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 $d=1$

Itai Cohen (830255)

Paul S. Shamble (4172164)

Ron R. Hoy (4172167)

Tsevi Beatus (4172170)

English

arXiv

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 Myrmarachne formicaria, 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

Shamble, Paul S.

Hoy, Ron R.

Cohen, Itai

Beatus, Tsevi

Dryad Digital Repository (Duke University)

Proceedings of the Royal Society B Biological Sciences

Data from: Walking like an ant: a quantitative and experimental approach to understanding locomotor mimicry in the jumping spider Myrmarachne formicaria

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 spiderMyrmarachne formicaria, 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.</jats:p

Paul S. Shamble

Ron R. Hoy

Itai Cohen

Tsevi Beatus

Crossref

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 Myrmarachne formicaria, 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.General READMEThis general README file provides an overall description of the other files present in this repository.Dryad data repo README.txtBehaviorData related to behavior experiments presented in the manuscript.HighSpeedData related to high-speed (limb kinematics) experiments presented in the manuscript.OverallMovementData related to overall movement experiments presented in the manuscript.Funding provided by: National Science FoundationCrossref Funder Registry ID: http://dx.doi.org/10.13039/100000001Award Number: IOS BRAIN EAGER grant 1546710, NSF-CBET 093333

ZENODO

Electronic Archiving System

We sharpen an estimate of [4] for the topological degree of continuous maps from a sphere Sdinto itself in the case d >= 2. This provides the answer for d >= 2 to a question raised by Brezis. The problem is still open for d = 1. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.CAM

Nguyên, Hoài-Minh

Infoscience - École polytechnique fédérale de Lausanne

A refined estimate for the topological degreeHoai-Minh Nguyen∗October 8, 2017AbstractWe sharpen an estimate of [4] for the topological degree of continuous maps from asphere Sd into itself in the case d ≥ 2. This provides the answer for d ≥ 2 to a questionraised by Brezis. The problem is still open for d = 1.AMS classification: 47H11, 55C25, 58C35.Keywords: topological degree, fractional Sobolev spaces.1 IntroductionMotivated by the theory of Ginzburg Landau equations (see, e.g., [1]), Bourgain, Brezis,and the author established in [4]:Theorem 1. Let d ≥ 1. For every 0 < δ <√2, there exists a positive constant C(δ) such that,for all g ∈ C(Sd,Sd), ∣deg g∣ ≤ C(δ) ∫Sd∫Sd∣g(x)−g(y)∣>δ1∣x − y∣2d dxdy. (1.1)Here and in what follows, for x ∈ Rd+1, ∣x∣ denotes its Euclidean norm in Rd+1.The constant C(δ) depends also on d but for simplicity of notation we omit d. Esti-mate (1.1) was initially suggested by Bourgain, Brezis, and Mironescu in [2]. It was provedin [3] in the case where d = 1 and δ is sufficiently small. In [9], the author improved (1.1) byestablishing that (1.1) holds for 0 < δ < `d = √2 + 2d+1 with a constant C(δ) independent ofδ. It was also shown there that (1.1) does not hold for δ ≥ `d.This note is concerned with the behavior of C(δ) as δ → 0. Brezis [7] (see also [6, Openproblem 3]) conjectured that (1.1) holds withC(δ) = Cδd, (1.2)∗EPFL SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerland, hoai-minh.nguyen@epfl.ch1for some positive constant C depending only on d. This conjecture is somehow motivated bythe fact that (1.1)-(1.2) holds “in the limit” as δ → 0. More precisely, it is known that (see[8, Theorem 2])limδ→0 ∫Sd ∫Sd∣g(x)−g(y)∣>δδd∣x − y∣2d dxdy =Kd∫Sd ∣∇g(x)∣d dx for g ∈ C1(Sd)for some positive constant Kd depending only on d and thatdeg g = 1∣Sd∣ ∫Sd Jac(g) for g ∈ C1(Sd,Sd),by Kronecker’s formula.In this note, we confirm Brezis’ conjecture for d ≥ 2. The conjecture is still open for d = 1.Here is the result of the note.Theorem 2. Let d ≥ 2. There exists a positive constant C = C(d), depending only on d, suchthat, for all g ∈ C(Sd,Sd),∣deg g∣ ≤ C ∫Sd∫Sd∣g(x)−g(y)∣>δδd∣x − y∣2d dxdy for 0 < δ < 1. (1.3)2 Proof of Theorem 2The proof of Theorem 2 is in the spirit of the approach in [4, 9]. One of the new ingredientsof the proof is the following result [10, Theorem 1], which has its roots in [5]:Lemma 1. Let d ≥ 1, p ≥ 1, let B be an open ball in Rd, and let f be a real bounded measurablefunction defined in B. We have, for all δ > 0,1∣B∣2 ∫B ∫B ∣f(x) − f(y)∣p dxdy ≤ Cp,d ⎛⎜⎜⎝∣B∣pd−1 ∫B∫B∣f(x)−f(y)∣>δδp∣x − y∣d+p dxdy + δp⎞⎟⎟⎠ , (2.1)for some positive constant Cp,d depending only on p and d.In Lemma 1, ∣B∣ denotes the Lebesgue measure of B.We are ready to presentProof of Theorem 2. We follow the strategy in [4, 9]. We first assume in addition thatg ∈ C1(Sd,Sd). Let B be the open unit ball in Rd+1 and let u ∶ B → B be the average extensionof g, i.e.,u(X) = ⨏B(x,r) g(s)ds for X ∈ B, (2.2)where x =X/∣X ∣, r = 2(1− ∣X ∣), and B(x, r) ∶= {y ∈ Sd; ∣y−x∣ ≤ r}. In this proof, ⨏D g(s)ds de-notes the equantity 1∣D∣ ∫D g(s)ds for a measurable subset D of Sd with positive (d-dimensional2Hausdorff) measure. Fix α = 1/2 and for every x ∈ Sd, let ρ(x) be the length of the largestradial interval coming from x on which ∣u∣ > α (possibly ρ(x) = 1). In particular, if ρ(x) < 1,then ∣⨏B(x,2ρ(x)) g(s)ds∣ = 1/2. (2.3)By [4, (7)], we have ∣deg g∣ ≤ C ∫Sdρ(x)<11ρ(x)d dx. (2.4)Here and in what follows, C denotes a positive constant which is independent of x, ξ , η, g,and δ, and can change from one place to another.We now implement ideas involving Lemma 1 applied with p = 1. We have, by (2.3),⨏B(x,2ρ(x))⨏B(x,2ρ(x)) ∣g(ξ) − g(η)∣dξ dη ≥ ⨏B(x,2ρ(x)) ∣g(ξ) − ⨏B(x,2ρ(x)) g(η)dη∣ dξ ≥ C.This yields, for some 1 ≤ j0 ≤ d + 1,⨏B(x,2ρ(x))⨏B(x,2ρ(x)) ∣gj0(ξ) − gj0(η)∣dξ dη ≥ C,where gj denotes the j-th component of g. It follows from (2.1) that, for some δ0 > 0 (δ0depends only on d) and for 0 < δ < δ0,ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj0(ξ)−gj0(η)∣>δδ∣ξ − η∣d+1 dξ dη ≥ C,which impliesd+1∑j=1 ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δδ∣ξ − η∣d+1 dξ dη ≥ C. (2.5)Sinceρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣ξ−η∣>C1ρ(x)δδ∣ξ − η∣d+1 dξ dη < C2(d + 1) ,if C1 > 0 is large enough (the largeness of C1 depends only on C and d), it follows from (2.5)thatd+1∑j=1 ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δ∣ξ−η∣≤Cρ(x)δδ∣ξ − η∣d+1 dξ dη ≥ C. (2.6)We derive from (2.4) and (2.6) that, for 0 < δ < δ0,∣deg g∣ ≤ C ∫Sdρ(x)<11ρ(x)2d−1 dx d+1∑j=1∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δ∣ξ−η∣≤Cρ(x)δδ∣ξ − η∣d+1 dξ dη.3This implies, by Fubini’s theorem, that, for 0 < δ < δ0,∣deg g∣ ≤ C d+1∑j=1 ∫Sd ∫Sd∣gj(ξ)−gj(η)∣>δδ∣ξ − η∣d+1 dξ dη ∫ρ(x)≥C∣ξ−η∣/δ2ρ(x)>∣x−ξ∣1ρ(x)2d−1 dx. (2.7)We have∫2ρ(x)>∣x−ξ∣ρ(x)≥C∣ξ−η∣/δ1ρ(x)2d−1 dx ≤ ∫2ρ(x)>∣x−ξ∣∣x−ξ∣>C∣ξ−η∣/δ1ρ(x)2d−1 dx + ∫ρ(x)≥C∣ξ−η∣/δ∣x−ξ∣≤C∣ξ−η∣/δ1ρ(x)2d−1 dx≤ ∫∣x−ξ∣>C∣ξ−η∣/δC∣x − ξ∣2d−1 dx + ∫∣x−ξ∣≤C∣ξ−η∣/δCδ2d−1∣ξ − η∣2d−1 dx.Finally, we use the assumption that d ≥ 2. Since d > 1, it follows that∫ρ(x)>∣x−ξ∣ρ(x)≥C∣ξ−η∣/δ1ρ(x)2d−1 dx ≤ Cδd−1∣ξ − η∣d−1 . (2.8)Combining (2.7) and (2.8) yields, for 0 < δ < δ0,∣deg g∣ ≤ C d+1∑j=1 ∫Sd ∫Sd∣gj(ξ)−gj(η)∣>δδd∣ξ − η∣2d dξ dη. (2.9)Assertion (1.3) is now a direct consequence of (2.9) for δ < δ0 and (1.1) for δ0 ≤ δ < 1.The proof in the case g ∈ C(Sd,Sd) can be derived from the case g ∈ C1(Sd,Sd) via astandard approximation argument. The details are omitted.Acknowledgement: The author warmly thanks Haim Brezis for communicating [7] andHaim Brezis and Itai Shafrir for interesting discussions.References[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices. Progress in Nonlinear Dif-ferential Equations and their Applications, 13, Birkha¨user Boston, 1994.[2] J. Bourgain, H. Brezis, P. Mironescu, Lifting, Degree, and Distributional Jacobian Re-visited, Comm. Pure Appl. Math., 58 (2005), 529-551.[3] J. Bourgain, H. Brezis, P. Mironescu, Complements to the paper ” Lifting, De-gree, and Distributional Jacobian Revisited”, https://hal.archives-ouvertes.fr/hal-00747668/document.4[4] J. Bourgain, H. Brezis, H-M. Nguyen, A new estimate for the topological degree, C. R.Math. Acad. Sci. Paris 340 (2005), 787–791.[5] J. Bourgain and H-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad.Sci. Paris 343 (2006), 75-80.[6] H. Brezis, New questions related to the topological degree. The unity of mathematics,137–154, Progr. Math. 244, Birkhauser, 2006.[7] H. Brezis, Private communication, 2006.[8] H-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006),689–720.[9] H-M. Nguyen, Optimal constant in a new estimate for the degree, J. Anal. Math. 101(2007), 367–395.[10] H-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial DifferentialEquations 41 (2011), 483–509.5