October  2017, 10(5): 1025-1042. doi: 10.3934/dcdss.2017054

Pattern dynamics of a delayed eco-epidemiological model with disease in the predator

1. 

Department of Computer Science and Technology, North University of China, Taiyuan Shan'xi 030051, China

2. 

Complex Systems Research Center, Shanxi University, Taiyuan Shan'xi 030051, China

* Corresponding author: Zhen Jin

Received  October 2016 Revised  January 2017 Published  June 2017

Fund Project: The work is supported by the National Natural Science Foundation of China under Grants (11331009,11671241 and 11301490), 131 Talents of Shanxi University, Program for the Outstanding Innovative Teams (OIT) of Higher Learning Institutions of Shanxi, and Natural Science Foundation of Shanxi Province Grant no. 201601D021002

The eco-epidemiology, combining interacting species with epidemiology, can describe some complex phenomena in real ecosystem. Most diseases contain the latent stage in the process of disease transmission. In this paper, a spatial eco-epidemiological model with delay and disease in the predator is studied. By mathematical analysis, the characteristic equations are derived, then we give the conditions of diffusion-driven equilibrium instability and delay-driven equilibrium instability, and find the ranges of existence of Turing patterns in parameter space. Moreover, numerical results indicate that a parameter variation has influences on time and spatially averaged densities of pattern reaching stationary states when other parameters are fixed. The obtained results may explain some mechanisms of phenomena existing in real ecosystem.

Citation: Jing Li, Zhen Jin, Gui-Quan Sun, Li-Peng Song. Pattern dynamics of a delayed eco-epidemiological model with disease in the predator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1025-1042. doi: 10.3934/dcdss.2017054
References:
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A. AbdelrazecJ. Bélair and C. Shan, Modeling the spread and control of dengue with limited public health resources, Mathematical Biosciences, 271 (2016), 136-145. doi: 10.1016/j.mbs.2015.11.004. Google Scholar

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R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford university press, Oxford, 1992.Google Scholar

[3]

J. L. AragonC. Varea and R. A. Barrio, Spatial patterning in modified Turing systems: Application to pigmentation patterns on marine fish, Forma, 13 (1998), 213-221. Google Scholar

[4]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Charles Griffin and Company Ltd, Bucks, 1975. Google Scholar

[5]

R. A. BarrioC. Varea and J. L. Aragón, A two-dimensional numerical study of spatial pattern formation in interacting Turing systems, Bulletin of mathematical biology, 61 (1999), 483-505. doi: 10.1006/bulm.1998.0093. Google Scholar

[6]

A. M. Bate and F. M. Hilker, Predator-prey oscillations can shift when diseases become endemic, Journal of Theoretical Biology, 316 (2013), 1-8. doi: 10.1016/j.jtbi.2012.09.013. Google Scholar

[7]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. doi: 10.1142/9789812798725. Google Scholar

[8]

C. BowmanA. B. Gumel and P. Van den Driessche, A mathematical model for assessing control strategies against West Nile virus, Bulletin of Mathematical Biology, 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002. Google Scholar

[9]

L. W. Buss, Competitive intransitivity and size-frequency distributions of interacting populations, Proceedings of the National Academy of Sciences, 77 (1980), 5355-5359. doi: 10.1073/pnas.77.9.5355. Google Scholar

[10]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 747-766. doi: 10.1016/S0362-546X(98)00126-6. Google Scholar

[11]

P. J. Cunningham and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139. doi: 10.2307/1932137. Google Scholar

[12]

H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Mathematical Biosciences, 68 (1984), 213-231. doi: 10.1016/0025-5564(84)90032-4. Google Scholar

[13]

N. S. GoelS. C. Maitra and E. W. Montroll, On the Volterra and other nonlinear models of interacting populations, Reviews of Modern Physics, 43 (1971), 231-276. doi: 10.1103/RevModPhys.43.231. Google Scholar

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K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27 (1989), 609-631. doi: 10.1007/BF00276947. Google Scholar

[15]

M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Analysis: Real World Applications, 11 (2010), 2224-2236. doi: 10.1016/j.nonrwa.2009.06.012. Google Scholar

[16]

H. W. Hethcote, A thousand and one epidemic models, in: S. A. Levin, Frontiers in mathematical biology, Leture Notes in Biomathematics, Springer Berlin Heidelberg, Berlin, (1994), 504-515. doi: 10.1007/978-3-642-50124-1_29. Google Scholar

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H. W. HethcoteW. Wang and L. Han, A predator-prey model with infected prey, Theoretical Population Biology, 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[18]

F. M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations, Journal of Theoretical Biology, 255 (2008), 299-306. doi: 10.1016/j.jtbi.2008.08.018. Google Scholar

[19]

Y. H. Hsieh and C. K. Hsiao, Predator-prey model with disease infection in both populations, Mathematical Medicine and Biology, 25 (2008), 247-266. doi: 10.1093/imammb/dqn017. Google Scholar

[20]

K. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118. Google Scholar

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C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. doi: 10.1126/science.284.5421.1826. Google Scholar

[22]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, Journal of Differential Equations, 231 (2006), 534-550. doi: 10.1016/j.jde.2006.08.001. Google Scholar

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Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105. Google Scholar

[24]

X. LiG. Hu and Z. Feng, A periodic and diffusive predator-prey model with disease in the prey, Discrete and Continuous Dynamical Systems-Series S, 10 (2017), 445-461. doi: 10.3934/dcdss.2017021. Google Scholar

[25]

L. LiZ. Jin and J. Li, Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Applied Mathematical Modelling, 40 (2016), 4765-4777. doi: 10.1016/j.apm.2015.12.003. Google Scholar

[26]

X. Lian, H. Wang and W. Wang, Delay-driven pattern formation in a reaction-diffusion predator-prey model incorporating a prey refuge, J. Stat. Mech. , 4 (2013), P04006, 16 pp. doi: 10.1088/1742-5468/2013/04/P04006. Google Scholar

[27]

Q. X. LiuP. M. J. Herman and W. M. Mooij, Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems, Nature communications, 5 (2014), 1-7. doi: 10.1038/ncomms6234. Google Scholar

[28]

R. T. LiuS. S. Liaw and P. K. Maini, Two-stage Turing model for generating pigment patterns on the leopard and the jaguar, Physical Review E, 74 (2006), 011914(1-8). doi: 10.1103/PhysRevE.74.011914. Google Scholar

[29]

A. Martin and S. Ruan, Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43 (2001), 247-267. doi: 10.1007/s002850100095. Google Scholar

[30]

M. Pascual, Diffusion-induced chaos in a spatial predator-prey system, Proceedings of the Royal Society of London B: Biological Sciences, 251 (1993), 1-7. doi: 10.1098/rspb.1993.0001. Google Scholar

[31]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. doi: 10.1086/282272. Google Scholar

[32]

S. Sen, P. Ghosh and S. S. Riaz et al. , Time-delay-induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80 (2009), 046212. doi: 10.1103/PhysRevE.80.046212. Google Scholar

[33]

A. R. E. SinclairS. Mduma and J. S. Brashares, Patterns of predation in a diverse predator-prey system, Nature, 425 (2003), 288-290. doi: 10.1038/nature01934. Google Scholar

[34]

L. A. de Souza and C. E. de Carvalho Freitas, Fishing sustainability via inclusion of man in predator-prey models: A case study in Lago Preto, Manacapuru, Amazonas, Ecological Modelling, 221 (2010), 703-712. doi: 10.1016/j.ecolmodel.2009.04.037. Google Scholar

[35]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, Journal of Mathematical Biology, 63 (2011), 557-574. doi: 10.1007/s00285-010-0381-5. Google Scholar

[36]

Y. SuJ. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, Journal of Differential Equations, 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. Google Scholar

[37]

G.-Q. Sun, Mathematical modeling of population dynamics with Allee effect, Nonlinear Dynamics, 85 (2016), 1-12. doi: 10.1007/s11071-016-2671-y. Google Scholar

[38]

G.-Q. SunZ. Jin and L. Li, Spatial patterns of a predator-prey model with cross diffusion, Nonlinear Dynamics, 69 (2012), 1631-1638. doi: 10.1007/s11071-012-0374-6. Google Scholar

[39]

G. SunZ. Jin and Q.-X. Liu, Pattern formation in a spatial S-I model with non-linear incidence rates, Journal of Statistical Mechanics: Theory and Experiment, 2007 (2007), P11011(1-14). Google Scholar

[40]

G.-Q. SunM. Jusup and Z. Jin, Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Physics of Life Reviews, 19 (2016), 43-47. doi: 10.1016/j.plrev.2016.08.002. Google Scholar

[41]

G. -Q. Sun, S. -L. Wang and Q. Ren, et al. , Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak, Scientific Reports, 5 (2015), 11246. doi: 10.1038/srep11246. Google Scholar

[42]

G.-Q. SunZ.-Y. Wu and Z. Jin, Influence of isolation degree of spatial patterns on persistence of populations, Nonlinear Dynamics, 83 (2016), 811-819. doi: 10.1007/s11071-015-2369-6. Google Scholar

[43]

G.-Q. SunJ. Zhang and L. P. Song, Pattern formation of a spatial predator-prey system, Applied Mathematics and Computation, 218 (2012), 11151-11162. doi: 10.1016/j.amc.2012.04.071. Google Scholar

[44]

K. Uriu and Y. Iwasa, Turing pattern formation with two kinds of cells and a diffusive chemical, Bulletin of Mathematical Biology, 69 (2007), 2515-2536. doi: 10.1007/s11538-007-9230-0. Google Scholar

[45]

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

[46]

E. Venturino, Epidemics in predator-prey models: disease in the predators, Mathematical Medicine and Biology, 19 (2002), 185-205. doi: 10.1093/imammb/19.3.185. Google Scholar

[47]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. doi: 10.1038/118558a0. Google Scholar

[48]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. Int. Explor. Mer., 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3. Google Scholar

[49]

K. A. J. White and C. A. Gilligan, Spatial heterogeneity in three species, plant-parasite-hyperparasite systems, Philosophical Transactions of the Royal Society of London B, 353 (1998), 543-557. doi: 10.1098/rstb.1998.0226. Google Scholar

[50]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences, 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[51]

Y. Xiao and L. Chen, Analysis of a three species eco-epidemiological model, Journal of Mathematical Analysis and Applications, 258 (2001), 733-754. doi: 10.1006/jmaa.2001.7514. Google Scholar

[52]

D. XiaoW. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, Journal of Mathematical Analysis and Applications, 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048. Google Scholar

[53]

Y. Xiao and F. Van Den Bosch, The dynamics of an eco-epidemic model with biological control, Ecological Modelling, 168 (2003), 203-214. doi: 10.1016/S0304-3800(03)00197-2. Google Scholar

[54]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[55]

P. Yu, Closed-form conditions of bifurcation points for general differential equations, International Journal of Bifurcation and Chaos, 15 (2005), 1467-1483. doi: 10.1142/S0218127405012582. Google Scholar

[56]

J. ZhangZ. Jin and G.-Q. Sun, Modeling seasonal rabies epidemics in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. Google Scholar

show all references

References:
[1]

A. AbdelrazecJ. Bélair and C. Shan, Modeling the spread and control of dengue with limited public health resources, Mathematical Biosciences, 271 (2016), 136-145. doi: 10.1016/j.mbs.2015.11.004. Google Scholar

[2]

R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford university press, Oxford, 1992.Google Scholar

[3]

J. L. AragonC. Varea and R. A. Barrio, Spatial patterning in modified Turing systems: Application to pigmentation patterns on marine fish, Forma, 13 (1998), 213-221. Google Scholar

[4]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Charles Griffin and Company Ltd, Bucks, 1975. Google Scholar

[5]

R. A. BarrioC. Varea and J. L. Aragón, A two-dimensional numerical study of spatial pattern formation in interacting Turing systems, Bulletin of mathematical biology, 61 (1999), 483-505. doi: 10.1006/bulm.1998.0093. Google Scholar

[6]

A. M. Bate and F. M. Hilker, Predator-prey oscillations can shift when diseases become endemic, Journal of Theoretical Biology, 316 (2013), 1-8. doi: 10.1016/j.jtbi.2012.09.013. Google Scholar

[7]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. doi: 10.1142/9789812798725. Google Scholar

[8]

C. BowmanA. B. Gumel and P. Van den Driessche, A mathematical model for assessing control strategies against West Nile virus, Bulletin of Mathematical Biology, 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002. Google Scholar

[9]

L. W. Buss, Competitive intransitivity and size-frequency distributions of interacting populations, Proceedings of the National Academy of Sciences, 77 (1980), 5355-5359. doi: 10.1073/pnas.77.9.5355. Google Scholar

[10]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 747-766. doi: 10.1016/S0362-546X(98)00126-6. Google Scholar

[11]

P. J. Cunningham and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139. doi: 10.2307/1932137. Google Scholar

[12]

H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Mathematical Biosciences, 68 (1984), 213-231. doi: 10.1016/0025-5564(84)90032-4. Google Scholar

[13]

N. S. GoelS. C. Maitra and E. W. Montroll, On the Volterra and other nonlinear models of interacting populations, Reviews of Modern Physics, 43 (1971), 231-276. doi: 10.1103/RevModPhys.43.231. Google Scholar

[14]

K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27 (1989), 609-631. doi: 10.1007/BF00276947. Google Scholar

[15]

M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Analysis: Real World Applications, 11 (2010), 2224-2236. doi: 10.1016/j.nonrwa.2009.06.012. Google Scholar

[16]

H. W. Hethcote, A thousand and one epidemic models, in: S. A. Levin, Frontiers in mathematical biology, Leture Notes in Biomathematics, Springer Berlin Heidelberg, Berlin, (1994), 504-515. doi: 10.1007/978-3-642-50124-1_29. Google Scholar

[17]

H. W. HethcoteW. Wang and L. Han, A predator-prey model with infected prey, Theoretical Population Biology, 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[18]

F. M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations, Journal of Theoretical Biology, 255 (2008), 299-306. doi: 10.1016/j.jtbi.2008.08.018. Google Scholar

[19]

Y. H. Hsieh and C. K. Hsiao, Predator-prey model with disease infection in both populations, Mathematical Medicine and Biology, 25 (2008), 247-266. doi: 10.1093/imammb/dqn017. Google Scholar

[20]

K. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118. Google Scholar

[21]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. doi: 10.1126/science.284.5421.1826. Google Scholar

[22]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, Journal of Differential Equations, 231 (2006), 534-550. doi: 10.1016/j.jde.2006.08.001. Google Scholar

[23]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105. Google Scholar

[24]

X. LiG. Hu and Z. Feng, A periodic and diffusive predator-prey model with disease in the prey, Discrete and Continuous Dynamical Systems-Series S, 10 (2017), 445-461. doi: 10.3934/dcdss.2017021. Google Scholar

[25]

L. LiZ. Jin and J. Li, Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Applied Mathematical Modelling, 40 (2016), 4765-4777. doi: 10.1016/j.apm.2015.12.003. Google Scholar

[26]

X. Lian, H. Wang and W. Wang, Delay-driven pattern formation in a reaction-diffusion predator-prey model incorporating a prey refuge, J. Stat. Mech. , 4 (2013), P04006, 16 pp. doi: 10.1088/1742-5468/2013/04/P04006. Google Scholar

[27]

Q. X. LiuP. M. J. Herman and W. M. Mooij, Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems, Nature communications, 5 (2014), 1-7. doi: 10.1038/ncomms6234. Google Scholar

[28]

R. T. LiuS. S. Liaw and P. K. Maini, Two-stage Turing model for generating pigment patterns on the leopard and the jaguar, Physical Review E, 74 (2006), 011914(1-8). doi: 10.1103/PhysRevE.74.011914. Google Scholar

[29]

A. Martin and S. Ruan, Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43 (2001), 247-267. doi: 10.1007/s002850100095. Google Scholar

[30]

M. Pascual, Diffusion-induced chaos in a spatial predator-prey system, Proceedings of the Royal Society of London B: Biological Sciences, 251 (1993), 1-7. doi: 10.1098/rspb.1993.0001. Google Scholar

[31]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. doi: 10.1086/282272. Google Scholar

[32]

S. Sen, P. Ghosh and S. S. Riaz et al. , Time-delay-induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80 (2009), 046212. doi: 10.1103/PhysRevE.80.046212. Google Scholar

[33]

A. R. E. SinclairS. Mduma and J. S. Brashares, Patterns of predation in a diverse predator-prey system, Nature, 425 (2003), 288-290. doi: 10.1038/nature01934. Google Scholar

[34]

L. A. de Souza and C. E. de Carvalho Freitas, Fishing sustainability via inclusion of man in predator-prey models: A case study in Lago Preto, Manacapuru, Amazonas, Ecological Modelling, 221 (2010), 703-712. doi: 10.1016/j.ecolmodel.2009.04.037. Google Scholar

[35]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, Journal of Mathematical Biology, 63 (2011), 557-574. doi: 10.1007/s00285-010-0381-5. Google Scholar

[36]

Y. SuJ. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, Journal of Differential Equations, 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. Google Scholar

[37]

G.-Q. Sun, Mathematical modeling of population dynamics with Allee effect, Nonlinear Dynamics, 85 (2016), 1-12. doi: 10.1007/s11071-016-2671-y. Google Scholar

[38]

G.-Q. SunZ. Jin and L. Li, Spatial patterns of a predator-prey model with cross diffusion, Nonlinear Dynamics, 69 (2012), 1631-1638. doi: 10.1007/s11071-012-0374-6. Google Scholar

[39]

G. SunZ. Jin and Q.-X. Liu, Pattern formation in a spatial S-I model with non-linear incidence rates, Journal of Statistical Mechanics: Theory and Experiment, 2007 (2007), P11011(1-14). Google Scholar

[40]

G.-Q. SunM. Jusup and Z. Jin, Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Physics of Life Reviews, 19 (2016), 43-47. doi: 10.1016/j.plrev.2016.08.002. Google Scholar

[41]

G. -Q. Sun, S. -L. Wang and Q. Ren, et al. , Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak, Scientific Reports, 5 (2015), 11246. doi: 10.1038/srep11246. Google Scholar

[42]

G.-Q. SunZ.-Y. Wu and Z. Jin, Influence of isolation degree of spatial patterns on persistence of populations, Nonlinear Dynamics, 83 (2016), 811-819. doi: 10.1007/s11071-015-2369-6. Google Scholar

[43]

G.-Q. SunJ. Zhang and L. P. Song, Pattern formation of a spatial predator-prey system, Applied Mathematics and Computation, 218 (2012), 11151-11162. doi: 10.1016/j.amc.2012.04.071. Google Scholar

[44]

K. Uriu and Y. Iwasa, Turing pattern formation with two kinds of cells and a diffusive chemical, Bulletin of Mathematical Biology, 69 (2007), 2515-2536. doi: 10.1007/s11538-007-9230-0. Google Scholar

[45]

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

[46]

E. Venturino, Epidemics in predator-prey models: disease in the predators, Mathematical Medicine and Biology, 19 (2002), 185-205. doi: 10.1093/imammb/19.3.185. Google Scholar

[47]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. doi: 10.1038/118558a0. Google Scholar

[48]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. Int. Explor. Mer., 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3. Google Scholar

[49]

K. A. J. White and C. A. Gilligan, Spatial heterogeneity in three species, plant-parasite-hyperparasite systems, Philosophical Transactions of the Royal Society of London B, 353 (1998), 543-557. doi: 10.1098/rstb.1998.0226. Google Scholar

[50]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences, 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[51]

Y. Xiao and L. Chen, Analysis of a three species eco-epidemiological model, Journal of Mathematical Analysis and Applications, 258 (2001), 733-754. doi: 10.1006/jmaa.2001.7514. Google Scholar

[52]

D. XiaoW. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, Journal of Mathematical Analysis and Applications, 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048. Google Scholar

[53]

Y. Xiao and F. Van Den Bosch, The dynamics of an eco-epidemic model with biological control, Ecological Modelling, 168 (2003), 203-214. doi: 10.1016/S0304-3800(03)00197-2. Google Scholar

[54]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[55]

P. Yu, Closed-form conditions of bifurcation points for general differential equations, International Journal of Bifurcation and Chaos, 15 (2005), 1467-1483. doi: 10.1142/S0218127405012582. Google Scholar

[56]

J. ZhangZ. Jin and G.-Q. Sun, Modeling seasonal rabies epidemics in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. Google Scholar

Figure 1.  Schematic diagrams of the cubic function $y(e)$ for $y_{1}>0$ in Theorem 3.3. (a) $y_{3}x y0$. (b) $y_{3}=0$ and $y_{2} < 0$. (c) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2} < 0$
Figure 2.  The bifurcation diagram of system (4) in parameter space $r-h$. (a) Parameters are $\beta_{1}=1.8$, $\mu=0.6$, $m=0.8$, $D_{1}=1$, $D_{2}=0.03$, $D_{3}=2$, $\tau=0.01$. (b) Parameters are $\beta_{1}=1.8$, $\mu=0.6$, $m=0.7$, $D_{1}=10$, $D_{2}=0.1$, $D_{3}=4$, $\tau=0.01$
Figure 3.  Coefficients of the dispersion relation of the characteristic equation (16) for $r=0.1$, $h=0.07$, $\beta_{1}=1.8$, $\mu=0.6$, $m=0.8$, $D_{1}=1$, $D_{2}=0.03$, $D_{3}=2$, $\tau=0.01$
Figure 4.  Coefficients of the dispersion relation of the characteristic equation (16) for $r=0.1$, $h=0.1$, $\beta_{1}=1.8$, $\mu=0.6$, $m=0.7$, $D_{1}=10$, $D_{2}=0.1$, $D_{3}=4$, $\tau=0.01$
Figure 5.  Schematic diagrams of the cubic function $y(e)$ for $y_{1}>0$ in Theorem 3.4. (a) $y_{2}^{2}-3y_{1}y_{3}\leq 0$. (b) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2}>0$. (c) $y_{3}=0$ and $y_{2}>0$. (d) $y_{3} < 0$. (e) $y_{3}=0$ and $y_{2} < 0$. (f) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2} < 0$
Figure 6.  Spatial patterns (top) and the corresponding spatially averaged population density (bottom). (a) Small "black-eye" pattern (r=0.1), (b) small "black-eye" pattern (r=0.15)
Figure 7.  Spatial patterns (top) and the corresponding spatially averaged population density (bottom). (a) Big "black-eye" pattern (h=0.1), (b) big "black-eye" pattern (h=0.14)
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