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Spatial distributions characterize the evolution of reaction-diffusion models of several physical, chemical, and biological systems. We present two novel algorithms for the efficient simulation of these models: Spatial -Leaping (S -Leaping), employing a unified acceleration of the stochastic simulation of reaction and diffusion, and Hybrid -Leaping (H -Leaping), combining a deterministic diffusion approximation with a -Leaping acceleration of the stochastic reactions. AbstractSpatial distributions characterize the evolution of reaction-diffusion models of sev-

We study the fluid dynamics of two fish-like bodies with synchronised swimming patterns. Our studies are based on two-dimensional simulations of viscous incompressible flows. We distinguish between motion patterns that are externally imposed on the swimmers and self-propelled swimmers that learn manoeuvres to achieve certain goals. Simulations of two rigid bodies executing pre-specified motion indicate that flow-mediated interactions can lead to substantial drag reduction and may even generate thrust intermittently. In turn we examine two self-propelled swimmers arranged in a leader-follower configuration, with a-priori specified body-deformations. We find that the swimming of the leader remains largely unaffected, while the follower experiences either an increase or decrease in swimming speed, depending on the initial conditions. Finally, we consider a follower that synchronises its motion so as to minimize its lateral deviations from the leader's path. The leader employs a steady gait while the follower uses a reinforcement learning algorithm to adapt its swimming-kinematics. We find that swimming in a synchronised tandem can yield up to about 30% reduction in energy expenditure for the follower, in addition to a 20% increase in its swimming-efficiency. The present results indicate that synchronised swimming of two fish can be energetically beneficial.

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