Spatial patterning can be crucially important for understanding the behavior of interacting populations. Here we investigate a simple model of parasite and host populations in which parasites are random walkers that must come into contact with a host
in order to reproduce. We focus on the spatial arrangement of parasites around a single host, and we derive using analytics and numerical simulations the necessary conditions placed on the parasite fecundity and lifetime for the populations long-term survival. We also show that the parasite population can be pushed to extinction by a large drift velocity, but, counterintuitively, a small drift velocity generally increases the parasite population.
We study a totally asymmetric simple exclusion process (TASEP) with one defect site, hopping rate $q<1$, near the system boundary. Regarding our system as a pair of uniform TASEPs coupled through the defect, we study various methods to match a emph{f
inite} TASEP and an emph{infinite} one across a common boundary. Several approximation schemes are investigated. Utilizing the finite segment mean-field (FSMF) method, we set up a framework for computing the steady state current $J$ as a function of the entry rate $% alpha $ and $q$. For the case where the defect is located at the entry site, we obtain an analytical expression for $J(alpha, q) $ which is in good agreement with Monte Carlo simulation results. When the defect is located deeper in the bulk, we refined the scheme of MacDonald, et.al. [Biopolymers, textbf{6}, 1 (1968)] and find reasonably good fits to the density profiles before the defect site. We discuss the strengths and limitations of each method, as well as possible avenues for further studies.
We study the effects of local inhomogeneities, i.e., slow sites of hopping rate $q<1$, in a totally asymmetric simple exclusion process (TASEP) for particles of size $ell geq 1$ (in units of the lattice spacing). We compare the simulation results of
$ell =1$ and $ell >1$ and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the emph{location} of the slow site for both $ell =1$ and $ell >1$ cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when $qto 0$. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.
