There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., Cox et al. (Astérisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:477–496, 2009, Electron J Probab 19:1–64, 2014), Durrett and Neuhauser (Ann Probab 22:289–333, 1994). These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge et al. (Electron J Probab 22:1–40, 2017) to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al. (Theoret Pop Biol 55(1999):270–282, 1999) there were two nontrivial stationary distributions.

在许多情况下，重新缩放的相互作用粒子系统已被证明会收敛到具有双稳态反应项的反应扩散方程 (RDE)，例如，参见 Cox 等人。(Astérisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:477–496, 2009, Electron J Probab 19:1–64, 2014), Durrett and Neuhauser (Ann Probab 22:281–9933) . 这些 RDE 具有行波解决方案。当波速非零时，块构造已被用于证明非平凡平稳分布的存在或不存在。在这里，我们遵循 Etheridge 等人的论文中的方法。（Electron J Probab 22:1–40, 2017）证明在 RDE 极限具有双稳态反应项且行波速度为 0 的各种示例中，人们可以更快地运行时间并进一步重新调整空间以通过平均曲率获得运动收敛。这开辟了证明具有快速搅拌的有性生殖模型具有不连续相变的可能性，以及在 Molofsky 等人研究的非线性选民模型相图的区域 2 中的可能性。（Theoret Pop Biol 55(1999):270–282, 1999）有两个非平凡的平稳分布。