Nonmonotone travelling waves in a single species reaction–diffusion equation with delay

By Teresa Faria, Sergei I. Trofimchuk. 2005

117 citations recorded by the source corpus.

We prove the existence of a continuous family of positive and generally non-monotone travelling fronts for delayed reaction-diffusion equations ut(t, x) = �u(t, x)−u(t, x)+ g(u(t − h, x)) (∗), when g ∈ C 2 (R+, R+) has exactly two fixed points: x1 = 0 and x2 = K > 0. Recently, non-monotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass type equations with diffusion fall within this subclass of (∗). As an example, we consider the diffusive Nicholson’s blowflies equation.

Subfields: Applied Mathematics, Numerical Analysis, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/ddd74f49888da8cdfd3260b46514a4137b2c8bb3

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Citations in the corpus

Monotone traveling wavefronts of the KPP-Fisher delayed equation (2010, methodology): “Nevertheless, as it was shown in [12] for the case of the Mackey-Glass type equations, the method of [11] may be refined to establish the existence of positive wavefronts as well.”
Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay (2015, background): “For details, we refer to [3,5,7,14,32,33], see also the summary in [13].”
Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays (2014, unknown): “Some other results have also been presented for (fast) traveling wave solutions of scalar equations (Faria and Trofimchuk [7] [8], Kwong and Ou [14]).”
On the existence of traveling waves for delayed reaction–diffusion equations☆ (2009, background): “Other related results for nonmonotone equations can also be found in [4,9,15].”
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity (2008, background): “[8,9], Gourley and Ruan [11], Li et al.”
Spreading speeds and travelling waves for non-monotone time-delayed lattice equations (2010, background): “This phenomenon has been observed both numerically and analytically (e.g. Gourley 2000; Kyrychko et al. 2005; Faria & Trofimchuk 2006).”
Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects (2008, methodology): “For the case b0 > e, that is, ve−v is nonmonotone on v ∈ [0, ln b0], we refer to [16,17,29].”