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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 and Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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. 

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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. 

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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. 

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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. 

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