Scale-dependent colocalization in a population of gyrotactic swimmers

We study the small scale clustering of gyrotactic swimmers transported by a turbulent flow, when the intrinsic variability of the swimming parameters within the population is considered. By means of extensive numerical simulations, we find that the variety of the population introduces a characteristic scale $R^{*}$ in its spatial distribution. At scales smaller than $R^{*}$ the swimmers are homogeneously distributed, while at larger scales an inhomogeneous distribution is observed with a fractal dimension close to what observed for a monodisperse population characterized by mean parameters. The scale $R^{*}$ depends on the dispersion of the population and it is found to scale linearly with the standard deviation both for a bimodal and for a Gaussian distribution. Our numerical results, which extend recent findings for a monodisperse population, indicate that in principle it is possible to observe small scale, fractal clustering in a laboratory experiment with gyrotactic cells.

Figure 1

Correlation dimension $D$ for a homogeneous population of gyrotactic swimmers as a function of the stability number $\mathrm{\Psi }$. Different lines correspond to different swimming numbers: $\mathrm{\Phi }=0.33$ (red crosses), $\mathrm{\Phi }=0.66$ (blue squares), $\mathrm{\Phi }=1.0$ (purple circles), and $\mathrm{\Phi }=3.0$ (black triangles). The error bars are estimated on the fluctuations of the dimension with the statistics.

Figure 2

Vertical section of the positions of two species of swimmers with ${\mathrm{\Psi }}_1=0.5$ (red), ${\mathrm{\Psi }}_2=0.667$ (blue), and ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=3.0$ in a turbulent flow.

Figure 3

Probability $P_{12}\left(r\right)$ to find two cells of different populations 1 and 2 at distance smaller than $r$ for different pairs of population parameters with $\mathrm{\Delta }\mathrm{\Psi }=0.0042$ (red crosses), $\mathrm{\Delta }\mathrm{\Psi }=0.021$ (blue squares), $\mathrm{\Delta }\mathrm{\Psi }=0.042$ (purple circles), and $\mathrm{\Delta }\mathrm{\Psi }=0.125$ (black triangles). Each population is composed by $6.4\times {10}^4$ individuals.

Figure 4

Crossover scale $R^{*}$ as a function of $\mathrm{\Delta }\mathrm{\Psi }$ for two different sets of populations with $\mathrm{\Phi }=3$ (red squares) and $\mathrm{\Phi }=1.5$ (blue circles). (Inset) The same data plotted as a function of $\mathrm{\Phi }\mathrm{\Delta }\mathrm{\Psi }$.

Figure 5

Crossover scale $R^{*}$ as a function of $\mathrm{\Delta }\mathrm{\Psi }$ for two different sets of populations with $\mathrm{\Psi }=0.4$ (red squares), $\mathrm{\Psi }=0.65$ (blue circles), and $\mathrm{\Psi }=0.85$ (pink triangles). (Inset) The same data plotted as a function of $\mathrm{\Psi }\mathrm{\Delta }\mathrm{\Phi }$.

Figure 6

Probability $P\left(r\right)$ to find two cells at distance smaller than $r$ for a population of gyrotactic swimmers with fixed $\mathrm{\Phi }=3$ and $\mathrm{\Psi }$ Gaussian distributed with $\overline{\mathrm{\Psi }}=0.583$ and ${\sigma }_{\mathrm{\Psi }}=0.008$ (red crosses), ${\sigma }_{\mathrm{\Psi }}=0.042$ (blue squares), ${\sigma }_{\mathrm{\Psi }}=0.083$ (purple circles), and ${\sigma }_{\mathrm{\Psi }}=0.166$ (black triangles). Each population is composed by $3\times {10}^5$ individuals. (Inset) Crossover scale $R^{*}$ as a function of ${\sigma }_{\mathrm{\Psi }}$.