• Previous Article
    Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise
  • PROC Home
  • This Issue
  • Next Article
    Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem
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), (1989), 311.

[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), (1991), 1287.

[3]

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

[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), (2001), 206.

[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), (1999), 65.

[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), (1992), 1552.

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1993).

[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), (2001), 489.

[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), (2001), 377.

[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), (2003), 55.

[11]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, {\em J. Math. Biol.} 36 (1998), (1998), 389.

[12]

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

[13]

P. Y. H. Pang and M. X. Wang, Stragey and stationary pattern in a three-species predator-prey model,, J. Differential Equations, (2004), 245.

[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), (2003), 919.

[15]

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

[16]

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

show all references

References:
[1]

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

[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), (1991), 1287.

[3]

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

[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), (2001), 206.

[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), (1999), 65.

[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), (1992), 1552.

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1993).

[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), (2001), 489.

[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), (2001), 377.

[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), (2003), 55.

[11]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, {\em J. Math. Biol.} 36 (1998), (1998), 389.

[12]

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

[13]

P. Y. H. Pang and M. X. Wang, Stragey and stationary pattern in a three-species predator-prey model,, J. Differential Equations, (2004), 245.

[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), (2003), 919.

[15]

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

[16]

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

[1]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501

[2]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[3]

Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303

[4]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

[5]

Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423

[6]

Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719

[7]

Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321

[8]

Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703

[9]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267

[10]

Meng Fan, Qian Wang. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 563-574. doi: 10.3934/dcdsb.2004.4.563

[11]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

[12]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823

[13]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

[14]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

[15]

Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303

[16]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[17]

Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693

[18]

Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92

[19]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

[20]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173

 Impact Factor: 

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]