• Ling, Zhi (School of Mathematical Science Yangzhou University) ;
  • Zhang, Lai (Department of Mathematics and Mathematical Statistics Umea University)
  • Received : 2012.12.03
  • Published : 2014.09.30


This paper is concerned with a reaction-diffusion single species model with harvesting on n-dimensional isotropically growing domain. The model on growing domain is derived and the corresponding comparison principle is proved. The asymptotic behavior of the solution to the problem is obtained by using the method of upper and lower solutions. The results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady state solution while it takes a negative effect on the asymptotic stability of the trivial solution, but the effect of the harvesting rate is opposite. The analytical findings are validated with the numerical simulations.


  1. E. J. Crampin, E. A. Gaffney, and P. K. Maini, Reaction and diffusion on growing domains: scenarios for robust pattern formation, Bull. Math. Biol. 61 (1999), no. 6, 1093-1120.
  2. E. J. Crampin, W. W. Hackborn, and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bull. Math. Biol. 64 (2002), no. 4, 746-769.
  3. R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 353-369.
  4. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
  5. G. Hetzer, A. Madzvamuse, and W. X. Shen, Characterization of turing diffusion-driven instability on evolving domains, Discrete Contin. Dyn. Syst. 32 (2012), no. 11, 3975-4000.
  6. A. Kolmogoroff, I. Petrovsky, and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, (French) Moscow Univ. Bull. Math. 1 (1937), 1-25.
  7. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1967.
  8. S. Seirin Lee and E. A. Gaffney, Aberrant behaviours of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays, Bull. Math. Biol. 72 (2010), no. 8, 2161-2179.
  9. J. A. Mackenzie and A. Madzvamuse, Analysis of stability and convergence of finite-difference methods for a reaction-diffusion problem on a one-dimensional growing domain, IMA J. Numer. Anal. 31 (2011), no. 1, 212-232.
  10. A. Madzvamuse, Stability analysis of reaction-diffusion systems with constant coeffi-cients on growing domains, Int. J. Dyn. Syst. Differ. Equ. 1 (2008), no. 4, 250-262.
  11. A. Madzvamuse, E. A. Gaffney, and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion, J. Math. Biol. 61 (2010), no. 1, 133-164.
  12. J. D. Murray, Mathematical Biology I: An Introduction, Springer, Berlin, 2002.
  13. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
  14. R. G. Plaza, F. Sachez-Garduno, P. Padilla, R. A. Barrio, and P. K. Maini, The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations 16 (2004), no. 4, 1093-1214.
  15. Q. L. Tang, L. Zhang, and Z. G. Lin, Asymptotic profile of species migrating on a growing habitat, Acta Appl. Math. 116 (2011), no. 2, 227-235.
  16. C. Varea, J. L. Aragon, and R. A. Barrio, Confined turing patterns in growing systems, Phys. Rev. E 56 (1997), no. 1, 1250-1253.