Model of Collective Fish Behavior with Hydrodynamic Interactions

Fish schooling is often modeled with self-propelled particles subject to phenomenological behavioral rules. Although fish are known to sense and exploit flow features, these models usually neglect hydrodynamics. Here, we propose a novel model that couples behavioral rules with far-field hydrodynamic interactions. We show that (1) a new “collective turning” phase emerges, (2) on average, individuals swim faster thanks to the fluid, and (3) the flow enhances behavioral noise. The results of this model suggest that hydrodynamic effects should be considered to fully understand the collective dynamics of fish.

Figure 1

(a) Sketch of two interacting swimmers, showing the heading ${\mathit{e}}_i^{\parallel }$ of swimmer $i$, its viewing angle ${\theta }_{ij}$, the relative alignment angle ${\varphi }_{ij}$, the interswimmer distance ${\rho }_{ij}$, and the polar coordinates in the reference frame of swimmer $j$ (${\mathit{e}}_j^{\rho }$, ${\mathit{e}}_j^{\theta }$). (b) Gray scale representation of the anisotropic visual perception, modeled by the term $(1+\cos {\theta }_{ij})$ in Eqs. (2)–(4).

Figure 2

(a)–(d) Plots of the swimmer positions and associated streamlines for different values of the parameters. On all figures, the dipole intensity is $I_f={10}^{-2}$, the scale bar corresponds to $10r_0$, and the color scale represents the instantaneous velocity. For clarity, swimmers are represented as “airfoils” of length $7r_0$. Four distinct dynamical phases are observed: (a) swarming for $I_n=0.8$, $I_{\parallel }=0.5$; (b) schooling for $I_n=0.5$, $I_{\parallel }=9$; (c) milling for $I_n=0.3$, $I_{\parallel }=1.5$; and (d) turning for $I_n=0.2$, $I_{\parallel }=4$. (e) Paths followed by each swimmer during 12 dimensionless time units, the final time corresponding to (d). For long-time dynamics, see Supplemental Material Fig. 1 and Movies 1–4 [41].

Figure 3

Phase diagram using the value of the polarization $P$ and the milling $M$ as a color code (d), for three cases: (a) no fluid (${\mathit{U}}_i={\mathrm{\Omega }}_i=0$), (b) fluid without induced rotation (${\mathit{U}}_i\ne 0$, ${\mathrm{\Omega }}_i=0$), and (c) full hydrodynamic model (${\mathit{U}}_i\ne 0$, ${\mathrm{\Omega }}_i\ne 0$). In all cases, the dipole intensity is $I_f={10}^{-2}$, the solid white line indicates the $M=0.4$ level, and the dashed line indicates the $P=0.5$ level. Solid black lines in (b),(c) show the $V$ levels and the solid blue line in (a) shows the milling-schooling transition line found in [20]. In (c), the black dots show the parameter values corresponding to Figs. 2.

Figure 4

Heat maps showing the probability of presence $p(\rho ,\theta )$ of all other swimmers in the framework of an individual: (a) swarming for $I_n=0.8$, $I_{\parallel }=0.5$; (b) schooling for $I_n=0.5$, $I_{\parallel }=9$; (c) milling for $I_n=0.3$, $I_{\parallel }=1.5$; and (d) turning for $I_n=0.2$, $I_{\parallel }=4$.

Figure 5

(a) Phase diagram in the absence of noise ($I_n=0$). The color code and the contour levels are the same as in Fig. 3. (b) Values of the polarization $P$ (red), milling $M$ (green), and mean velocity $V$ (black) for two cases: no fluid ($I_f=0$) and low noise ($I_n=0.05$) with solid lines; full hydrodynamic model ($I_f={10}^{-2}$) and no noise ($I_n=0$) with open symbols.