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State transformations of time-varying delay systems and their applications to state observer design
A periodic and diffusive predator-prey model with disease in the prey
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 |
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.
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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