Most of classical theoretical ecology is based on the assumption that organisms in a community can be naturally partitioned into groups of individuals that can be treated as identical. At the same time, mounting experimental evidence from studies of microbial communities raises the intriguing question whether this intuition is an accurate description of the microbial world. This work builds on Mac Arthur’s model of competitive coexistence on multiple resources to construct a framework that does not rely on postulated existence of species as fundamental ecological variables. In one parameter regime, effective “species” with a core and accessory genome naturally appear in this model as emergent concepts. However, the same model allows a smooth transition to a highly diverse regime where the species formalism becomes inadequate. An alternative description is proposed based on the dynamical modes of population fluctuations. This approach provides a naturally hierarchical description of community dynamics which is well-defined even when the species description breaks down. The relevance of this framework for understanding the complexity of naturally observed microbial communities is discussed.

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

In pattern-forming developmental systems, cells commonly interpret graded input signals, known as morphogens. Morphogens often pattern tissues through cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute suboptimal use of the positional information provided by the input morphogen because noise is added at each tier. However, the conventional theory neglects the role of the format in which information is encoded. We argue that the relevant performance measure is not solely the amount of information carried by the morphogen, but the amount of information that can be accessed by the downstream network. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo.

In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to the velocity of the relaxed equations, while other variables of the relaxing equations converge strongly to the corresponding variables of the relaxed equations. We prove that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepared data, the convergence of the velocity is strong near t = 0. The strong convergence rates of all variables are also estimated.

We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.