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June  2017, 10(3): 445-461. doi: 10.3934/dcdss.2017021

A periodic and diffusive predator-prey model with disease in the prey

Xiaoling Li 1, , Guangping Hu 1,, , Zhaosheng Feng 2, and  Dongliang Li 3,

1. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA
3. 

College of Atmospheric Science, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Guangping Hu

Received  December 2015 Revised  October 2016 Published  February 2017

In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a $ SI $ model when the predator species is absent. Byintroducing the basic reproduction number for the $ SI $ model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.

Keywords: Seasonality, reaction-diffusion equations, predator-prey model, disease, uniform persistence, global extinction.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92D25, 92D30.
Citation: 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
References:
[1]

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Z. J. Du, X. Chen and Z. Feng, Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type Ⅱ functional response and harvesting terms, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 1203-1214. doi: 10.3934/dcdss.2014.7.1203. Google Scholar

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

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J. K. Hale, Asymptotic Behavior of Dissipative Systems Mathematical Surveys and Monographs. Volume 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025-0-8218-1527-X. Google Scholar

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M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dynam. Diff. Eqns, 13 (2001), 107-131. doi: 10.1023/A:1009044515567. Google Scholar

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J. J. Li and W. J. Gao, Analysis of a prey-predator model with disease in prey, Appl. Math. Comput, 217 (2010), 4024-4035. doi: 10.1016/j.amc.2010.10.009. Google Scholar

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S. X. Pan, Minimal wave speeds of delayed dispersal predator-prey systems with stage structure, Electron. J. Differential Equations, 2016 (2016), 1-16. Google Scholar

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L. P. Peng, Z. Feng and C. J. Liu, Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers, Discrete Contin. Dyn. Syst., 34 (2014), 4807-4826. doi: 10.3934/dcds.2014.34.4807. Google Scholar

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R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

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R. O. Peterson and R. E. Page, Wolf density as a predictor of predation rate, Swedish Wildlife Research, suppl. 1 (1987), 771-773. Google Scholar

[20]

S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, 15 (2009), 293-316. doi: 10.1201/9781420059861.ch15. Google Scholar

[21]

B. G. Sampath Aruna Pradeep and W. Ma, Global stability of a delayed mosquito-transmitted disease model with stage structure, Electron. J. Differential Equations, 2015 (2015), 1-19. Google Scholar

[22]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math, 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar

[24]

E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402. doi: 10.1216/rmjm/1181072471. Google Scholar

[25]

E. Venturino, The effects of diseases on competing species, Math. Biosci, 174 (2001), 111-131. doi: 10.1016/S0025-5564(01)00081-5. Google Scholar

[26]

E. Venturino, Epidemics in predator-prey model: Disease in the predators, IMA J. Math. Med. Biol, 19 (2002), 185-205. doi: 10.1093/imammb/19.3.185. Google Scholar

[27]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[28]

J. Wu, Spatial structure: partial differential equations models, in Mathematical epidemiology, Lecture Notes in Math. , Springer, Berlin, 1945 (2008), 191-203. doi: 10.1007/978-3-540-78911-6_8. Google Scholar

[29]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci, 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[30]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, Appl. Math. Comput, 131 (2002), 397-414. doi: 10.1016/S0096-3003(01)00156-4. Google Scholar

[31]

L. Zhang and Z.-C. Wang, Spatial dynamics of a diffusive predator-prey model with stage structure, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1831-1853. doi: 10.3934/dcdsb.2015.20.1831. Google Scholar

[32]

L. Zhang, Z.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Differential Equations, 258 (2015), 3011-3036. doi: 10.1016/j.jde.2014.12.032. Google Scholar

[33]

X. Zhang, Y. Huang and P. Weng, Permanence and stability of a diffusive predator-prey model with disease in the prey, Comput. Math. Appl, 68 (2014), 1431-1445. doi: 10.1016/j.camwa.2014.09.011. Google Scholar

[34]

X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[35]

Y. Zhou, W. Zhang, S. Yuan and H. Hu, Persistence and extinction in stochastic SIRS models with general nonlinear incidence rate, Electron. J. Differential Equations, 2014 (2014), 1-17. Google Scholar

show all references

References:
[1]

N. Bairagi, P. K. Roy and J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions-A comparative study, J. Theor. Biol, 248 (2007), 10-25. doi: 10.1016/j.jtbi.2007.05.005. Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003. doi: 10.1002/0470871296-0-471-49301-5. Google Scholar

[3]

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

[4]

J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model, Ecol. Model, 151 (2002), 15-28. doi: 10.1016/S0304-3800(01)00415-X. Google Scholar

[5]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications. Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1992. Google Scholar

[6]

Z. J. Du, X. Chen and Z. Feng, Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type Ⅱ functional response and harvesting terms, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 1203-1214. doi: 10.3934/dcdss.2014.7.1203. Google Scholar

[7]

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

[8]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Mathematical Surveys and Monographs. Volume 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025-0-8218-1527-X. Google Scholar

[9]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1991. Google Scholar

[10]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dynam. Diff. Eqns, 13 (2001), 107-131. doi: 10.1023/A:1009044515567. Google Scholar

[11]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial. Diff. Eqns, 22 (1997), 413-433. doi: 10.1080/03605309708821269. Google Scholar

[12]

J. J. Li and W. J. Gao, Analysis of a prey-predator model with disease in prey, Appl. Math. Comput, 217 (2010), 4024-4035. doi: 10.1016/j.amc.2010.10.009. Google Scholar

[13]

Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186. doi: 10.3934/dcdsb.2009.12.169. Google Scholar

[14]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal, 37 (2005), 251-275. doi: 10.1137/S0036141003439173. Google Scholar

[15]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc, 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[16]

S. X. Pan, Minimal wave speeds of delayed dispersal predator-prey systems with stage structure, Electron. J. Differential Equations, 2016 (2016), 1-16. Google Scholar

[17]

L. P. Peng, Z. Feng and C. J. Liu, Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers, Discrete Contin. Dyn. Syst., 34 (2014), 4807-4826. doi: 10.3934/dcds.2014.34.4807. Google Scholar

[18]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

[19]

R. O. Peterson and R. E. Page, Wolf density as a predictor of predation rate, Swedish Wildlife Research, suppl. 1 (1987), 771-773. Google Scholar

[20]

S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, 15 (2009), 293-316. doi: 10.1201/9781420059861.ch15. Google Scholar

[21]

B. G. Sampath Aruna Pradeep and W. Ma, Global stability of a delayed mosquito-transmitted disease model with stage structure, Electron. J. Differential Equations, 2015 (2015), 1-19. Google Scholar

[22]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math, 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar

[24]

E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mountain J. Math, 24 (1994), 381-402. doi: 10.1216/rmjm/1181072471. Google Scholar

[25]

E. Venturino, The effects of diseases on competing species, Math. Biosci, 174 (2001), 111-131. doi: 10.1016/S0025-5564(01)00081-5. Google Scholar

[26]

E. Venturino, Epidemics in predator-prey model: Disease in the predators, IMA J. Math. Med. Biol, 19 (2002), 185-205. doi: 10.1093/imammb/19.3.185. Google Scholar

[27]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[28]

J. Wu, Spatial structure: partial differential equations models, in Mathematical epidemiology, Lecture Notes in Math. , Springer, Berlin, 1945 (2008), 191-203. doi: 10.1007/978-3-540-78911-6_8. Google Scholar

[29]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci, 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[30]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, Appl. Math. Comput, 131 (2002), 397-414. doi: 10.1016/S0096-3003(01)00156-4. Google Scholar

[31]

L. Zhang and Z.-C. Wang, Spatial dynamics of a diffusive predator-prey model with stage structure, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1831-1853. doi: 10.3934/dcdsb.2015.20.1831. Google Scholar

[32]

L. Zhang, Z.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Differential Equations, 258 (2015), 3011-3036. doi: 10.1016/j.jde.2014.12.032. Google Scholar

[33]

X. Zhang, Y. Huang and P. Weng, Permanence and stability of a diffusive predator-prey model with disease in the prey, Comput. Math. Appl, 68 (2014), 1431-1445. doi: 10.1016/j.camwa.2014.09.011. Google Scholar

[34]

X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[35]

Y. Zhou, W. Zhang, S. Yuan and H. Hu, Persistence and extinction in stochastic SIRS models with general nonlinear incidence rate, Electron. J. Differential Equations, 2014 (2014), 1-17. Google Scholar

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