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 and 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. 

[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. 

[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. 

[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. 

[5]

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

[6]

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

[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. 

[8]

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

[9]

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

[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. 

[11]

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

[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. 

[13]

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

[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. 

[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. 

[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. 

[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. 

[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. 

[19]

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

[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. 

[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. 

[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. 

[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. 

[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. 

[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. 

[26]

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

[27]

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

[28]

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

[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. 

[30]

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

[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. 

[32]

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

[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. 

[34]

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

[35]

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

[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. 

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. 

[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. 

[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. 

[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. 

[5]

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

[6]

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

[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. 

[8]

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

[9]

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

[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. 

[11]

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

[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. 

[13]

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

[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. 

[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. 

[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. 

[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. 

[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. 

[19]

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

[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. 

[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. 

[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. 

[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. 

[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. 

[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. 

[26]

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

[27]

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

[28]

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

[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. 

[30]

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

[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. 

[32]

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

[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. 

[34]

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

[35]

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

[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. 

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