-
Previous Article
The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system
- CPAA Home
- This Issue
-
Next Article
Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating
Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction
1. | Department of Mathematics, Korea University, Anam-dong, Seoul, 02841, South Korea |
2. | Department of Mathematics, Korea University, Sejong-ro Sejong, 30019, South Korea |
We consider a predator-prey system with a ratio-dependent functional response when a prey population is infected. First, we examine the global attractor and persistence properties of the time-dependent system. The existence of nonconstant positive steady-states are studied under Neumann boundary conditions in terms of the diffusion effect; namely, pattern formations, arising from diffusion-driven instability, are investigated. A comparison principle for the parabolic problem and the Leray-Schauder index theory are employed for analysis.
References:
[1] |
R. Arditi and L. R. Ginzburg,
Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[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 ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.
doi: 10.2307/1940007. |
[4] |
O. Arino, A. El. abdllaoui, J. Mikram and J. Chattopadhyay,
Infection in prey population may act as a
biological control in raito-dependent predator-prey models, Nonlinearity, 17 (2004), 1101-1116.
doi: 10.1088/0951-7715/17/3/018. |
[5] |
E. Beltrami and T. Carroll,
Modelling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32 (1994), 857-863.
doi: 10.1007/BF00168802. |
[6] |
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), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[7] |
J. Chattopadhyay and O. Arino,
A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766.
doi: 10.1016/S0362-546X(98)00126-6. |
[8] |
J. Chattopadhyay and S. Pal,
Viral infection of phytoplankton-zooplankton system-a mathematical modeling, Ecol. Modelling, 151 (2002), 15-28.
doi: 10.1016/S0304-3800(01)00415-X. |
[9] |
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.
doi: 10.1006/tpbi.1999.1414. |
[10] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. |
[11] |
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.
doi: 10.2307/1940008. |
[12] |
K. Hadeler and H. Freedman,
Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631.
doi: 10.1007/BF00276947. |
[13] |
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), 377-396.
doi: 10.1007/s002850100100. |
[14] |
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), 55-83.
doi: 10.1016/S0025-5564(02)00127-X. |
[15] |
Y. Kuang and E. Beretta,
Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050. |
[16] |
Y. Lou and W. M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[17] |
C. S. Lin, W. M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[18] |
Z. Lin and M. Pedersen,
Stability in a diffusive food-chain model with Michaelis-Menten functional
response, Nonlinear Anal., 57 (2004), 421-433.
doi: 10.1016/j.na.2004.02.022. |
[19] |
L. Nirenberg,
Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1974. |
[20] |
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, 33 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[21] |
P. Y. H. Pang and M. X. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[22] |
C. V. Pao,
Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[23] |
C. V. Pao,
Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903.
doi: 10.1016/0362-546X(95)00058-4. |
[24] |
K. Ryu and I. Ahn,
Positive solutions to ratio-dependent predator-prey interacting systems, J. Differential Equations, 218 (2005), 117-135.
doi: 10.1016/j.jde.2005.06.020. |
[25] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations,
$2^{nd}$ edition, Springer-Verlag, New York, 1994. |
[26] |
Y. Xiao and L. Chen,
A ratio-dependent predator-prey model with disease in the prey, Appl. Maths. Comp., 131 (2002), 397-414.
doi: 10.1016/S0096-3003(01)00156-4. |
[27] |
Y. Xiao and L. Chen,
Analysis of a three species eco-epidemiological model, J. Math. Anal.
Appl., 258 (2001), 733-754.
doi: 10.1006/jmaa.2001.7514. |
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.
doi: 10.1016/S0022-5193(89)80211-5. |
[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 ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.
doi: 10.2307/1940007. |
[4] |
O. Arino, A. El. abdllaoui, J. Mikram and J. Chattopadhyay,
Infection in prey population may act as a
biological control in raito-dependent predator-prey models, Nonlinearity, 17 (2004), 1101-1116.
doi: 10.1088/0951-7715/17/3/018. |
[5] |
E. Beltrami and T. Carroll,
Modelling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32 (1994), 857-863.
doi: 10.1007/BF00168802. |
[6] |
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), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[7] |
J. Chattopadhyay and O. Arino,
A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766.
doi: 10.1016/S0362-546X(98)00126-6. |
[8] |
J. Chattopadhyay and S. Pal,
Viral infection of phytoplankton-zooplankton system-a mathematical modeling, Ecol. Modelling, 151 (2002), 15-28.
doi: 10.1016/S0304-3800(01)00415-X. |
[9] |
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.
doi: 10.1006/tpbi.1999.1414. |
[10] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. |
[11] |
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.
doi: 10.2307/1940008. |
[12] |
K. Hadeler and H. Freedman,
Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631.
doi: 10.1007/BF00276947. |
[13] |
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), 377-396.
doi: 10.1007/s002850100100. |
[14] |
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), 55-83.
doi: 10.1016/S0025-5564(02)00127-X. |
[15] |
Y. Kuang and E. Beretta,
Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050. |
[16] |
Y. Lou and W. M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[17] |
C. S. Lin, W. M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[18] |
Z. Lin and M. Pedersen,
Stability in a diffusive food-chain model with Michaelis-Menten functional
response, Nonlinear Anal., 57 (2004), 421-433.
doi: 10.1016/j.na.2004.02.022. |
[19] |
L. Nirenberg,
Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1974. |
[20] |
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, 33 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[21] |
P. Y. H. Pang and M. X. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[22] |
C. V. Pao,
Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[23] |
C. V. Pao,
Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903.
doi: 10.1016/0362-546X(95)00058-4. |
[24] |
K. Ryu and I. Ahn,
Positive solutions to ratio-dependent predator-prey interacting systems, J. Differential Equations, 218 (2005), 117-135.
doi: 10.1016/j.jde.2005.06.020. |
[25] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations,
$2^{nd}$ edition, Springer-Verlag, New York, 1994. |
[26] |
Y. Xiao and L. Chen,
A ratio-dependent predator-prey model with disease in the prey, Appl. Maths. Comp., 131 (2002), 397-414.
doi: 10.1016/S0096-3003(01)00156-4. |
[27] |
Y. Xiao and L. Chen,
Analysis of a three species eco-epidemiological model, J. Math. Anal.
Appl., 258 (2001), 733-754.
doi: 10.1006/jmaa.2001.7514. |
[1] |
Jing Li, Zhen Jin, Gui-Quan Sun, Li-Peng Song. Pattern dynamics of a delayed eco-epidemiological model with disease in the predator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1025-1042. doi: 10.3934/dcdss.2017054 |
[2] |
Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026 |
[3] |
Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189 |
[4] |
Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136 |
[5] |
Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209 |
[6] |
Lopo F. de Jesus, César M. Silva, Helder Vilarinho. Random perturbations of an eco-epidemiological model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 257-275. doi: 10.3934/dcdsb.2021040 |
[7] |
Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035 |
[8] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[9] |
Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045 |
[10] |
Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849 |
[11] |
Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295 |
[12] |
Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173 |
[13] |
R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2791-2815. doi: 10.3934/dcdsb.2021160 |
[14] |
Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 |
[15] |
Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082 |
[16] |
Andrei Korobeinikov. Global properties of a general predator-prey model with non-symmetric attack and consumption rate. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1095-1103. doi: 10.3934/dcdsb.2010.14.1095 |
[17] |
Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 |
[18] |
Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101 |
[19] |
H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221 |
[20] |
Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]