November  2018, 23(9): 3817-3836. doi: 10.3934/dcdsb.2018082

A non-autonomous predator-prey model with infected prey

1. 

College of Mathematics and Statistics, Northeast Petroleum University, Daqing, 163318, China

2. 

College of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

3. 

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

1Corresponding author

Received  September 2016 Revised  December 2017 Published  March 2018

A non-constant eco-epidemiological model with SIS-type infectious disease in prey is formulated and investigated, it is assumed that the disease is endemic in prey before the invasion of predator and that predation is more likely on infected prey than on the uninfected. Sufficient conditions for both permanence and extinction of the infected prey, and the necessary conditions for the permanence of the infected prey are established. It is shown that the predation preference to infected prey may even increase the possibility of disease endemic, and that the introduction of new resource for predator could be helpful for it to eradicate the infected prey. Numerical simulations have been performed to verify/extend our analytical results.

Citation: Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[2]

A. M. Bate and F. M. Hilker, Disease in group-defending prey can benefit predators, Theor Ecol., 7 (2014), 87-100. doi: 10.1007/s12080-013-0200-x. 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]

P. Chesson, Understanding the role of environmental variation in population and community dynamics, Theoret. Population. Biol., 64 (2003), 253-254. doi: 10.1016/j.tpb.2003.06.002. Google Scholar

[5]

J. P. Cohn, Saving the Salton Sea-Researchers work to understand its problems and provide possible solutions, Biosci., 50 (2000), 295-301. Google Scholar

[6]

A. P. Dobson, The population biology of parasite-induced changes in host behavior, Q. Rev. Biol., 30 (1988), 139-165. doi: 10.1086/415837. Google Scholar

[7]

M. FanY. Michael and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170 (2001), 199-208. doi: 10.1016/S0025-5564(00)00067-5. Google Scholar

[8]

M. Fan and Y. Kuang, Dynamics of a non-autonomous predator-prey system with the Beddington-Deangelis functional response, Math. Anal. Appl., 295 (2004), 15-39. doi: 10.1016/j.jmaa.2004.02.038. Google Scholar

[9]

M. FanQ. Wang and X. F. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proc. Roy. Soc. Edinborgh Seet., 133 (2003), 97-118. doi: 10.1017/S0308210500002304. Google Scholar

[10]

X. FengZ. Teng and L. Zhang, The permanence for nonautonomous n-species Lotka-Volterra competitive systems with feedback controls, Rocky Mountain J. Math., 38 (2008), 1355-1376. doi: 10.1216/RMJ-2008-38-5-1355. Google Scholar

[11]

M. Friend, Avian disease at the Salton Sea, Hydrobiologia, 161 (2002), 293-306. doi: 10.1007/978-94-017-3459-2_21. Google Scholar

[12]

G. GriffithsA. WilbyM. Crawley and M. Thomas, Density-dependent effects of predator species-richness in diversity-function studies, Ecology, 89 (2008), 2986-2993. doi: 10.1890/08-0685.1. Google Scholar

[13]

H. W. HethcoteW. WangL. Han and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[14]

J. C. Holmes and W. M. Bethel, Modifications of intermediate host behaviour by parasites, In: Canning, E. V., Wright, C. A. (Eds.), Behavioural Aspects of Parasite Transmission, Suppl I to Zool. f. Linnean Soc., 51 (1972), 123-149. Google Scholar

[15]

M. KoopmansB. WilbrinkM. ConynG. NatropH. van der Nat and H. Vennema, Transmission of H7N7 avian influenza A virus to human beings during a large outbreak in commercial poultry farms in the Netherlands, Lancet., 363 (2004), 587-593. doi: 10.1016/S0140-6736(04)15589-X. Google Scholar

[16]

J. R. Krebs, Optimal foraging: decision rules for predators, In: Krebs, J. R., Davies, N.B. (Eds.), Behavioural Ecology: an Evolutionary approach, First ed. Blackwell Scientific Publishers, Oxford, (1978), 23-63. Google Scholar

[17]

Y. LiJ. WangB. SunJ. TangX. Xie and S. Pang, Modeling and analysis of the secondary routine dose against measles in China, Adv. Difference Equ., 89 (2017), 1-14. doi: 10.1186/s13662-017-1125-2. Google Scholar

[18]

S. LiuL. Chen and Z. Liu, Extionction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667-684. doi: 10.1016/S0022-247X(02)00329-3. Google Scholar

[19]

S. Liu, X. Xie and J. Tang, Competing population model with nonlinear intraspecific regulation and maturation delays, Int. J. Biomath., 5(2012), 1260007, 22 pp. doi: 10.1142/S1793524512600078. Google Scholar

[20]

Y. LuD. Li and S. Liu, Modeling of hunting strategies of the predators in susceptible and infected prey, Appl. Math. Comput., 284 (2016), 268-285. doi: 10.1016/j.amc.2016.03.005. Google Scholar

[21]

Y. LuK. Pawelek and S. Liu, A stage-structured predator-prey model with predation over juvenile prey, Appl. Math. Comput., 297 (2017), 115-130. doi: 10.1016/j.amc.2016.10.035. Google Scholar

[22]

X. NiuL. Zhang and Z. Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Appl. Math. Model., 35 (2011), 457-470. doi: 10.1016/j.apm.2010.07.010. Google Scholar

[23]

C. PackerR. D. HoltP. J. HudsonK. D. Lafferty and A. P. Dobson, Keeping the herds healthy and alert: Implications of predator control for infectious disease, Ecol. Lett., 6 (2003), 797-802. doi: 10.1046/j.1461-0248.2003.00500.x. Google Scholar

[24]

R. O. Peterson and R. E. Page, The rise and fall of Isle Royale wolves, 1975-1986, J. Mamm., 69 (1988), 89-99. doi: 10.2307/1381751. Google Scholar

[25]

G. P. Samanta, Analysis of a delay nonautonomous predator-prey system with disease in the prey, Nonlinear Anal. Model. Control, 15 (2010), 97-108. Google Scholar

[26]

A. ShiP. CrowleyM. McPeekJ. Petranka and K. Strohmeier, Predation, competition and prey communities: A review of field experiments, Ann. Rev. Ecol. Semantics, 16 (1985), 269-311. doi: 10.1146/annurev.es.16.110185.001413. Google Scholar

[27]

M. SieberH. Malchow and F. M. Hilker, Disease-induced modification of prey competition in eco-epidemiological models, Ecological Complexity, 18 (2014), 74-82. doi: 10.1016/j.ecocom.2013.06.002. Google Scholar

[28]

X. Wang, S. Liu and X. Song, Dynamic of a non-autonomous HIV-1 infection model with delays, Int. J. Biomath., 6 (2013), 59–84. doi: 10.11421S1793524513500307. Google Scholar

[29]

X. WangS. Liu and L. Rong, Permanence and extinction of a nonautonomous HIV-1 model with time delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1783-1800. doi: 10.3934/dcdsb.2014.19.1783. Google Scholar

[30]

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

[31]

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

[32]

Y. YangJ. Yin and C. Jin, Existence and attractivity of time periodic solutions for Nicholson's blowflies model with nonlinear diffusion, Math. Methods Appl. Sci., 37 (2014), 1736-1754. doi: 10.1002/mma.2932. Google Scholar

[33]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z. Google Scholar

[34]

T. Zhang and Z. Teng, Permanence and extinction for a non-autonomous SIRS epidemic model with time delay, Appl. Math. Model., 33 (2009), 1058-1071. doi: 10.1016/j.apm.2007.12.020. Google Scholar

[35]

T. ZhangZ. Teng and S. Gao, Threshold conditions for a non-autonomous epidemic model with vaccination, Appl. Anal., 87 (2008), 181-199. doi: 10.1080/00036810701772196. Google Scholar

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[2]

A. M. Bate and F. M. Hilker, Disease in group-defending prey can benefit predators, Theor Ecol., 7 (2014), 87-100. doi: 10.1007/s12080-013-0200-x. 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]

P. Chesson, Understanding the role of environmental variation in population and community dynamics, Theoret. Population. Biol., 64 (2003), 253-254. doi: 10.1016/j.tpb.2003.06.002. Google Scholar

[5]

J. P. Cohn, Saving the Salton Sea-Researchers work to understand its problems and provide possible solutions, Biosci., 50 (2000), 295-301. Google Scholar

[6]

A. P. Dobson, The population biology of parasite-induced changes in host behavior, Q. Rev. Biol., 30 (1988), 139-165. doi: 10.1086/415837. Google Scholar

[7]

M. FanY. Michael and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170 (2001), 199-208. doi: 10.1016/S0025-5564(00)00067-5. Google Scholar

[8]

M. Fan and Y. Kuang, Dynamics of a non-autonomous predator-prey system with the Beddington-Deangelis functional response, Math. Anal. Appl., 295 (2004), 15-39. doi: 10.1016/j.jmaa.2004.02.038. Google Scholar

[9]

M. FanQ. Wang and X. F. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proc. Roy. Soc. Edinborgh Seet., 133 (2003), 97-118. doi: 10.1017/S0308210500002304. Google Scholar

[10]

X. FengZ. Teng and L. Zhang, The permanence for nonautonomous n-species Lotka-Volterra competitive systems with feedback controls, Rocky Mountain J. Math., 38 (2008), 1355-1376. doi: 10.1216/RMJ-2008-38-5-1355. Google Scholar

[11]

M. Friend, Avian disease at the Salton Sea, Hydrobiologia, 161 (2002), 293-306. doi: 10.1007/978-94-017-3459-2_21. Google Scholar

[12]

G. GriffithsA. WilbyM. Crawley and M. Thomas, Density-dependent effects of predator species-richness in diversity-function studies, Ecology, 89 (2008), 2986-2993. doi: 10.1890/08-0685.1. Google Scholar

[13]

H. W. HethcoteW. WangL. Han and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[14]

J. C. Holmes and W. M. Bethel, Modifications of intermediate host behaviour by parasites, In: Canning, E. V., Wright, C. A. (Eds.), Behavioural Aspects of Parasite Transmission, Suppl I to Zool. f. Linnean Soc., 51 (1972), 123-149. Google Scholar

[15]

M. KoopmansB. WilbrinkM. ConynG. NatropH. van der Nat and H. Vennema, Transmission of H7N7 avian influenza A virus to human beings during a large outbreak in commercial poultry farms in the Netherlands, Lancet., 363 (2004), 587-593. doi: 10.1016/S0140-6736(04)15589-X. Google Scholar

[16]

J. R. Krebs, Optimal foraging: decision rules for predators, In: Krebs, J. R., Davies, N.B. (Eds.), Behavioural Ecology: an Evolutionary approach, First ed. Blackwell Scientific Publishers, Oxford, (1978), 23-63. Google Scholar

[17]

Y. LiJ. WangB. SunJ. TangX. Xie and S. Pang, Modeling and analysis of the secondary routine dose against measles in China, Adv. Difference Equ., 89 (2017), 1-14. doi: 10.1186/s13662-017-1125-2. Google Scholar

[18]

S. LiuL. Chen and Z. Liu, Extionction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667-684. doi: 10.1016/S0022-247X(02)00329-3. Google Scholar

[19]

S. Liu, X. Xie and J. Tang, Competing population model with nonlinear intraspecific regulation and maturation delays, Int. J. Biomath., 5(2012), 1260007, 22 pp. doi: 10.1142/S1793524512600078. Google Scholar

[20]

Y. LuD. Li and S. Liu, Modeling of hunting strategies of the predators in susceptible and infected prey, Appl. Math. Comput., 284 (2016), 268-285. doi: 10.1016/j.amc.2016.03.005. Google Scholar

[21]

Y. LuK. Pawelek and S. Liu, A stage-structured predator-prey model with predation over juvenile prey, Appl. Math. Comput., 297 (2017), 115-130. doi: 10.1016/j.amc.2016.10.035. Google Scholar

[22]

X. NiuL. Zhang and Z. Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Appl. Math. Model., 35 (2011), 457-470. doi: 10.1016/j.apm.2010.07.010. Google Scholar

[23]

C. PackerR. D. HoltP. J. HudsonK. D. Lafferty and A. P. Dobson, Keeping the herds healthy and alert: Implications of predator control for infectious disease, Ecol. Lett., 6 (2003), 797-802. doi: 10.1046/j.1461-0248.2003.00500.x. Google Scholar

[24]

R. O. Peterson and R. E. Page, The rise and fall of Isle Royale wolves, 1975-1986, J. Mamm., 69 (1988), 89-99. doi: 10.2307/1381751. Google Scholar

[25]

G. P. Samanta, Analysis of a delay nonautonomous predator-prey system with disease in the prey, Nonlinear Anal. Model. Control, 15 (2010), 97-108. Google Scholar

[26]

A. ShiP. CrowleyM. McPeekJ. Petranka and K. Strohmeier, Predation, competition and prey communities: A review of field experiments, Ann. Rev. Ecol. Semantics, 16 (1985), 269-311. doi: 10.1146/annurev.es.16.110185.001413. Google Scholar

[27]

M. SieberH. Malchow and F. M. Hilker, Disease-induced modification of prey competition in eco-epidemiological models, Ecological Complexity, 18 (2014), 74-82. doi: 10.1016/j.ecocom.2013.06.002. Google Scholar

[28]

X. Wang, S. Liu and X. Song, Dynamic of a non-autonomous HIV-1 infection model with delays, Int. J. Biomath., 6 (2013), 59–84. doi: 10.11421S1793524513500307. Google Scholar

[29]

X. WangS. Liu and L. Rong, Permanence and extinction of a nonautonomous HIV-1 model with time delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1783-1800. doi: 10.3934/dcdsb.2014.19.1783. Google Scholar

[30]

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

[31]

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

[32]

Y. YangJ. Yin and C. Jin, Existence and attractivity of time periodic solutions for Nicholson's blowflies model with nonlinear diffusion, Math. Methods Appl. Sci., 37 (2014), 1736-1754. doi: 10.1002/mma.2932. Google Scholar

[33]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z. Google Scholar

[34]

T. Zhang and Z. Teng, Permanence and extinction for a non-autonomous SIRS epidemic model with time delay, Appl. Math. Model., 33 (2009), 1058-1071. doi: 10.1016/j.apm.2007.12.020. Google Scholar

[35]

T. ZhangZ. Teng and S. Gao, Threshold conditions for a non-autonomous epidemic model with vaccination, Appl. Anal., 87 (2008), 181-199. doi: 10.1080/00036810701772196. Google Scholar

Figure 1.  Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. Here $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos t$; $(c)$: $\eta_1(t) = 3+\cos t$; $(d)$: $\eta_1(t) = 4.5+\cos t$
Figure 2.  Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos(\sqrt{t})$; $(c)$: $\eta_1(t) = 3+\cos(\sqrt{t})$; $(d)$: $\eta_1(t) = 4.5+\cos(\sqrt{t})$
Figure 3.  Basic behavior of solutions of model (49) with different intrinsic growth rate $r(t)$ for predator $y$. $(a)$: $r(t)\equiv 0$; $(b)$: $r(t) = \sin t+13$; $(c)$: $r(t) = \sin t+19$.Here we set $\eta_1(t) = \cos t+1$, and all the other parameters are same as those for FIG. 1
[1]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167

[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]

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

[4]

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

[5]

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

[6]

H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221

[7]

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

[8]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268

[9]

Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

[10]

Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831

[11]

Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028

[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]

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

[14]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[15]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059

[16]

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

[17]

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

[18]

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

[19]

Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11

[20]

Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (119)
  • HTML views (481)
  • Cited by (0)

Other articles
by authors

[Back to Top]