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March  2018, 17(2): 375-389. doi: 10.3934/cpaa.2018021

Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction

Wonlyul Ko 1, and  Inkyung Ahn 2,,

1. 

Department of Mathematics, Korea University, Anam-dong, Seoul, 02841, South Korea
2. 

Department of Mathematics, Korea University, Sejong-ro Sejong, 30019, South Korea

Received  February 2017 Revised  August 2017 Published  March 2018

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.

Keywords: Non-constant positive steady states, eco-epidemiological model, predator-prey interaction, global attractor, pattern formation.
Mathematics Subject Classification: Primary:35J60, 35K57;Secondary:35B36.
Citation: Wonlyul Ko, Inkyung Ahn. Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction. Communications on Pure & Applied Analysis, 2018, 17 (2) : 375-389. doi: 10.3934/cpaa.2018021
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.  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.  doi: 10.2307/1940007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[12]

K. Hadeler and H. Freedman, Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631.  doi: 10.1007/BF00276947.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  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.  doi: 10.1016/S0022-5193(89)80211-5.  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.  doi: 10.2307/1940007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[12]

K. Hadeler and H. Freedman, Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631.  doi: 10.1007/BF00276947.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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