= Solution
The necessary and sufficient condition is
$$
\boxed{\sum_i p_i>0\quad\Longleftrightarrow\quad\rho(\Lambda)>1,}
$$
where $\rho$ is the <spectral radius> of the <nonnegative matrix>. No irreducibility is required. We use the given nonnegative-vector characterization of its Perron <eigenvalue>.
For necessity, suppose $p\ne0$ is the survival <fixed point> and let $S=\{i:p_i>0\}$. Since $1-e^{-z}<z$ for every $z>0$, its equation gives $(\Lambda p)_i>p_i$ for every $i\in S$. There are finitely many such coordinates, so
$$
\alpha=\min_{i\in S}\frac{(\Lambda p)_i}{p_i}>1.
$$
Outside $S$ the inequality $\Lambda p\ge\alpha p$ still holds, because its right side is zero and its left side is nonnegative. The provided <Perron–Frobenius theorem> characterization therefore gives $\rho(\Lambda)\ge\alpha>1$. In particular survival is impossible at $\rho=1$ for this Poisson offspring law.
Conversely suppose $\rho>1$. The given characterization supplies a nonzero $v\ge0$ with $\Lambda v\ge\rho v$; normalize it so $\max_i v_i=1$. For sufficiently small $\eta>0$,
$$
T_i(\eta v)\ge1-e^{-\eta\rho v_i}\ge\eta\rho v_i-\tfrac12\eta^2\rho^2v_i^2\ge\eta v_i.
$$
For example any $0<\eta\le\min(1,2(\rho-1)/\rho^2)$ works. Coordinates with $v_i=0$ satisfy the inequality automatically. Starting at $x_0=\eta v$, iterate $x_{t+1}=T(x_t)$. Monotonicity makes the sequence increase, and $T$ takes nonnegative <vectors> into $[0,1]^l$, so it converges to a <fixed point> $x_\infty\ge\eta v\ne0$. Part (a) gives $p\ge x_\infty$, proving survival from at least one type. This proves the <spectral survival criterion for multitype Poisson branching> for reducible as well as irreducible reproduction <matrices>.
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