A uniqueness result on detecting a prey in a spider orb-web

TL;DR

This paper proves a mathematical uniqueness result for identifying the prey's impact location on a spider orb-web using dynamic measurements of web vibrations, modeled as a continuous membrane with fibrous structure.

Contribution

It introduces a novel inverse problem framework for localizing prey on a spider web and establishes a uniqueness theorem based on limited dynamic measurements.

Findings

Uniqueness of prey localization from measurements on a small ring

Modeling the web as a fibrous membrane with pre-stress

Applicable to axially-symmetric orb-webs under transverse load

Abstract

We consider the inverse problem of localizing a prey hitting a spider orb-web from dynamic measurements taken near the center of the web, where the spider is supposed to stay. The actual discrete orb-web, formed by a finite number of radial and circumferential threads, is modelled as a continuous membrane. The membrane has a specific fibrous structure, which is inherited from the original discrete web, and it is subject to tensile pre-stress in the referential configuration. The transverse load describing the prey's impact is assumed of the form $g(t)f(x)$, where $g(t)$ is a known function of time and $f(x)$ is the unknown term depending on the position variable $x$. For axially-symmetric orb-webs supported at the boundary and undergoing infinitesimal transverse deformations, we prove a uniqueness result for $f(x)$ in terms of measurements of the transverse dynamic displacement taken on an arbitrarily small and thin ring centered at the origin of the web, for a sufficiently large interval of time.