In this paper we investigate the spread of advantageous genes in two variants of the F-KPP model with dormancy. The first variant, in which dormant individuals do not move in space and instead form localized seed banks, has recently been introduced in Blath, Hammer and Nie (2020). However, there, only a relatively crude upper bound for the critical speed of potential travelling wave solutions has been provided. The second model variant is new and describes a situation in which the dormant forms of individuals are subject to motion, while the active individuals remain spatially static instead. 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 the 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. Interestingly, the slow-down effect of dormancy on the speed of propagation of beneficial alleles is often more serious in model variant II (the spore model) than in variant I (the seed bank model), and this can be understood mathematically via probabilistic representations of solutions in terms of (two variants of) on/off branching Brownian motions. Our proofs make rather heavy use of probabilistic tools in the tradition of McKean (1975), Bramson (1978), Neveu (1987), Lalley and Sellke (1987), Champneys et al (1995) and others. However, the two-compartment nature of the model and the special forms of dormancy also pose obstacles to the classical formalism, giving rise to a variety of open research questions that we briefly discuss at the end of the paper.
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