Monotone wave-fronts in a structured population model with distributed maturation delay

We analyse a stage-structured reaction–diffusion model for a single species on an infinite 1D domain. Recognising that not all individuals may take the same amount of time to mature, the maturation delay is incorporated via a probability distribution function, leading to a distributed delay system. The system is non-local in space, because individuals may have moved while immature. A detailed investigation of travelling front solutions connecting the extinction state with the positive equilibrium is carried out, focussing attention on the minimum speed and the qualitative form of the profile, which appears always to be monotone. A rigorous proof of existence is provided for a special, but realistic, choice of the probability distribution function representing the maturation delay. Numerical simulations of the initial value problem are also presented.