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Population dynamics with age-dependent diffusion and death rates

Published online by Cambridge University Press:  21 February 2013

M. AL-JARARHA  and
CHUNHUA OU
M. AL-JARARHA
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada, A1C 5S7 email: [email protected]
CHUNHUA OU
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada, A1C 5S7 email: [email protected]
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Abstract

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability.

Keywords

Steady-state solutionTravelling wave solutionAge-structured modelBirth function34K1035B3535K5592D25
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 4 , August 2013 , pp. 471 - 500
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Al-Omari, J. F. M. & Gourley, S. A. (2002) Monotone traveling fronts in an age-structured reaction-diffusion model of a single species. J. Math. Biol. 45, 294–312.CrossRefGoogle Scholar
[2]Diekmann, O. (1978) Thresholds and traveling waves for the geographical spread of infection. J. Math Biol. 6, 109–130.Google Scholar
[3]Diekmann, O. & Kapper, H. G. (2003) On the bounded solutions of a nonlinear convolution equations. Nonlinear Anal. Theory Method Appl. 2, 721–737.Google Scholar
[4]Gourley, S. A. & Kuang, Y. (2003) Wavefronts and global stability in a time-delayed population with stage structure. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2034), 1563–1579.Google Scholar
[5]Gourley, S. A. & So, J. W.-H. (2003) Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay. Proc. R. Soc. Edinburgh 133A, 527–548.Google Scholar
[6]Gourley, S. A., So, J. W.-H. & Wu, J. (2004) Non- locality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics. J. Math. Sci. 124 (4), 5119–5153.Google Scholar
[7]Gourley, S. A. & Wu, J. (2006) Delayed non-local diffusive systems in biological invasion and disease spread. In: Nonlinear Dynamics and Evolution Equations, Vol. 48, Fields Institute Communications; American Mathematical Society, Providence, RI, pp. 137–200.Google Scholar
[8]Hale, J. K. (1988) Asymptotic Behavior of Dissipative Systems (Math. Survey and Monographs), Vol. 25, American Mathematical Society, Providence, RI.Google Scholar
[9]Hirsch, M. W. (1984) The dynamical systems approach to differential equations. AMS 11 (1), 1–64.Google Scholar
[10]Huddleston, J. V. (1983) Population dynamics with age and time-dependent birth and death rates. Bull. Math. Biol. 45 (5), 827–836.Google Scholar
[11]Jin, Y. & Zhao, X.-Q. (2009) Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure. SIAM J. Math. Anal. 40 (6), 2496–2516.Google Scholar
[12]Liang, D., So, J. W.-H., Zhang, F. & Zou, X. (2003) Population dynamics models with nonlocal delay on bounded domains and their numerical computation. Differ. Equ. Dyn. Syst. 11 (1&2), 117–139.Google Scholar
[13]Liang, D. & Wu, J. (2003) Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects. J. Nonlinear Sci. 13, 289–310.Google Scholar
[14]Mei, M. & So, J. W.-H. (2008) Stability of strong travelling waves for non-local time-delayed reaction diffusion equation. Proc. R. Soc. Edinburgh 138A, 551–568.Google Scholar
[15]Metz, J. A. J. & Diekmann, O. (1986) The Dynamics of Physiologically Structured Populations. Metz, J. A. J. & Dekmann, O. (editors), Springer-Verlag, Berlin, Heidelberg, Germany.Google Scholar
[16]Ou, C. & Wu, J. (2005) Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model. Nonlinear Anal. 63 (3), 364–387.Google Scholar
[17]Ou, C. & Wu, J. (2006) Spatial spread of rabies revisited: Influence of age-dependent diffusion on nonlinear dynamics. SIAM J. Appl. Math. 67 (1), 138–163.Google Scholar
[18]Ou, C. & Wu, J. (2007) Persistence of wavefronts in delayed nonlocal reaction-diffusion equation. J. Differ. Equ. 235, 219–261.Google Scholar
[19]Ou, C. & Wu, J. (2007) Traveling wavefronts in a delayed food-limited population model. SIAM J. Math. Anal. 39 (1), 103–125.Google Scholar
[20]Smith, H. L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math Surveys and Monographs. Vol. 41, American Mathematical Society, Providence, RI.Google Scholar
[21]Smith, H. & Thieme, H. (1991) Strongly order preserving semiflows generated by functional differential equations. J. Differ. Equ. 93, 332–363.Google Scholar
[22]So, J. W.-H., Wu, J. & Zou, X. (2001) A reaction-diffusion model for a single species with age-structured. I Travelling wavefronts on unbounded domains. Proc. R. Soc. Lond. A 457, 1841–1853.CrossRefGoogle Scholar
[23]Thieme, H. R. (1979) Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306, 94–121.Google Scholar
[24]Thieme, H. R. (1993) Persistence under relaxed point-disspativity with applications to an endemic model. SIAM J. Math. Anal. 24 (2), 407–435.Google Scholar
[25]Thieme, H. R. & Zhao, X.-Q. (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model. J. Differ. Equ. 195, 430–470.Google Scholar
[26]Vaidya, N. K. & Wu, J. (2008) Modeling spruce budworm population revisited: Impact of physiological structure on outbreak control. Bull. Math. Biol. 70, 769–784.Google Scholar
[27]Wu, J. (1996) Theory and Applications of Partial Functional Differential Equations (Applied Mathematical Sciences). 119, Springer-Verlag, New York, NY.Google Scholar
[28]Xu, D. & Zhao, X.-Q. (2003) A nonlocal reaction-diffusion population model with stage structure. Can. Appl. Math. Q. 11 (3), 303–319.Google Scholar
[29]Zhao, X.-Q. (1996) Global attractivity and stability in some monotone discrete dynamical systems. Bull. Austral. Math Soc. 53, 305–324.CrossRefGoogle Scholar
[30]Zhao, X.-Q. (2003) Dynamical Systems in Population Biology. Springer-Verlag, New York, NY.Google Scholar