A Trade-Off Between Simplicity and Robustness? Illustration on a Lattice-Gas Model of Swarming

Abstract

We re-examine a cellular automaton model of swarm formation. The local rule is stochastic and defined simply as a force that aligns particles with their neighbours. This lattice-gas cellular automaton was proposed by Deutsch to mimic the self-organisation process observed in various natural systems (birds, fishes, bacteria, etc.). We explore the various patterns the self-organisation process may adopt. We observe that, according to the values of the two parameters that define the model, the alignment sensitivity and density of particles, the system may display a great variety of patterns. We analyse this surprising diversity of patterns with numerical simulations. We ask where this richness comes from. Is it an intrinsic characteristic of the model or a mere effect of the modelling simplifications?

Chapter’s Appendix: Simulation of Reflecting Borders in a LGCA

The simulation of the reflecting borders boundary conditions is applied as follows.

We initialise the system by letting each channel contain a particle with probability $d$. There are exceptions: (a) The border cells are all empty. (b) For the cells situated immediately next to the northern, eastern, southern and western borders cells, we respectively empty the North, East, South and West channels. Formally, we use ${\mathcal L}= \left\{ {0},\dots ,{X}\right\} \times \left\{ {0},\dots ,{Y}\right\}$ and:

• $\mathcal {B}= \{ (i,j) \in {\mathcal L}, i=0 \text{ or }\ i=X-1 \text{ or }\ j=0 \text{ or }\ j=Y-1 \}$,

• $\mathcal {B}_{\mathrm {N}}= \{ (i,j) \in {\mathcal L}, j=Y-1\}$,

• $\mathcal {B}_{\mathrm {E}}= \{ (i,j) \in {\mathcal L}, i=X-1\}$,

• $\mathcal {B}_{\mathrm {S}}= \{ (i,j) \in {\mathcal L}, j=0\}$,

• $\mathcal {B}_{\mathrm {W}}= \{ (i,j) \in {\mathcal L}, i=0\}$.

By noting the initial condition $x \in {Q^{\mathcal L}}$ for each $(i,j) \in {\mathcal L}$, taking $x(i,j) = (q_{\mathrm {\mathtt {n}}}, q_{\mathrm {\mathtt {e}}}, q_{\mathrm {\mathtt {s}}}, q_{\mathrm {\mathtt {w}}} )$, we have $q_{\mathrm {\mathtt {n}}}=0 \text{ if }\ (i,j) \in \mathcal {B} \cup {\mathcal {B}}_{\mathrm {N}}$, $q_{\mathrm {\mathtt {e}}}=0 \text{ if }\ (i,j) \in \mathcal {B} \cup \mathcal {B}_{\mathrm {E}}$, etc. We call this last set of 4 conditions, the integrity condition.

The integrity condition guarantees that no particle will travel to a border cell (in $\mathcal {B}$). It is easy to see that the propagation step preserves the integrity condition, but not the interaction step. Our method thus consists in checking if the north channel of a cell of $\mathcal {B}_{\mathrm {N}}$ is occupied. In this case, the particle is re-affected among the free channels of the cell, with a uniform probability. All happens as if the particle has “bounced” on a northern wall. Clearly, such a rearrangement is always possible as this cell cannot contain four particles. The same procedures is applied for the three other directions.