Invasion front dynamics of interactive populations in environments with barriers

Abstract

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 (1D) environment, we found that a Langevin equation as [Formula: see text] describing invasion front position. We then obtained how [Formula: see text] and [Formula: see text] depend on population interactions and environmental barrier intensity. In two dimensional (2D) environment, for the average interface position movements we found a Langevin equation as [Formula: see text]. Similar to the 1D case, we calculate how [Formula: see text] and [Formula: see text] respond to population interaction and environmental barrier intensity. Finally, the study of invasion front morphology through dynamic scaling analysis showed that growth exponent, [Formula: see text], depends on both population interaction and environmental barrier intensity. Saturated interface width, [Formula: see text], versus width of the 2D environment (L) also exhibits scaling behavior. Our findings show 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 irregularities of tumor border.

Figure 1

(a) Schematic illustration of the single-species model in one dimension. A unit randomly will be selected and duplication and migration trial happen independently. If the selected unit contains a species and two nearest neighbors are empty, the species duplicates into one and can migrate to the other one. The blue arrow shows the direction of upcoming migration to an empty nearest neighbor and the cyan arrow shows upcoming duplication into an empty nearest neighbor. The red arrow shows a failed trial to occupy an empty nearest neighbor. While this attempt has failed, the strength of the barrier has decreased by 1. In a simple single-species model, after $n=N$ trials, the unit becomes occupiable. (b) Schematic illustration of the two-species model in one dimension. In a randomly selected unit, each species tries to occupy an empty nearest neighbor. We should have $\zeta n_1+n_2>N$ or $n_1+\zeta n_2>N$ for a unit to become occupiable. Migration and duplication happen similar to the single-species model. If the selected unit contains both species, one will be selected randomly for duplication (migration) first and then the other will be selected. However, since we do not include spatial exclusion, selection does not have a relevant role in the majority of cases.

Figure 2

(a) Realization of $X-\overline{X}$ versus time for different values of N for the single-species model. (b) $⟨{(X-\overline{X})}^2⟩$ versus time for different values of N. The linear behavior in log/log diagram and the slope of one ensures the random walk like behavior of fluctuations and thus we can write: $⟨{(X-\overline{X})}^2⟩=D_{f}t$. (c) Invasion front velocity and diffusion constant versus environmental barrier intensity, N. For the large values of N, we have $V_f\propto N^{-{\gamma }_V}$ with ${\gamma }_V=1\pm 0.05$ and $D_f\propto N^{-{\gamma }_D}$ with ${\gamma }_D=2\pm 0.05$.

Figure 3

(a) The effect of interaction term, $\zeta$ on normalized diffusion constant, $N^2{D}_f$ for different values of N. As it shows, interaction term affects diffusion constant differently. (b) $D_f$ versus N for different interactions. Interestingly, ${\gamma }_D$ depends on $\zeta$. Inset shows $N^{{\gamma }_D}{D}_f$ versus $\zeta$ for different values of N. (c) Normalized invasion velocity versus $\zeta$ for different values of N. (d) V versus N for different interactions. ${\gamma }_V$ also depends on $\zeta$. Based on this figure, competitive populations are more sensitive to environmental stresses. (e) ${\gamma }_D$ and ${\gamma }_V$ versus $\zeta$. While ${\gamma }_V$ monotonically decreases by $\zeta$, ${\gamma }_D$ has the maximum at $\zeta \sim 0$.

Figure 4

(a) Schematic illustration of the two-species model in 2D. All arrows represent the same process as their 1D counterparts. (b) $D_H$ versus L for different values of $\zeta$ compared to the single population model. As one may expect, $D_H$ behaves as $\sim L^{-1}$ for all interactions similarly. (c) $D_H$ versus N for $L=20$ and different values of $\zeta$ and single population model. (d) Average velocity of interface, $V_H$, versus N.

Figure 5

(a) Growth exponent,$\beta$ versus N for the single population model and two interactive populations with different values for $\zeta$. As this figure shows, N decreases beta only for non-competitive populations ($\zeta \ge 0$). (b) $W_{\mathit{sat}}$ versus $\zeta$ for $N=10$. This figure shows that for $\zeta =-1$ $W_{\mathit{sat}}$ is much larger that other values of $\zeta$ which means that for competitive populations, invasion front might be much rough. (c) $W_{\mathit{sat}}$ versus L. The slop of log/log diagram gives us the roughness exponent. (d) $W_{\mathit{sat}}$ versus N. For $\zeta =-1$ we have ${\alpha }_N=0.40$ and for $\zeta \ge 0$, we have ${\alpha }_N=0.61\pm 0.02$.