The establishment of polyploid populations is constrained by minority cytotype exclusion, whereby newly formed polyploids are lost when rare in predominantly diploid populations. Here, we consider this problem through a continuous-time approximation of a discrete model of tetraploid establishment. The spatio-temporal dynamics of sexually reproducing mixed-ploidy populations is then formally investigated using a reaction-diffusion framework, which allows us to determine the conditions for spatial invasion of tetraploids. We first study the local dynamics of gamete frequencies in populations composed of diploid, triploid, and tetraploid cytotypes, where \(\upsilon \) denotes the per-generation proportion of unreduced (diploid) gametes and \(\phi \) denotes the relative viable-gamete contribution of triploid cytotypes. The conjugacy between models of cytotype and gamete frequency dynamics is formally established. Then, we show that the system admits a bistable structure and characterize its equilibria. In one spatial dimension, the continuous-time approximation yields a closed-form traveling-wave solution and a wave speed that scales with the standard deviation of distances between mother and offspring birth locations, \(\sigma \) . Extending the analysis to radially symmetric geometry, we show that successful establishment from a localized tetraploid patch requires a critical nucleus of radius \(R_{c}\) , for which we derive an asymptotic approximation \(R_{c} \sim \upsilon ^{-1}\sigma \sqrt{(1 - \phi )/2}\) for small \(\upsilon \) . We confirm these analytical predictions through numerical simulations of the continuous-time model. Our analyses reveal that sufficiently large founding patches can nucleate expanding bistable waves without intrinsic tetraploid fitness advantages, or reproductive strategies that circumvent frequency-dependent selection in sexually reproducing populations.
