September  2018, 17(5): 1899-1920. doi: 10.3934/cpaa.2018090

Existence and asymptotic behaviors of traveling waves of a modified vector-disease model

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

2. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA

* Corresponding author

Received  July 2017 Revised  November 2017 Published  April 2018

Fund Project: This work is supported by NSF of China (Grants No. 11471146,11771185 and 11671176).

In this paper, we are concerned with the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We establish the existence of traveling wave solutions for the modified vector-disease model without delay, then explore the existence of traveling fronts for the model with a special local delay convolution kernel by employing the geometric singular perturbation theory and the linear chain trick. Finally, we deal with the local stability of the steady states, the existence and asymptotic behaviors of traveling wave solutions for the model with the convolution kernel of a special non-local delay.

Citation: Zengji Du, Zhaosheng Feng. Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1899-1920. doi: 10.3934/cpaa.2018090
References:
[1]

P. AshwinM. V. Bartuccelli and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.   Google Scholar

[2]

P. W. Bates and F. X. Chen, Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 26 (1999), 1-19.   Google Scholar

[3]

X. Chen and Z. J. Du, Existence of Positive Periodic Solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst., 17 (2018), 67-80.   Google Scholar

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.   Google Scholar

[5]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.   Google Scholar

[6]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.   Google Scholar

[7]

Z. J. DuZ. Feng and X.N. Zhang, Traveling wave phenomena of n-dimensional diffusive predatorprey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.   Google Scholar

[8]

F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters, Discrete Contin. Dyn. Syst., 17 (2007), 787-806.   Google Scholar

[9]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.   Google Scholar

[10]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.   Google Scholar

[11]

N. Fenichel, Geometric singluar perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.   Google Scholar

[12]

Q. T. GanR. XuY. L. Li and R. X. Hu, Traveling waves in an infectious disease model with a fixed latent period and a spatio-temporal delay, Math. Comput. Modeling, 53 (2011), 814-823.   Google Scholar

[13]

R. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.   Google Scholar

[14]

R. Gardner and J. Smoller, The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index, J. Differential Equations, 47 (1983), 133-161.   Google Scholar

[15]

S. A. Gourley and M. A. J. Chaplain, Travelling fronts in a food-limited population model with time delay, Proc. R. Soc. Edinb A, 132 (2002), 75-89.   Google Scholar

[16]

S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A, 459 (2003), 1563-1579.   Google Scholar

[17]

S. A. Gourley and S. G. Ruan, Convergence and traveling wave fronts in functional differential equations with nonlocal terms: A competition model, SIAM. J. Math. Anal., 35 (2003), 806-822.   Google Scholar

[18]

J. Huang and X. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 243-256.   Google Scholar

[19]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, In Dynamical systems (ed. R. Johnson). Lecture Notes in Mathematics, vol. 1609, Springer, 1995. Google Scholar

[20]

Y. Kuang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with allee effects and disease-modified fitness, Discrete Contin. Dyn. Syst. B, 19 (2014), 89-130.   Google Scholar

[21]

C. Z. Li and H. P. Zhu, Canard cycles for predatorCprey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.   Google Scholar

[22]

W. T. LiN. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.   Google Scholar

[23]

W. T. LiZ. C. Wang and J. H. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.   Google Scholar

[24]

G. Y. Lv and M. X. Wang, Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model, Nonlinear Anal., 11 (2010), 2035-2043.   Google Scholar

[25]

G. Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. (RWA), 11 (2010), 1323-1329.   Google Scholar

[26]

S. Ma, Asymptotic stability of traveling waves in a discrete convolution model for phase transitions, J. Math. Anal. Appl., 308 (2005), 240-256.   Google Scholar

[27]

M. B. A. Mansour, Traveling wave solutions for a singularly perturbed Burgers-KdV equation, Pramana, Indin Acad. Sci., 73 (2009), 799-806.   Google Scholar

[28]

M. A. Pozio, Some conditions for global asymptotic stability of equilibria of integrodifferential equations, J. Math. Anal. Appl., 95 (1983), 501-527.   Google Scholar

[29]

S. G. Ruan and D. M. Xiao, Stability of steady states and existence of traveling wave in a vector disease model, Proc. Roy. Soc. Edinburgh, 134A (2004), 991-1011.   Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.   Google Scholar

[31]

Z. C. WangW. T. Li and S. G. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.   Google Scholar

[32]

J. Wu and X. Zou, Travelling wave fronts of reaction diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.   Google Scholar

[33]

H. XiangY. L. Tang and H. F. Huo, A viral model with intracellular delay and humoral immunity, Bull. Malays. Math. Sci. Soc., 40 (2017), 1011-1023.   Google Scholar

[34]

J. M. Zhang, Existence of traveling waves in a modelified vector-disease model, Appl. Math. Model., 33 (2009), 626-632.   Google Scholar

[35]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.   Google Scholar

[36]

X. Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019.   Google Scholar

show all references

References:
[1]

P. AshwinM. V. Bartuccelli and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.   Google Scholar

[2]

P. W. Bates and F. X. Chen, Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 26 (1999), 1-19.   Google Scholar

[3]

X. Chen and Z. J. Du, Existence of Positive Periodic Solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst., 17 (2018), 67-80.   Google Scholar

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.   Google Scholar

[5]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.   Google Scholar

[6]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.   Google Scholar

[7]

Z. J. DuZ. Feng and X.N. Zhang, Traveling wave phenomena of n-dimensional diffusive predatorprey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.   Google Scholar

[8]

F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters, Discrete Contin. Dyn. Syst., 17 (2007), 787-806.   Google Scholar

[9]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.   Google Scholar

[10]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.   Google Scholar

[11]

N. Fenichel, Geometric singluar perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.   Google Scholar

[12]

Q. T. GanR. XuY. L. Li and R. X. Hu, Traveling waves in an infectious disease model with a fixed latent period and a spatio-temporal delay, Math. Comput. Modeling, 53 (2011), 814-823.   Google Scholar

[13]

R. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.   Google Scholar

[14]

R. Gardner and J. Smoller, The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index, J. Differential Equations, 47 (1983), 133-161.   Google Scholar

[15]

S. A. Gourley and M. A. J. Chaplain, Travelling fronts in a food-limited population model with time delay, Proc. R. Soc. Edinb A, 132 (2002), 75-89.   Google Scholar

[16]

S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A, 459 (2003), 1563-1579.   Google Scholar

[17]

S. A. Gourley and S. G. Ruan, Convergence and traveling wave fronts in functional differential equations with nonlocal terms: A competition model, SIAM. J. Math. Anal., 35 (2003), 806-822.   Google Scholar

[18]

J. Huang and X. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 243-256.   Google Scholar

[19]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, In Dynamical systems (ed. R. Johnson). Lecture Notes in Mathematics, vol. 1609, Springer, 1995. Google Scholar

[20]

Y. Kuang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with allee effects and disease-modified fitness, Discrete Contin. Dyn. Syst. B, 19 (2014), 89-130.   Google Scholar

[21]

C. Z. Li and H. P. Zhu, Canard cycles for predatorCprey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.   Google Scholar

[22]

W. T. LiN. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.   Google Scholar

[23]

W. T. LiZ. C. Wang and J. H. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.   Google Scholar

[24]

G. Y. Lv and M. X. Wang, Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model, Nonlinear Anal., 11 (2010), 2035-2043.   Google Scholar

[25]

G. Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. (RWA), 11 (2010), 1323-1329.   Google Scholar

[26]

S. Ma, Asymptotic stability of traveling waves in a discrete convolution model for phase transitions, J. Math. Anal. Appl., 308 (2005), 240-256.   Google Scholar

[27]

M. B. A. Mansour, Traveling wave solutions for a singularly perturbed Burgers-KdV equation, Pramana, Indin Acad. Sci., 73 (2009), 799-806.   Google Scholar

[28]

M. A. Pozio, Some conditions for global asymptotic stability of equilibria of integrodifferential equations, J. Math. Anal. Appl., 95 (1983), 501-527.   Google Scholar

[29]

S. G. Ruan and D. M. Xiao, Stability of steady states and existence of traveling wave in a vector disease model, Proc. Roy. Soc. Edinburgh, 134A (2004), 991-1011.   Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.   Google Scholar

[31]

Z. C. WangW. T. Li and S. G. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.   Google Scholar

[32]

J. Wu and X. Zou, Travelling wave fronts of reaction diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.   Google Scholar

[33]

H. XiangY. L. Tang and H. F. Huo, A viral model with intracellular delay and humoral immunity, Bull. Malays. Math. Sci. Soc., 40 (2017), 1011-1023.   Google Scholar

[34]

J. M. Zhang, Existence of traveling waves in a modelified vector-disease model, Appl. Math. Model., 33 (2009), 626-632.   Google Scholar

[35]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.   Google Scholar

[36]

X. Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019.   Google Scholar

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