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In this paper we investigate the spread of advantageous genes in two variantsof the F-KPP model with dormancy. The first variant, in which dormantindividuals do not move in space and instead form "localized seed banks", hasrecently been introduced in Blath, Hammer and Nie (2020). However, there, onlya relatively crude upper bound for the critical speed of potential travellingwave solutions has been provided. The second model variant is new and describesa situation in which the dormant forms of individuals are subject to motion,while the "active" individuals remain spatially static instead. This can bemotivated e.g. by spore dispersal of fungi, where the "dormant" spores aredistributed by wind, water or insects, while the "active" fungi are locallyfixed. For both models, we establish the existence of monotone travelling wavesolutions, determine the corresponding critical wave speed in terms of themodel parameters, and characterize aspects of the asymptotic shape of the wavesdepending on the decay properties of the initial condition. Interestingly, theslow-down effect of dormancy on the speed of propagation of beneficial allelesis often more serious in model variant II (the "spore model") than in variant I(the "seed bank model"), and this can be understood mathematically viaprobabilistic representations of solutions in terms of (two variants of)"on/off branching Brownian motions". Our proofs make rather heavy use ofprobabilistic tools in the tradition of Watanabe (1967), McKean (1975), Bramson(1978), Neveu (1987), Lalley and Sellke (1987), Champneys et al (1995) andothers. However, the two-compartment nature of the model and the special formsof dormancy also pose obstacles to the classical formalism, giving rise to avariety of open research questions that we briefly discuss at the end of thepaper.

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