For wildlife to spread into the area, you need plants there first. Predators won't spread until prey starts living there. And prey uses plants for shelter from Predators and for food.
The diagram below shows the different mechanisms for seed dispersal -- wind, falling to the ground and germinating, falling to the ground and hitchhiking in animals' guts, or on fur or one clothing in the case of humans.
Wind dispersal is the farthest-reaching mechanism if there are winds present. Especially since your only viable plants verge on one side of a desolate zone.
The efficiency, and therefore the rate of dispersal is very complicated determined by the seed casing itself, wind speed, direction, gusting, probability of landing in a good spot, and not getting eaten before the plant reaches maturity and generates new seed.
You have a small advantage to accelerate the dispersal -- indirectly. The non-desert flora surrounding the desolate region will disperse seeds into the zone. These will attract birds and rodents, who will spread seeds in their scat and fur.
The topic of the mathematics of modeling seed dispersal is an ongoing topic for seedologists -- or whatever the discipline is called.
So unless you want to do a lot of very complicated probabilistic calculations, you can use a simple algebraic term and guess at an answer.
$$Rate = \chi \times d \frac{km | miles}{year|plant seeding cycle}$$
where $\chi$ is the probability a sufficient density of seeds will populate the adjacent area
and
where d is the average dispersal distance of seeds.
values for $\chi$ seem valid from 5% to 40% depending on species, weather, wind, the population of seed eaters, etc per seeding cycle.
values for d seem valid from 1 (1.6 km) mile to 10 miles (16 km)
