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Globally Asymptotic Stability of a Delayed Integro-Differential Equation With Nonlocal Diffusion

Published online by Cambridge University Press:  20 November 2018

Peixuan Weng  and
Li Liu
Peixuan Weng
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 510631, P. R. China. e-mail: [email protected], [email protected]
Li Liu
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 510631, P. R. China. e-mail: [email protected], [email protected]
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Abstract

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We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As an application, we discuss an example for a population model with age structure.

Keywords

45J0535K5792D25integro-differential equationnonlocal diffusionequilibriumglobally asymptotic stabilitypopulation model with age structure
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 2 , 01 June 2017 , pp. 436 - 448
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Bates, P. W., Fife, P. C., and X. Wang, Traveling waves in a convolution model for phase transition. Arch. Rational Mech. Anal. 138(1997), 105–136. http://dx.doi.org/10.1007/s002050050037 Google Scholar
[2] J., Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. 132(2004), 2433–2439. http://dx.doi.org/10.1090/S0002-9939-04-07432-5 Google Scholar
[3] Chen, X. F., Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equation. Adv. Differential Equations 2(1997), 125–160. Google Scholar
[4] Faria, T., W. Zh. Huang, and Wu, J. H., Travelling waves for delayed reaction-diffusion equations with global response. Proc. R. Soc. A 462(2006), 229–261. http://dx.doi.org/10.1098/rspa.2OO5.1554 Google Scholar
[5] Fife, P. and X. Wang, A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions. Adv. Differential Equations 3(1998), 85–110. Google Scholar
[6] Hale, J. K., Theory of functional differential equations. Springer-Verlag, New York, 1977. Google Scholar
[7] Kuang, Y., Delay differential equations with applications in population dynamics, Academic Press, Boston, MA, 1993. Google Scholar
[8] Liu, L. and Weng, P. X., A nonlocal diffusion model of a single species with age structure. J. Math. Anal. Appl. 432(2015), 38–52. http://dx.doi.org/10.1016/j.jmaa.2015.06.052 Google Scholar
[9] Martin, R. H. and Smith, H. L., Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321(1990), 1–44. Google Scholar
[10] Murray, J. D., Mathematical biology. Second ed. Springer-Verlag, Berlin, 1998. Google Scholar
[11] Medlock, J. and M. Kot, Spreading disease: integro-differential equations old and new. Math. Biosci. 184(2003), 201–222. http://dx.doi.org/10.1016/S0025-5564(03)00041-5 Google Scholar
[12] Pan, S. X., Li, W. T., and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay. Nonlinear Anal. TMA 72(2010), 3150–3158. http://dx.doi.org/10.101 6/j.na.2009.12.008 Google Scholar
[13] So, J.W.-H., J. Wu, and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on the unbounded domains. Proc. R. Soc. Lond. Ser. A. 457(2001), 1841–1853. http://dx.doi.org/1 0.1098/rspa.2OO1.0789 Google Scholar
[14] Wang, Z.-C. and W.-T. Li, Dynamics of a nonlocal delayed reaction-diffusion equation without quasi-monotonicity. Proc. Roy. Soc. Edinburgh Sect. A 140(2010), 1081–1109. http://dx.doi.org/10.101 7/S0308210509000262 Google Scholar
[15] Wu, S. L., Weng, P. X., and Ruan, S. G., Spatial dynamics of a lattice population model with two age classes and maturation delay,. European J. Appl. Math. 26(2015), 61–91. http://dx.doi.org/10.1017/S0956792514000333 Google Scholar
[16] Wu, S. L. and Liu, S. Y., Traveling waves for delayed non-local diffusion equation with crossing-monostability. Appl. Math. Comput. 217(2010), 1435–1444. Google Scholar
[17] Xu, Z. Q. and Weng, P. X., Traveling waves in a convolution model with infinite distributed delay and non-monotonicity. Nonlinear Anal. RWA 12(2011), 633–647. http://dx.doi.org/10.1016/j.nonrwa.2010.07.006 Google Scholar
[18] Xu, Z. Q. and Xiao, D. M., Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays. Differential Integral Equations 11-12(2014), 1073–1106.Google Scholar
[19] Xu, Z. Q. and Xiao, D. M., Regular traveling waves for a nonlocal diffusion equation. J. Differential Equations 258(2015), 191–223. http://dx.doi.org/10.101 6/j.jde.2O14.09.008 Google Scholar
[20] Zhang, G. B., Traveling waves in a nonlocal dispersal population model with age-structure. Nonlinear Anal. 74(2011), 5030–5047. http://dx.doi.org/!0.1016/j.na.2O11.04.069 Google Scholar