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2013, 2013(special): 11-19. doi: 10.3934/proc.2013.2013.11

Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey

1. 

Department of Information and Mathematics, Korea University, Jochiwon 339-700, South Korea, South Korea

2. 

Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 360-764, South Korea

Received  September 2012 Revised  January 2013 Published  November 2013

In this paper, we consider a ratio-dependent predator-prey model with disease in the prey under Neumann boundary condition. we construct a global attractor region for all time-dependent non-negative solutions of the system and investigate the asymptotic behavior of positive constant solution. Furthermore, we also study the asymptotic behavior of the non-negative equilibria.
Citation: Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11
References:
[1]

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

[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. Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, New York, 1993.  Google Scholar

[8]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol. 42 (2001), no. 6, 489-506.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36 (1998), no. 4, 389-406.  Google Scholar

[12]

Z. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal. 57(2004) 421-433.  Google Scholar

[13]

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.  Google Scholar

[14]

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 (4) (2003), 919-942.  Google Scholar

[15]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal. 26(1996), 1889-1903.  Google Scholar

[16]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, Appl. Maths. Comp. 131(2002), 397-414.  Google Scholar

show all references

References:
[1]

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

[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. Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, New York, 1993.  Google Scholar

[8]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol. 42 (2001), no. 6, 489-506.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36 (1998), no. 4, 389-406.  Google Scholar

[12]

Z. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal. 57(2004) 421-433.  Google Scholar

[13]

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.  Google Scholar

[14]

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 (4) (2003), 919-942.  Google Scholar

[15]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal. 26(1996), 1889-1903.  Google Scholar

[16]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, Appl. Maths. Comp. 131(2002), 397-414.  Google Scholar

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