  We examine a system of interacting random walks with leftward drift on
$\mathbb{Z}$, which begins with a single active particle at the origin and some
distribution of inactive particles on the positive integers. Inactive particles
become activated when landed on by other particles, and all particles beginning
at the same point posses equal leftward drift. Once activated, the trajectories
of distinct particles are independent. This system belongs to a broader class
of problems involving interacting random walks on rooted graphs, referred to
collectively as the frog model. Additional conditions that we impose on our
model include that the number of frogs (i.e. particles) at positive integer
points is a sequence of independent random variables which is increasing in
terms of the standard stochastic order, and that the sequence of leftward
drifts associated with frogs originating at these points is decreasing. Our
results include sharp conditions with respect to the sequence of random
variables and the sequence of drifts, that determine whether the model is
transient (meaning the probability infinitely many frogs return to the origin
is $0$) or non-transient. Several, more specific, versions of the model
described will also be considered, and a cleaner, more simplified set of sharp
conditions will be established for each case.

Probability