How the interplay of dormancy and selection affects the wave of advance of an advantageous gene

We investigate the spread of advantageous genes in two variants of the F-KPP model with dormancy. In the first model, dormant individuals do not move in space and instead form ‘localized seed banks’. In the second model, dormant forms of individuals are subject to motion, while the ‘active’ (reproducing) individuals remain spatially static. This can be motivated e.g. by spore dispersal of fungi, where the ‘dormant’ spores are distributed by wind, water or insects, while the ‘active’ fungi are locally fixed. For both models, we establish existence of monotone travelling wave solutions, determine the corresponding critical wave speed in terms of the model parameters, and characterize aspects of the asymptotic shape of the waves depending on the decay properties of the initial condition. We find that the slow-down effect of dormancy on the speed of propagation of beneficial alleles is more serious in variant II (the ‘spore model’) than in variant I (the ‘seed bank model’). Mathematically, this can be understood via probabilistic representations of solutions in terms of (two variants of) ‘on/off branching Brownian motion’. A variety of open research questions are briefly discussed at the end of the paper. • Dormancy reduces the speed of advance of an advantageous gene significantly. • The slow-down effect is larger in spore models compared to seed bank models. • These F-KPP-type models admit on/off branching Brownian motions as moment duals. • The speed of travelling waves can be precisely quantified via the model parameters. • Travelling waves behave asymptotically as their linearizations.