The main goal of this small book is to present new results on the existence of non-trivial stationary distributions and the possibility of extinction of one type for a large family of spin-flip systems which are called voter model perturbations. Indeed, the processes of interest, \(\xi_t\in\{0,1\}^{\mathbb{Z}^d}\), are systems with rates \(c^0_{\varepsilon}(s,\xi)=c^v(x,\xi)+\varepsilon^2c_{\varepsilon}^{\ast}(x,\xi)\geq 0\), \(\;x\in \mathbb{Z}^d\), \(\xi\in \{0,1\}^{\mathbb{Z}^d}\), where \(c^v\) is a voter flip rate associated with a symmetric probability kernel and \(c_{\varepsilon}^{\ast}\) is a translation invariant signed perturbation of particular type. When space and time are rescaled, the system converges to the solution of a reaction diffusion equation in dimension \(d\geq 3\). From the properties of this PDE, combining with some methods arising from a low density super-Brownian limit theorem and a block construction, the authors obtain asymptotically sharp conditions under which coexistence holds or one type takes over, respectively, in a voter model perturbation for small enough \(\varepsilon\). Four examples are considered in the case when their parameters are close to the voter model,
(i) a stochastic spatial Lotka-Volterra system from [C. Neuhauser and S. W. Pacala, Ann. Appl. Probab. 9, No. 4, 1226–1259 (1999; Zbl 0948.92022)],
(ii) a model of evolutionary games due to H. Ohtsuki et al. [“A simple rule for the evolution of cooperation on graphs and social networks”, Nature 441, 502–505 (2006; doi:10.1038/nature04605)],
(iii) a continuous time version of the nonlinear model of J. Molofsky et al. [Theor. Popul. Biol. 55, No. 3, 270–282 (1999; Zbl 0951.92032)], and
(iv) a voter model in which opinion changes are followed by an independent exponentially distributed latent period during which votes will note change again.