Invasion front dynamics of interactive populations in environments with barriers

Invading populations normally comprise different subpopulations that interact while trying to overcome existing barriers against their way to occupy new areas. However, the majority of studies so far only consider single or multiple population invasion into areas where there is no resistance against the invasion. Here, we developed a model to study how cooperative/competitive populations invade in the presence of a physical barrier that should be degraded during the invasion. For one dimensional environment, we found that a Langevin equation as $dX/dt=V_ft+\sqrt{D_f}\eta(t)$ describes invasion front position. We then obtained how $V_f$ and $D_f$ depend on population interactions and environmental barrier intensity. For the 2D case, for the average interface position we found a Langevin equation as $dH/dt=V_Ht+\sqrt{D_H}\eta(t)$. Similar to the 1D case, we found how $V_H$ and $D_H$ respond to population interaction and environmental barrier intensity. Finally, the study of invasion front morphology through dynamic scaling analysis showed that growth exponent, $\beta$, depends on both population interaction and environmental barrier intensity. Saturated interface width, $W_{sat}$, versus width of the 2D environment ($L$) also exhibits scaling behavior. Comparing results for the 2D environment revealed that competition among subpopulations leads to more rough invasion fronts. Considering the wide range of shreds of evidence for clonal diversity in cancer cell populations, our findings suggest that interactions between such diverse populations can potentially participate in the geometry of the tumor border.