A refined estimate for the topological degree

We sharpen an estimate of Bourgain, Brezis, and Nguyen for the topological degree of continuous maps from a sphere $\mathbb{S}^d$ into itself in the case $d \ge 2$. This provides the answer for $d \ge 2$ to a question raised by Brezis. The problem is still open for $d=1$

OverallMovement

Data related to overall movement experiments presented in the manuscript

Behavior

Data related to behavior experiments presented in the manuscript

HighSpeed

Data related to high-speed (limb kinematics) experiments presented in the manuscript

General README

This general README file provides an overall description of the other files present in this repository

Table S1, Figure S1 and Figure S2 from Walking like an ant: a quantitative and experimental approach to understanding locomotor mimicry in the jumping spider <i>Myrmarachne formicaria</i>

Protective mimicry, in which a palatable species avoids predation by being mistaken for an unpalatable model, is a remarkable example of adaptive evolution. These complex interactions between mimics, models and predators can explain similarities between organisms beyond the often-mechanistic constraints typically invoked in studies of convergent evolution. However, quantitative studies of protective mimicry typically focus on static traits (e.g. colour and shape) rather than on dynamic traits like locomotion. Here, we use high-speed cameras and behavioural experiments to investigate the role of locomotor behaviour in mimicry by the ant-mimicking jumping spider <i>Myrmarachne formicaria</i>, comparing its movement to that of ants and non-mimicking spiders. Contrary to previous suggestions, we find mimics walk using all eight legs, raising their forelegs like ant antennae only when stationary. Mimics exhibited winding trajectories (typical wavelength = 5–10 body lengths), which resemble the winding patterns of ants specifically engaged in pheromone-trail following, although mimics walked on chemically inert surfaces. Mimics also make characteristically short (approx. 100 ms) pauses. Our analysis suggests that this makes mimics appear ant-like to observers with slow visual systems. Finally, behavioural experiments with predatory spiders yield results consistent with the protective mimicry hypothesis. These findings highlight the importance of dynamic behaviours and observer perception in mimicry