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

Title & Authors
LONG-TIME PROPERTIES OF PREY-PREDATOR SYSTEM WITH CROSS-DIFFUSION
Shim Seong-A;

Abstract
Using calculus inequalities and embedding theorems in $\small{R^1}$, we establish $\small{W^1_2}$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) \$d_1\;
Keywords
prey-predator system;cross-diffusion;self-diffusion;calculus inequalities;uniform bound;Liapunov functional;convergence;
Language
English
Cited by
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GLOBAL EXISTENCE OF SOLUTIONS TO THE PREY-PREDATOR SYSTEM WITH A SINGLE CROSS-DIFFUSION,;

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CONVERGENCE PROPERTIES OF PREDATOR-PREY SYSTEMS WITH FUNCTIONAL RESPONSE,;

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[ W12 ]-ESTIMATES ON THE PREY-PREDATOR SYSTEMS WITH CROSS-DIFFUSIONS AND FUNCTIONAL RESPONSES,;

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1.
Turing Patterns in a Predator-Prey System with Self-Diffusion, Abstract and Applied Analysis, 2013, 2013, 1
2.
CONVERGENCE PROPERTIES OF PREDATOR-PREY SYSTEMS WITH FUNCTIONAL RESPONSE, Honam Mathematical Journal, 2008, 30, 3, 411
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

2.
H. Amann, Dynamic theory of quasilinear parabolic equations, III. Global Existence, Math Z. 202 (1989), 219-250

3.
H. Amann, Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems, Differential and Integral Equations 3 (1990), No. 1, 13-75

4.
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

5.
N. Boudiba and M. Pierre, Global existence for coupled reaction-diffusion systems, J. Math. Anal. Appl. 250 (2000), 1-12

6.
P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z. 194 (1987), 375-396

7.
A. Friedman, Partial differential equations, Holt, Rinehart and Winston, New York, 1969

8.
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

9.
K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314

10.
K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, in press

11.
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

12.
Y. Lou and W. -M. Ni, Diffusion, Self-Diffusion and Cross-Diffusion, Journal of Differential Equations 131 (1996), 79-131

13.
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

14.
L. Nirenberg, On elliptic partial differential equations, Ann. Scuo. Norm. Sup. Pisa 13(3) (1959), 115-162

15.
K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations 6 (1996), 1099-1122

16.
A. Okubo and L. A. Levin, Diffusion and Ecological Problems : modern perspective, Interdisciplinary Applied Mathematics, 2nd ed., Vol. 14, Springer, New York, 2001

17.
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

18.
W. H. Ruan, A competing reaction-diffusion system with small cross-diffusion coefficients, Can. Appl. Math. Quart. 7 (1999), 69-91

19.
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

20.
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

21.
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biology 79 (1979), 83-99

22.
S. -A. Shim, Uniform Boundedness and Convergence of Solutions to Cross-Diffusion Systems, J. Differential Equations 185 (2002), 281-305

23.
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

24.
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

25.
A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis, Theory, Methods & Applications 21 (1993), No. 8, 603-630