  Habitat loss is one of the biggest threats facing plant species nowadays. We
formulate a simple mathematical model of seed dispersal on reduced habitats to
discuss survival of the species in relation to the habitat size and seeds
production rate. Seeds get dispersed around the mother plant via several agents
in a random way. In our model seeds landing sites are distributed according to
a homogeneous Poisson point process with a constant rate on $\mathbb{R}$. We
will assume that each seed will successfully germinate and grow into a new
plant with the same characteristics as the mother plant. The time is discrete,
scaled according to generations of plants or can represent years, since annual
plants go through an entire growing cycle during one year. Then we will assume
there are two symmetric barriers with respect to the origin and consider that
the growth can not evolve past the barriers. Imposing barriers correspond to
the physical limitation of the habitat. We appeal to tools of Probability
Theory to formalize and study such a model, which can be seen as a
discrete-time one-dimensional branching random walk with barriers. By means of
coupling techniques and the comparison with suitably constructed multi-type
branching processes we localize the critical parameter of the process around
which there is survival with positive probability or extinction almost surely.
In addition, we consider a discrete-space version of the model for which exact
results are also obtained.
