A primer of swarm equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model.The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain $\delta$-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.Comment: 38 pages, submitted to SIAM J. Appl. Dyn. Sy

Presence of the “Threatened” \u3ci\u3eTrimerotropis Huroniana\u3c/i\u3e (Orthoptera: Acrididae) in Relation to the Occurrence of Native Dune Plant Species and the Exotic \u3ci\u3eCentaurea Biebersteinii\u3c/i\u3e

Trimerotropis huroniana Wlk. is a “Threatened” species in Michigan and Wisconsin with a distribution limited to open dune systems in the northern Great Lakes region of North America. Pitfall traps were utilized in the Grand Sable Dunes of Pictured Rocks National Lakeshore, MI, along with an herbaceous plant survey, to identify the relationship of T. huroniana with native dune plant species, Ammophila breviligulata Fern. (American beachgrass, Poaceae), Artemisia campestris L. (field sagewort, Asteraceae), and the exotic invasive plant Centaurea biebersteinii DC. [=Centaurea maculosa, spotted knapweed, Lamarck] (Asteraceae). The absence of C. biebersteinii resulted in an increased likelihood of capturing T. huroniana. This was most likely due to the increased likelihood of encountering A. campestris in areas without C. biebersteinii. The occurrence of A. breviligulata was independent of C. biebersteinii presence. A significant positive linear relationship occurred between the percent cover of A. campestris and the traps that captured T. huroniana. There was no significant relationship between A. breviligulata percent cover and the traps that captured T. huroniana. The occurrence and distribution of T. huroniana is closely related to the presence and abundance of A. campestris. Habitat conservation and improvement for T. huroniana should include increases in A. campestris populations through the removal of C. biebersteinii

The supervised hierarchical Dirichlet process

We propose the supervised hierarchical Dirichlet process (sHDP), a nonparametric generative model for the joint distribution of a group of observations and a response variable directly associated with that whole group. We compare the sHDP with another leading method for regression on grouped data, the supervised latent Dirichlet allocation (sLDA) model. We evaluate our method on two real-world classification problems and two real-world regression problems. Bayesian nonparametric regression models based on the Dirichlet process, such as the Dirichlet process-generalised linear models (DP-GLM) have previously been explored; these models allow flexibility in modelling nonlinear relationships. However, until now, Hierarchical Dirichlet Process (HDP) mixtures have not seen significant use in supervised problems with grouped data since a straightforward application of the HDP on the grouped data results in learnt clusters that are not predictive of the responses. The sHDP solves this problem by allowing for clusters to be learnt jointly from the group structure and from the label assigned to each group.Comment: 14 page