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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

 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.

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:
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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

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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