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

Xiaoling Li; Guangping Hu; Zhaosheng Feng; Dongliang Li

doi:10.3934/dcdss.2017021

Previous Article
Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales
DCDS-S Home
This Issue
Next Article
State transformations of time-varying delay systems and their applications to state observer design

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]	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

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

[1]	Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[2]	Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[3]	Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks & Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863
[4]	Sebastién Gaucel, Michel Langlais. Some remarks on a singular reaction-diffusion system arising in predator-prey modeling. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 61-72. doi: 10.3934/dcdsb.2007.8.61
[5]	Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703
[6]	Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501
[7]	Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
[8]	Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653
[9]	Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[10]	Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[11]	Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[12]	Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289
[13]	Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877
[14]	Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51
[15]	Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[16]	Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[17]	Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083
[18]	Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303
[19]	Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807
[20]	Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors