Travelling wavefronts in the diffusive single species model with Allee effect and distributed delay. (English) Zbl 1041.92027

Summary: We consider a diffusive single species model with Allee effect and distributed delay time. Special attention is paid to the existence of travelling wavefront solutions. First, we show that such fronts exist when the convolution kernel assumes a strong generic delay kernel and the delay is sufficiently small. Then, in the non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times, we show that such fronts still exist for weak generic delay kernels and small delay. The approach used in this paper is the geometric singular perturbation theory.


92D25 Population dynamics (general)
35B25 Singular perturbations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Gopalsamy, K.; Ladas, G., On the oscillation and asymptotic behavior of \(Ṅ(t)=N(t)[a+bN(t−τ)−cN^2(t−τ)]\), Quart. Appl. Math., 48, 3, 433-440 (1990) · Zbl 0719.34118
[2] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Populations Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, MA · Zbl 0752.34039
[3] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston, MA · Zbl 0777.34002
[4] Wu, J.; Zhao, X., Permanence and convergence in multi-species competition systems with delay, Proc. Am. Soc., 126, 1709-1714 (1998) · Zbl 0894.34062
[5] Schaaf, K., Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. Am. Math. Soc., 302, 587-615 (1987) · Zbl 0637.35082
[6] Britton, N. F., Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50, 1663-1688 (1990) · Zbl 0723.92019
[7] Wu, J., Theory and applications of partial functional differential equations (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0870.35116
[8] Wu, J.; Zou, X., Traveling wavefronts of reaction diffusion systems with delay, J. Dynam. Different. Equat., 13, 651-687 (2001) · Zbl 0996.34053
[9] Ma, S., Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differen. Equat., 171, 293-314 (2001) · Zbl 0988.34053
[10] Gourley, S. A., Wavefront solutions of a diffusive delay model for populations of Daphnia magna, Comp. Math. Appl., 42, 10-11, 1421-1430 (2001) · Zbl 0998.92029
[11] So, J. W.H.; Zou, X., Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput., 122, 385-395 (2001) · Zbl 1027.35051
[12] Huang, J.; Zou, X., Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271, 2, 455-466 (2002) · Zbl 1017.35116
[13] Gourley, S. A.; Chaplain, M. A.J., Travelling fronts in a food-limited population model with time delay, Proc. Roy. Soc. Edinburg A, 132, 75-89 (2002) · Zbl 1006.35051
[14] Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equat., 31, 53-98 (1979) · Zbl 0476.34034
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