Asymptotic growth conjecture for the first three-patch seasonal system

Assume $a>0>b$ and $a+2b<0$. For the seasonal system denoted by (47ab), let $\Lambda(m,T)$ be its growth rate, with migration rate $m$ and period $T$. Asymptotic growth conjecture. For every $T>0$ and every $m>0$,

$$\Lambda(\infty,T)=b,$$

and

$$\Lambda(m,\infty)=\frac{a+b-m}{2}.$$

These formulas concern the infinite-migration and infinite-period limits, respectively. The source states them as conjectures because its hypotheses do not determine the relevant limits for this system.