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.

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is CΔt, where Δt is the time-step size. Secondly, we prove the convergence rate CΔt2 for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate CΔt.

In this paper we study segmented strings in AdS3 coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the SL(2) Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

Two Dehn surgeries on a knot are called purely cosmetic, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in S^3, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation.