Mixed-order phase transition of the contact process near multiple junctions

Metapopulation model for a prey-predator system: Nonlinear migration due to the finite capacities of patches

Many species live in spatially separated patches, and individuals can migrate between patches through paths. In real ecosystems, the capacities of patches are finite. If a patch is already occupied by the individuals of some species, then the migration into the patch is impossible. In the present paper, we deal with prey-predator system composed of two patches. Each patch contains a limited number of cells, where the cell is either empty or occupied by an individual of prey or predator. We introduce "swapping migration" defined by the exchange between occupied and empty cells. An individual can migrate, only when there are empty cells in the destination patch. Reaction-migration equations in prey-predator system are presented, where the migration term forms nonlinear function of densities. We numerically solve equilibrium densities, and find that the population dynamics are largely affected by nonlinear migration. Not only extinction points but also the responses to the environmental changes crucially depend on the patch capacities.

Epidemics of random walkers in metapopulation model for complete, cycle, and star graphs

We present the metapopulation dynamic model for epidemic spreading of random walkers between subpopulations. A subpopulation is represented by a node on a graph. Each agent or individual is either susceptible (S) or infected (I). All agents move by random walk on the graph; namely, each agent randomly determines the destination of migration. The reaction-diffusion equations are presented as ordinary differential equations, not partial differential equations. To evaluate the risk of each subpopulation (node), we obtain the solutions of reaction-diffusion equations analytically and numerically for small, complete, cycle and star graphs. If a graph is homogeneous, or if every node has the same degree, then the solution never changes for any nodes. However, when a graph is heterogeneous, the infection density in equilibrium differs entirely among nodes. For example, on star graphs, the hub seems to be a supply source of disease because the infection density at the hub is much higher than that at the other nodes. On every graph, the epidemic thresholds are identical for all nodes.

Nonuniversal and anomalous critical behavior of the contact process near an extended defect

We consider the contact process near an extended surface defect, where the local control parameter deviates from the bulk one by an amount of λ(l)-λ(∞)=Al^{-s}, with l being the distance from the surface. We concentrate on the marginal situation s=1/ν_{⊥}, where ν_{⊥} is the critical exponent of the spatial correlation length, and study the local critical properties of the one-dimensional model by Monte Carlo simulations. The system exhibits a rich surface critical behavior. For weaker local activation rates A<A_{c}, the phase transition is continuous, having an order-parameter critical exponent, which varies continuously with A. For stronger local activation rates A>A_{c}, the phase transition is of mixed order: the surface order parameter is discontinuous; at the same time the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. The mixed-order transition regime is analogous to that observed recently at a multiple junction and can be explained by the same type of scaling theory.

Mixed-order phase transition in a colloidal crystal

Mixed-order phase transitions display a discontinuity in the order parameter like first-order transitions yet feature critical behavior like second-order transitions. Such transitions have been predicted for a broad range of equilibrium and nonequilibrium systems, but their experimental observation has remained elusive. Here, we analytically predict and experimentally realize a mixed-order equilibrium phase transition. Specifically, a discontinuous solid-solid transition in a 2D crystal of paramagnetic colloidal particles is induced by a magnetic field [Formula: see text] At the transition field [Formula: see text], the energy landscape of the system becomes completely flat, which causes diverging fluctuations and correlation length [Formula: see text] Mean-field critical exponents are predicted, since the upper critical dimension of the transition is [Formula: see text] Our colloidal system provides an experimental test bed to probe the unconventional properties of mixed-order phase transitions.