Criteria for dynamic similarity in bouncing gaits

Animals of different sizes tend to move in a dynamically similar manner when travelling at speeds corresponding to equal values of a dimensionless parameter (DP) called the Froude number. Consequently, the Froude number has been widely used for defining equivalent speeds and predicting speeds of locomotion by extinct species and on other planets. However, experiments using simulated reduced gravity have demonstrated that equality of the Froude number does not guarantee dynamic similarity. This has cast doubt upon the usefulness of the Froude number in locomotion research. Here we use dimensional analysis of the planar spring-mass model, combined with Buckingham's Pi-Theorem, to demonstrate that four DPs must be equal for dynamic similarity in bouncing gaits such as trotting, hopping and bipedal running. This can be reduced to three DPs by applying the constraint of maintaining a constant average speed of locomotion. Sensitivity analysis indicates that all of these DPs are important for predicting dynamic similarity. We show that the reason humans do not run in a dynamically similar manner at equal Froude number in different levels of simulated reduced gravity is that dimensionless leg stiffness decreases as gravity increases. The reason that the Froude number can predict dynamic similarity in Earth gravity is that dimensionless leg stiffness and dimensionless vertical landing speed are both independent of size. In conclusion, although equal Froude number is not sufficient for dynamic similarity, it is a necessary condition. Therefore, to detect fundamental differences in locomotion, animals of different sizes should be compared at equal Froude number, so that they can be as close to dynamic similarity as possible. More generally, the concept of dynamic similarity provides a powerful framework within which similarities and differences in locomotion can be interpreted.     

A synergetic theory of quadrupedal gaits and gait transitions

We present a theoretical analysis of the patterns of interlimb co-ordination in the gaits of quadrupedal locomotion. Introducing as collective variables a set of relative phases that describe the co-ordination patterns, we classify gaits by their symmetry properties, which can be expressed as invariances under groups of transformations. We define dynamics of the collective variables, on which we impose symmetry restrictions. The stable observable gait patterns correspond to atractors of these dynamics. A non-trivial consequence of this theoretical viewpoint is that gait transitions can take the form of non-equilibrium phase transitions that are accompanied by loss of stability. We show how various types of such phase transitions involving hysteresis, slowing down and fluctuation enhancement can occur. Also the difference between smooth and abrupt transitions is given theoretical foundation. While existing experimental evidence is consistent with the theory developed here, we propose new experimental measures that can serve to test the present theoretical framework. Finally, the influence of underlying symmetries of the dynamics on the nature of the gait patterns and their stability is analyzed. For example, breaking of a front-hind symmetry can lead to a change from absolute to relative co-ordination in the sense of von Holst (1939, Ergebnisse der Physiologie 42, 228). Also, differential stability of straight and reverse gaits results from thus lowering the symmetry.     

A possible energy-saving role for the major fascia of the thigh in running quadrupedal mammals

Experiments were performed to determine the mechanical importance of the fascia lata in stopping the hind limb during its rearward extension and reversing the direction of leg swing. Samples of fascia lata from a number of different mammals were subjected to tensile tests. Tangent Young's moduli reached about 0.5 GPa and stresses at failure about 50 MPa for fascia from each of the species examined. Energy losses incurred in a loading-unloading cycle were generally about 20%. The moment arms of the fascia lata, in combination with its muscle, about the hip and knee joints were determined and the extension of the fascia lata while its muscle is active was estimated. Calculations suggest that the fascia lata could help to reverse the backward swing of the hind limb by recoiling elastically shortly after the foot leaves the ground. Substantial savings of internal kilnetic energy could be made.     

Allometry and curvature in the long bones of quadrupedal mammals

The allometric relationships between basic structural proportions in long bones are examined in the humerus, radius, femur and tibia for a diverse group of 42 terrestrial quadrupedal mammals that span a size range from 0.02–6000 kg. Non-linear scaling is found for length vs. diameter in the tibia and radius, suggesting that the mechanical constraints on the skeleton differ within large and small body-size mammals. Curvature normalized to mid-shaft radius scales differently in the different long bones. Curvature is poorly related to size in the proximal limb bones (humerus and femur) while it decreases systematically with size in the tibia (mass exponent −0.13). The scaling of normalized curvature in the radius is unique among long bones. Variability of curvature in the radius is reduced at any size in comparison to that found in the other long bones. Normalized curvature is constant within the small body size group (0.02 to approximately 100 kg) while it decreases sharply with size within animals over 100 kg body mass. The unusual scaling found in the radius is probably the result of this bone's close alignment with the extrinsic forces which act on it during locomotion. The change in scaling within the radius for animals of different size may be indicative of more general size-dependent mechanical trade-offs which are masked by the complex loading circumstances of the other long bones.     

Symmetrical gaits and center of mass mechanics in small-bodied,primitive mammals

Widely accepted relationships between gaits (footfall patterns) and center of mass mechanics have been formulated from observations for cursorial mammals. However, sparse data on smaller or more generalized forms suggest a fundamentally different relationship. This study explores locomotor dynamics in one eutherian and five metatherian (marsupials) mammals—all small-bodied (<2 kg) with generalized body plans that utilize symmetrical gaits. Across our sample, trials conforming to vaulting mechanics occurred least frequently (<10% of all trials) while bouncing mechanics was obtained most commonly (60%); the remaining trials represented mixed mechanics. Contrary to the common situation in large mammals, there was no evidence for discrete gait switching within symmetrical gaits as speed increased. This was in part due to the common practice of grounded running. The adaptive advantage of different patterns of center-of-mass motion and their putative energy savings remain questionable in small-bodied mammals.     

陆地碳循环的动态非平衡假说

生态系统维持物质与能量的动态平衡是地球系统孕育与维持生物多样性的重要基础。自工业革命以来, 人类活动导致陆地生态系统的碳循环转变为动态非平衡,进而使陆地生态系统的结构与功能出现许多难以预测的变化动态。本文阐释了陆地生态系统碳循环的动态非平衡假说。该假说构建于陆地碳循环内部过程的四点基本特征和五类外部驱动因素。基于这些内部特征与外部因素, 本文归纳了陆地生态系统碳循环动态非平衡在不同时间与空间尺度的表达现象, 并从观测、实验与模型的角度讨论了其检测方法。陆地生态系统碳循环的动态非平衡假说不仅有助于我们理解复杂的陆地碳循环现象, 也为预测未来陆地碳汇动态提供了新的理论框架。     

中国哺乳动物地理分布的多元相似性聚类分析

中国的地形复杂、地貌多样,动物的地理分布亦表现出明显的区域间差异。本研究通过新近文献调研,获知中国现生哺乳动物计692种（隶于13目56科246属）;进而建立物种及其地理分布数据库。采用多元相似性聚类分析（multivariate similarity clustering analysis, MSCA）方法,探讨了中国哺乳动物的地理分布特征。结果表明：（1）中国的现生哺乳动物可划分为176个调查地理单元（investigated geographical unit, IGU）,进而归并为63个基础地理单元（basic geographical unit, BGU）;（2）当相似性系数取0.31、0.25和0.15时,上述63个BGU可分别归并为10个、7个和2个基础地理单元群;据此可对中国的哺乳动物地理分布作出2界7区10亚区的3级区划。本文建议,世界动物地理区划中古北界与东洋界在中国境内的分界线走向为：南迦巴瓦峰西部-玛卿岗日-秦岭北部-黄土高原南缘-大别山-淮河南-长江中下游一线。     

A hypothesis for the evolution of sex

Summary Sex is one of the most creative of the major transitions in Evolution and its existence allows faster and wider mobility of species in the ‘History of Life’. We postulate that sex evolved from prokaryotes in the tail of the fitness distribution curve for a given environment. Once sex was established we have the potential for the evolution of species and the rich flowering of organisms in a relatively short period of time.