq.1996.0157
  12. Y. Lou, W. -M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Systems 4 (1998), no. 2, 193-203 https://doi.org/10.3934/dcds.1998.4.193
  13. K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations 6 (1996), 1099-1122
  14. A. Okubo and L. A. Levin, Diffusion and Ecological Problems : modern perspective, Interdisciplinary Applied Mathematics, 2nd ed., Vol. 14, Springer, New York, 2001
  15. W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl. 197 (1996), 558-578 https://doi.org/10.1006/jmaa.1996.0039
  16. K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion system with competitive dynamics, J. Math. Anal. Appl. 283 (2003), 46-65 https://doi.org/10.1016/S0022-247X(03)00162-8
  17. N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biology 79 (1979), 83-99 https://doi.org/10.1016/0022-5193(79)90258-3
  18. S. -A. Shim, Uniform Boundedness and Convergence of Solutions to the Systems with Cross-Diffusions Dominated by Self-Diffusions, Nonlinear Analysis, Real World Applications 4 (2003), 65-86 https://doi.org/10.1016/S1468-1218(02)00014-7
  19. A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis, Theory, Methods & Applications 21 (1993), No. 8, 603-630
  20. H. Amann, Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems, Differential and Integral Equations 3 (1990), No. 1, 13-75
  21. K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems 9 (2003), 1049-1061 https://doi.org/10.3934/dcds.2003.9.1049
  22. S. -A. Shim, Uniform Boundedness and Convergence of Solutions to the Systems with a Single Non-zero Cross-Diffusion, J. Math. Anal. Appl. 279 (2003), No. 1, 1-21 https://doi.org/10.1016/S0022-247X(03)00045-3
  23. W. H. Ruan, A competing reaction-diffusion system with small cross-diffusion coefficients, Can. Appl. Math. Quart. 7 (1999), 69-91
  24. L. Nirenberg, On elliptic partial differential equations, Ann. Scuo. Norm. Sup. Pisa 13(3) (1959), 115-162
  25. S. -A. Shim, Uniform Boundedness and Convergence of Solutions to Cross-Diffusion Systems, J. Differential Equations 185 (2002), 281-305 https://doi.org/10.1006/jdeq.2002.4169

Cited by

  1. Turing Patterns in a Predator-Prey System with Self-Diffusion vol.2013, 2013, https://doi.org/10.1155/2013/891738
  2. CONVERGENCE PROPERTIES OF PREDATOR-PREY SYSTEMS WITH FUNCTIONAL RESPONSE vol.30, pp.3, 2008, https://doi.org/10.5831/HMJ.2008.30.3.411
  3. A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross- diffusion pp.0749159X, 2018, https://doi.org/10.1002/num.22315