# American Institute of Mathematical Sciences

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. [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. [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] 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. [5] J. P. Cohn, Saving the Salton Sea-Researchers work to understand its problems and provide possible solutions, Biosci., 50 (2000), 295-301. [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. [7] M. Fan, Y. 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. [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. [9] M. Fan, Q. 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. [10] X. Feng, Z. 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. [11] M. Friend, Avian disease at the Salton Sea, Hydrobiologia, 161 (2002), 293-306. doi: 10.1007/978-94-017-3459-2_21. [12] G. Griffiths, A. Wilby, M. 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. [13] H. W. Hethcote, W. Wang, L. 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. [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. [15] M. Koopmans, B. Wilbrink, M. Conyn, G. Natrop, H. 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. [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. [17] Y. Li, J. Wang, B. Sun, J. Tang, X. 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. [18] S. Liu, L. 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. [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. [20] Y. Lu, D. 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. [21] Y. Lu, K. 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. [22] X. Niu, L. 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. [23] C. Packer, R. D. Holt, P. J. Hudson, K. 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. [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. [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. [26] A. Shi, P. Crowley, M. McPeek, J. 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. [27] M. Sieber, H. 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. [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. [29] X. Wang, S. 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. [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. [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. [32] Y. Yang, J. 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. [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. [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. [35] T. Zhang, Z. Teng and S. Gao, Threshold conditions for a non-autonomous epidemic model with vaccination, Appl. Anal., 87 (2008), 181-199. doi: 10.1080/00036810701772196.

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##### 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. [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. [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] 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. [5] J. P. Cohn, Saving the Salton Sea-Researchers work to understand its problems and provide possible solutions, Biosci., 50 (2000), 295-301. [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. [7] M. Fan, Y. 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. [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. [9] M. Fan, Q. 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. [10] X. Feng, Z. 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. [11] M. Friend, Avian disease at the Salton Sea, Hydrobiologia, 161 (2002), 293-306. doi: 10.1007/978-94-017-3459-2_21. [12] G. Griffiths, A. Wilby, M. 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. [13] H. W. Hethcote, W. Wang, L. 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. [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. [15] M. Koopmans, B. Wilbrink, M. Conyn, G. Natrop, H. 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. [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. [17] Y. Li, J. Wang, B. Sun, J. Tang, X. 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. [18] S. Liu, L. 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. [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. [20] Y. Lu, D. 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. [21] Y. Lu, K. 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. [22] X. Niu, L. 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. [23] C. Packer, R. D. Holt, P. J. Hudson, K. 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. [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. [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. [26] A. Shi, P. Crowley, M. McPeek, J. 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. [27] M. Sieber, H. 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. [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. [29] X. Wang, S. 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. [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. [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. [32] Y. Yang, J. 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. [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. [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. [35] T. Zhang, Z. Teng and S. Gao, Threshold conditions for a non-autonomous epidemic model with vaccination, Appl. Anal., 87 (2008), 181-199. doi: 10.1080/00036810701772196.
Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. Here $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos t$; $(c)$: $\eta_1(t) = 3+\cos t$; $(d)$: $\eta_1(t) = 4.5+\cos t$
Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos(\sqrt{t})$; $(c)$: $\eta_1(t) = 3+\cos(\sqrt{t})$; $(d)$: $\eta_1(t) = 4.5+\cos(\sqrt{t})$
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
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