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

- Journal title : Communications of the Korean Mathematical Society
- Volume 21, Issue 4, 2006, pp.701-717
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2006.21.4.701

Title & Authors

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

Ko, Won-Lyul; Ahn, In-Kyung;

Ko, Won-Lyul; Ahn, In-Kyung;

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.

Keywords

positive solution;ratio-dependent;fixed point index;upper/lower solution;cross-diffusion;

Language

English

Cited by

1.

[ W^{1}_{2} ]-ESTIMATES ON THE PREY-PREDATOR SYSTEMS WITH CROSS-DIFFUSIONS AND FUNCTIONAL RESPONSES,;

References

1.

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol. 139 (1989), 311-326

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

3.

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratiodependent consumption, Ecology 73 (1992), 1544-1551

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

5.

E. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1976), 131-151

6.

W. Feng and X. Lu, Permanence effect in a three-species food chain model, Appl. Anal. 54 (1994), no. 3-4, 195-209

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

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

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

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

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

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

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

15.

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Diff. Eqs. 197 (2004), 315-348

16.

N. Lakos, Existence of steady-state soludions for a one-predator two-prey system, SIAM J. Math. Analysis 21 (1990), 647-659

17.

A. Leung, A study of 3-species prey-predator reaction-diffusions by monotone schemes, J. Math. Analysis Applic. 100 (1984), 583-604

18.

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1998), 143-166

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

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

22.

Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations 154 (1999), no. 1, 157-190

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

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

26.

K. Ryu and I. Ahn, Positive solutions to ratio-dependent predator-prey interacting systems, J. Differ. Equations 218 (2005), 117-135

27.

N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83-99

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