Monotone traveling wavefronts of the KPP-Fisher delayed equation

Journal of Differential Equations

Volumn 250, Issue 4, 2011, Pages 1767-1787

  Gomez, Adrian ^a   Trofimchuk, Sergei ^a  

^a UNIVERSIDAD DE TALCA   (Chile)

Author keywords

Existence; Heteroclinic solutions; KPP Fisher delayed reaction diffusion equation; Monotone positive traveling wave; Uniqueness

Indexed keywords

EID: 78650995938     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.11.011     Document Type: Article

References (41)

1. Ablowitz M.J., Zeppetella A. Explicit solution of Fisher's equation for a special wave speed. Bull. Math. Biol. 1979, 41:835-840.
2. Ai S. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differential Equations 2007, 232:104-133.
3. Aguerrea M., Trofimchuk S., Valenzuela G. Uniqueness of fast traveling fronts in a single species reaction-diffusion equation with delay. Proc. R. Soc. A 2008, 464:2591-2608.
4. Ashwin P., Bartuccelli M.V., Bridges T.J., Gourley S.A. Travelling fronts for the KPP equation with spatio-temporal delay. Z. Angew. Math. Phys. 2002, 53:103-122.
5. Boumenir A., Nguyen V.M. Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations. J. Differential Equations 2008, 244:1551-1570.
6. Bramson M. Convergence of solutions of the Kolmogorov equation to traveling waves. Mem. Amer. Math. Soc. 1983, 44(285).
7. Carr J., Chmaj A. Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. 2004, 132:2433-2439.
8. Coville J., Dávila J., Martínez S. Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differential Equations 2008, 244:3080-3118.
9. Daleckii J.L., Krein M.G. Stability of Solutions of Differential Equations in Banach Space. Transl. Math. Monogr. 1974, vol. 43. Amer. Math. Soc., Providence, RI.
10. Diekmann O., Kaper H.G. On the bounded solutions of a nonlinear convolution equation. Nonlinear Anal. 1978, 2:721-737.
11. Eastham M.S.P. The Asymptotic Solution of Linear Differential Systems. London Math. Soc. Monogr. Ser. 1989, Clarendon Press, Oxford.
12. Faria T., Huang W., Wu J. Traveling waves for delayed reaction-diffusion equations with non-local response. Proc. R. Soc. A 2006, 462:229-261.
13. Faria T., Trofimchuk S. Non-monotone traveling waves in a single species reaction-diffusion equation with delay. J. Differential Equations 2006, 228:357-376.
14. Faria T., Trofimchuk S. Positive travelling fronts for reaction-diffusion systems with distributed delay. Nonlinearity 2010, 23:2457-2481.
15. Fisher R.A. The wave of advance of advantageous gene. Ann. Eugen. 1937, 7:355-369.
16. Friesecke G. Exponentially growing solutions for a delay-diffusion equation with negative feedback. J. Differential Equations 1992, 98:1-18.
17. Gopalsamy K., He X.-Z., Sun D.Q. Oscillations and convergence in a diffusive delay logistic equation. Math. Nachr. 1993, 164:219-237.
18. Gourley S.A. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 2000, 41:272-284.
19. Hadeler K.P. Transport, reaction, and delay in mathematical biology, and the inverse problem for traveling fronts. J. Math. Sci. 2008, 149:1658-1678.
20. Hutchinson G.E. Circular causal systems in ecology. Ann. New York Acad. Sci. 1948, 50:221-246.
21. Kobayashi K. On the semilinear heat equation with time-lag. Hiroshima Math. J. 1977, 7:459-472.
22. Kolmogorov A., Petrovskii I., Piskunov N. Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem. Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh. 1937, 1:1-26.
23. Kwong M.K., Ou C. Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J. Differential Equations 2010, 249:728-745.
24. Lau K.-S. On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J. Differential Equations 1985, 59:44-70.
25. Luckhaus S. Global boundedness for a delay-differential equation. Trans. Amer. Math. Soc. 1986, 294:767-774.
26. Ma S. Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem. J. Differential Equations 2001, 171:294-314.
27. Mallet-Paret J. The Fredholm alternative for functional differential equations of mixed type. J. Dynam. Differential Equations 1999, 11:1-48.
28. Ou C., Wu J. Traveling wavefronts in a delayed food-limited population model. SIAM J. Math. Anal. 2007, 39:103-125.
29. Pan S. Asymptotic behavior of traveling fronts of the delayed Fisher equation. Nonlinear Anal. Real World Appl. 2009, 10:1173-1182.
30. Raugel G., Kirchgässner K. Stability of fronts for a KPP-system. II. The critical case. J. Differential Equations 1998, 146:399-456.
31. Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations 1995, World Scientific Publishing, River Edge, NJ.
32. Schaaf K. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Amer. Math. Soc. 1987, 302:587-615.
33. Trofimchuk E., Tkachenko V., Trofimchuk S. Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay. J. Differential Equations 2008, 245:2307-2332.
34. Trofimchuk E., Alvarado P., Trofimchuk S. On the geometry of wave solutions of a delayed reaction-diffusion equation. J. Differential Equations 2009, 246:1422-1444.
35. Wang Z.-C., Li W.T., Ruan S. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. J. Differential Equations 2006, 222:185-232.
36. Wang Z.-C., Li W.T., Ruan S. Traveling fronts in monostable equations with nonlocal delayed effects. J. Dynam. Differential Equations 2008, 20:573-607.
37. Widder D.V. The Laplace Transform. Princeton Math. Ser. 1941, vol. 6. Princeton University Press, 406 pp.
38. Wu J., Zou X. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynam. Differential Equations 2001, 13:651-687.
39. Yanagida E. Irregular behavior of solutions for Fisher's equation. J. Dynam. Differential Equations 2007, 19:895-914.
40. Yoshida K. The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology. Hiroshima Math. J. 1982, 12:321-348.
41. Zou X. Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type. J. Comput. Appl. Math. 2002, 146:309-321.