The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$

We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $\pi\in(0,1)$; conditionally on $\pi=p$, its lifetime $\Xi$ satisfies $$P(\Xi\ge k\mid \pi=p)=p^{k^{\gamma}},\qquad k\in\mathbb{N}_0,\gamma>0.$$ The law of $\pi$ has right edge density $$f_\pi(u)\sim(1-u)^{\beta-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1),$$ with $\beta>0$ and $L$ slowly varying; let $\eta$ denote the common law of the i.i.d. initial occupation numbers $\{\eta_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $\gamma-$dependent threshold $$\beta_c:=\frac{1}{2\gamma}.$$ If $\beta>\beta_c$ and $E(\eta)<\infty$, the process becomes extinct almost surely; if $\beta<\beta_c$ and $P(\eta=0)<1$, it survives with positive probability. At the boundary $\beta=\beta_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2\gamma})$. The case $\gamma=1$ (geometric lifetimes) recovers the benchmark $\beta_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.