Most actin wave models are based on local nonlinear processes that involve positive and negative feedbacks between interacting species (A). In an extended system with spatial coupling, such as diffusive transport (B), this may give rise to propagating waves (C), where is the wave propagation (group) speed. Examples of actin waves and models in D. discoideum (D–F) and Xenopus oocytes (G–I) showing kymographs of the experimentally observed waves (D, G), model schematics (E, H), and simulations (F, I). The models proposed by Arai et al., 2010; Shibata et al., 2012; Shibata et al., 2013 consider active PTEN, PIP2, and PIP3, and assume conservation of PTEN. The models in Goryachev et al., 2016; Michaud et al., 2022 are based on Rho (RD, RT) and its GAP (RGA-3/4) interacting with F-actin (F). (D–F) were modified from Arai et al., 2010, Figures 1D, 5A and D, respectively. (G–I) were modified from Michaud et al., 2022, Figures 2C, 7A and E, respectively.