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Travelling front solutions of a nonlocal Fisher equation. (English) Zbl 0982.92028

Summary: We consider a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher’s equation is a particular case. We consider travelling wavefront solutions connecting the two uniform states of the equation. We show that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist. We also construct expressions for the front and its evolution from initial data, showing that the main difference between our front and that of Fisher’s equation is that for sufficiently strong nonlocality our front is non-monotone and has a very prominent hump.

MSC:

92D25 Population dynamics (general)
35Q80 Applications of PDE in areas other than physics (MSC2000)
34C60 Qualitative investigation and simulation of ordinary differential equation models
35K57 Reaction-diffusion equations
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