Alastair M. Rucklidge

Cyclic dominance in evolutionary games: A review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator–prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock–paper–scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg–Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

A Heteroclinic Model of Geodynamo Reversals and Excursions

The Earth’s magnetic field is by and large a steady dipole, but its history has been punctuated by intermittent excursions and reversals. This is at least superficially similar to the behaviour of differential equations containing structurally stable heteroclinic cycles. We present a model of the geodynamo that is based on the symmetries of velocity fields in a rotating spherical shell, and that contains such a cycle. Patterns of excursions and reversals that resemble the geomagnetic record can be obtained by introducing small symmetry-breaking terms.

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasizing how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced. 76W05, 76F10

Chaos in magnetoconvection

The partial differential equations (PDEs) for two-dimensional incompressible convection in a strong vertical magnetic field have a codimension-three bifurcation when the parameters are chosen so that the bifurcations to steady and oscillatory convection coincide and the limit of narrow rolls is taken. The third-order set of ordinary differential equations (ODEs) that govern the behaviour of the PDEs near this bifurcation are derived using perturbation theory. These ODEs are the normal form of the codimension-three bifurcation; as such, they prove to be an excellent predictor of the behaviour of the PDEs. This is the first time that a detailed comparison has been made between the chaotic behaviour of a set of PDEs and that of the corresponding set of model ODEs, in a parameter regime where the ODEs are expected to provide accurate approximations to solutions of the PDEs. Most significantly, the transition from periodic orbits to a chaotic Lorenz attractor predicted by the ODEs is recovered in the PDEs, making this one of the few situations in which the nature of chaotic oscillations observed numerically in PDEs can be established firmly. Including correction terms obtained from the perturbation calculation enables the ODEs to track accurately the bifurcations in the PDEs over an appreciable range of parameter values. Numerical calculations suggest that the T-point (where there are heteroclinic connections between a saddle point and a pair of saddle-foci), which is associated with the transition from a Lorenz attractor to a quasi-attractor in the normal form, is also found in the PDEs. Further numerical simulations of the PDEs with square rolls confirm the existence of chaotic oscillations associated with a heteroclinic connection between a pair of saddle-foci.

Compressible magnetoconvection in three dimensions : pattern formation in a strongly stratified layer

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur.

Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state. We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors. The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications.

Solar Magnetoconvection (Invited Review)

In recent years the study of how magnetic fields interact with thermal convection in the Sun has made significant advances. These are largely due to the rapidly increasing computer power and its application to more physically relevant parameters regimes and to more realistic physics and geometry in numerical models. Here we present a survey of recent results following one line of investigations and discuss and compare the results of these with observed phenomena.

Pulsating waves in nonlinear magnetoconvection

Numerical experiments on compressible magnetoconvection reveal a new type of periodic oscillation, associated with alternating streaming motion. Analogous behaviour in a Boussinesq fluid is constrained by extra symmetry. A low-order model confirms that these pulsating waves appear via a pitchfork-Hopf-gluing bifurcation sequence from the steady state.

Quasicrystalline order and a crystal-liquid state in a soft-core fluid.

A two-dimensional system of soft particles interacting via a two-length-scale potential is studied. Density functional theory and Brownian dynamics simulations reveal a fluid phase and two crystalline phases with different lattice spacing. Of these the larger lattice spacing phase can form an exotic periodic state with a fraction of highly mobile particles: a crystal liquid. Near the transition between this phase and the smaller lattice spacing phase, quasicrystalline structures may be created by a competition between linear instability at one scale and nonlinear selection of the other.

When does cyclic dominance lead to stable spiral waves

Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the systems phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.