Exponential convergence-rate conjecture for the invading species

Let $X_t$ be the population state of the Moran model, let $e_{i_0}$ denote the equilibrium corresponding to species $i_0$, let $E$ be the set of environments, and let $\boldsymbol{s}$ be the environment process. Assume that $\Lambda_{i_0}<0$.

Exponential convergence-rate conjecture. For every initial population state $x$ with $x^{i_0}\neq 0$ and every $s\in E$,

$$\mathbb{P}_{(x,\boldsymbol{s})}\left(\limsup_{t\to\infty}\frac{1}{t}\log\lVert X_t-e_{i_0}\rVert\leq\Lambda_{i_0}\right)=1.$$

The preceding theorem shows that a species with negative invasion rate can invade the community with positive probability, whereas a species with positive invasion rate cannot invade. This conjecture predicts the almost-sure exponential rate at which trajectories approach the equilibrium associated with the invading species.