The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with longrange inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the wellknown Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.he activator-inhibitor principle, originally conceived by Turing in 1952 (1), provides a potential theoretical mechanism for the occurrence of regular patterns in biology (2-6) and chemistry (7-9), although experimental evidence in particular for biological systems has remained scarce (3, 4, 10). In the past decades, this principle has been applied to a wide range of ecological systems, including arid bush lands (11-15), mussel beds (16, 17), and boreal peat lands (18,19). The principle, in which a local positive activating feedback interacts with large-scale inhibitory feedback to generate spatial differentiation in growth, birth, mortality, respiration, or decay, explains the spontaneous emergence of regular spatial patterns in ecosystems even under near-homogeneous starting conditions. Physical theory offers an alternative mechanism for pattern formation, proposed by Cahn and Hilliard in 1958 (20). They identified that density-dependent rates of dispersal can lead to separation of a mixed fluid into two phases that are separated in distinct spatial regions, subsequently leading to pattern formation. The principle of density-dependent dispersal, switching between dispersion and aggregation as local density increases, has become a central mathematical explanation for phase separation in many fields (21) such as multiphase fluid flow (22), mineral exsolution and growth (23), and biological applications (24-28). Although aggregation due to individual motion is a commonly observed phenomenon within ecology, application of the principles of phase separation to explain pattern formation in ecological systems is absent both in terms of theory and experiments (25,26).Here, we apply the concept of phase separation to the formation of spatial patterns in the distribution of aggregating mussels. On intertidal flats, establishing mussel beds exhibit spatial self-org...

We present a model for a synthetic gene oscillator and consider the coupling of the oscillator to a periodic process that is intrinsic to the cell. We investigate the synchronization properties of the coupled system, and show how the oscillator can be constructed to yield a significant amplification of cellular oscillations. We reduce the driven oscillator equations to a normal form, and analytically determine the amplification as a function of the strength of the cellular oscillations. The ability to couple naturally occurring genetic oscillations to a synthetically designed network could lead to possible strategies for entraining and/or amplifying oscillations in cellular protein levels.

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In this article we construct, both asymptotically and numerically, multibump, blow-up, self-similar solutions to the complex Ginzburg-Landau equation (CGL) in the limit of small dissipation. Through a careful asymptotic analysis, involving a balance of both algebraic and exponential terms, we determine the parameter range over which these solutions may exist. Most intriguingly, we determine a branch of solutions that are not perturbations of solutions to the nonlinear Schrödinger equation (NLS); moreover, they are not monotone, but they are stable. Furthermore, these axisymmetric ring-like solutions exist over a broader parameter regime than the monotone profile.

The blood brain barrier (BBB) is the main barrier that separates the blood from the brain. Because of the BBB, the drug concentration-time profile in the brain may be substantially different from that in the blood. Within the brain, the drug is subject to distributional and elimination processes: diffusion, bulk flow of the brain extracellular fluid (ECF), extra-intracellular exchange, bulk flow of the cerebrospinal fluid (CSF), binding and metabolism. Drug effects are driven by the concentration of a drug at the site of its target and by drug-target interactions. Therefore, a quantitative understanding is needed of the distribution of a drug within the brain in order to predict its effect. Mathematical models can help in the understanding of drug distribution within the brain. The aim of this review is to provide a comprehensive overview of system-specific and drug-specific properties that affect the local distribution of drugs in the brain and of currently existing mathematical models that describe local drug distribution within the brain. Furthermore, we provide an overview on which processes have been addressed in these models and which have not. Altogether, we conclude that there is a need for a more comprehensive and integrated model that fills the current gaps in predicting the local drug distribution within the brain.