`When you say "Jump!"; I say "How far ?"': non-local jumping for stochastic lattice-based position jump simulations.

Position jump models of diffusion are a valuable tool in biology, but stochastic simulations can be very computationally intensive, especially when the number of particles involved grows large. It will be seen that time-savings can be made by allowing particles to jump with a range of distances and rates, rather than being restricted to moving to adjacent boxes on the lattice. Since diffusive systems can often be described with a PDE in the diffusive limit when particle numbers are large, we also discuss the derivation of equivalent boundary conditions for the discrete, non-local system, as well as variations on the basic scheme such as biased jumping and hybrid systems.