The stability–complexity relationship at age 40: a random matrix perspective

Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May's work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May's celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so-called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.     

Extending the quasi-neutral concept

In caricature, the equilibrium paradigm of community ecology states that plant communities are stable entities consisting of competing species - and that such species coexist because each has its “niche”. This paradigm, in its extreme, has been dead for some time. Nevertheless, it has yet to be replaced by a credible “non” — equilibrium paradigm. The quasi-neutral concept of plant communities, proposed by Kristjan Zobel,Folia Geobot. 36: 3–8, 2001, possesses some of the key ingredients of a nonequilibrium theory of diversity. It recognizes that there are inescapable relationships between diversity at different scales, that similarity can influence the rate of competitive exclusion, that successional change is typically associated with changes in life form, and that rarefaction (i.e. “sampling artifacts”) can have strong effects on fine-scale diversity. However, the current formulation of the quasi-neutral concept is incomplete in that it relies on an unrealistic definition of community, it assumes that random sampling means that species richness at one scale will be linearly related to richness at finer scales, it ignores the possibility of fine-scale processes producing broader-scale patterns, and it avoids the subject of fine-scale environmental heterogeneity. But the most serious limitation of the quasi-neutral concept is that similarity of species alone is not sufficient to allow indefinite coexistence. I present the results of a simple simulation to demonstrate that: (1) identical species will eventually be lost from communities due to stochastic “drift”, (2) slight variations in reproductive rates accelerate this loss, but (3) adding a miniscule “cost of commonness” to the model allows the indefinite coexistence of species. I conclude that the quasi-neutral model cannot work without some kind of trade-off.     

Alternative (un)stable states in a stochastic predator–prey model

Stochastic models sometimes behave qualitatively differently from their deterministic analogues. We explore the implications of this in ecosystems that shift suddenly from one state to another. This phenomenon is usually studied through deterministic models with multiple stable equilibria under a single set of conditions, with stability defined through linear stability analysis. However, in stochastic systems, some unstable states can trap stochastic dynamics for long intervals, essentially masquerading as additional stable states. Using a predator–prey model, we demonstrate that this effect is sufficient to make a stochastic system with one stable state exhibit the same characteristics as an analogous system with alternative stable states. Although this result is surprising with respect to how stability is defined by standard analyses, we show that it is well-anticipated by an alternative approach based on the system's “quasi-potential.” Broadly, understanding the risk of sudden state shifts will require a more holistic understanding of stability in stochastic systems.     

Microbiota and mitobiota. Putting an equal sign between mitochondria and bacteria

The recent revival of old theories and setting them on modern scientific rails to a large extent are also relevant to mitochondrial science. Given the widespread belief that mitochondria are symbionts of ancient bacterial origin, the processes inherent to mitochondrial physiology can be revised based on their comparative analysis with possible involvement of bacteria. Such comparison combined with discussion of the role of microbiota in pathogenesis allows discussion of the role of “mitobiota” (we introduce this term) as the combination of different phenotypic manifestations of mitochondria in the organism reflecting pathological changes in the mitochondrial genome. When putting an equal sign between mitochondria and bacteria, we find similarity between the mitochondrial and bacterial theories of cancer. The presence of the term “bacterial infection” suggests “mitochondrial infection”, and mitochondrial (oxidative) theory of aging can in some way be transformed into a “bacterial theory of aging”. The possible existence of such processes and the data confirming their presence are discussed in this review. If such a comparison has the right to exist, the homeostasis of “mitobiota” is of not lesser physiological importance than homeostasis of microbiota, which has been so intensively discussed recently.     

Studies on the reaction mechanism of DT diaphorase. Intermediary plateau and trough regions in the initial velocity vs substrate concentration curves.

The steady-state kinetics of DT diaphorase [NAD(P)H dehydrogenase (quinone): EC 1.6.99.2]from rat liver has been studied using 2,6-dichloroindophenol as electron acceptor and NADH as electron donor. The υ vs [S]curves revealed intermediary plateau and “trough” regions when NADH and dichloroindophenol were the varying substrates. It has been demonstrated by statistical analysis that equations involving fourth-power terms of the substrate concentrations are able to describe the “trough.” These equations have been previously shown in the literature to be capable of describing intermediary plateau regions also. Changes in protein concentration did not affect these curves substantially, whereas changes in pH caused shifts in the positions of their intermediary plateau regions. Increase of the ionic strength by addition of 0.25–0.5 m KCl changed the intermediary plateau region into a “trough” in the υ vs [NADH] plot, whereas the “trough” of the υ vs [dichloroindophenol]plot was transformed into an intermediary plateau. Changes in the temperature of the assay system also led to changes in the kinetic curves. Incubation of the enzyme at 23 °C for 150 min caused disappearance of the intermediary plateau and “trough” regions in the υ vs [NADH]or [dichloroindophenol]plots, respectively. These appeared again after cooling at 0 ° C.The effect of pH, ionic strength, and temperature are discussed within the framework of two main models chosen by statistical analysis: (1) the reaction is catalyzed by a four-site enzyme exhibiting positive and negative cooperativity with respect to the catalytic (kinetic) activity of the sites involved; (2) two enzyme species each possessing two sites are present. These two are either independent or interconnected via a slow isomerization step.     

Extinction and exponential growth in random environments.

The influence of randomly varying environments on unrestricted population growth and extinction is analyzed by means of branching processes with random environments (BPRE). A main theme is the interplay between environmental and sampling (or “demographic”) variability. If the two sources of variationg are of comparable magnitude, the environmental variation will dominate except as regards the event of extinction.A diffusion approximation of BPRE is proposed to study the situation of a large population with small environmental variance and mean offspring size near one.Comments on the ecological literature as well as on the relation of the results to previous work involving stochastic differential equations are also given.     

Testing the Role of Cursorial Specializations as Adaptive Key Innovations in Paleocene-Eocene Ungulates of North America

Episodes of rapid faunal turnover in the fossil record are often used to examine processes driving macroevolutionary changes, such as competitive exclusion. The sudden appearance in the earliest Eocene of North America of artiodactyls and perissodactyls, and subsequent decline of endemic “condylarths” constitutes such an episode. It has been suggested that the specializations for high speed locomotion (cursoriality) that are present in artiodactyls and perissodactyls were key innovations of these orders accounting for their success in the Eocene and onwards. A quantitative geometric morphometric analysis of distal femoral articular morphology was used to examine changes in locomotor specializations in North American ungulates across the Paleocene-Eocene boundary. “Condylarths” were found to have displayed a broad range of locomotor adaptions, including cursoriality. The early Eocene had the broadest disparity in terms of taxonomic and locomotor contributions to morphological diversity. Changes in locomotor variety were associated with the disappearance of arboreal taxa, primarily “condylarths.” The initial impact of artiodactyls and perissodactyls in North America on existing locomotor diversity was limited and does not support a competitive exclusion hypothesis.     

Logical-Semantic Analysis and the Development of L.S. Vygotsky's Ideas About “Units” and “Elements” of Psychological Systems

Based on an analysis of L.S. Vygotsky's concepts of “units” and “elements” of psychological systems, this article highlights five of their attributes. It shows that these attributes are logically symmetrical, since in their wording they can be converted into one another by negation or by replacing some words with their opposites. This suggests that the concepts of the “unit” and “element” of a system are different poles of one theoretical construct of the activity of human psychology. Thus methods for the study of psychological systems by breaking them down into elements or by separating them into units can be seen as complementary. The article describes differences among the concepts of “unit,” “minimal unit,” and “cell” of a psychological system. It reviews several problems that are solvable using the “method of units,” as well as some concepts of the theory of psychological systems that are understood as holistic, conceptual, and active processes and/or results of human interaction with the world. Among the examples of such systems are “systems of psychological functions” (according to Vygotsky), as well as separate activities (according to A.N. Leontiev), human actions and operations (interactions with the world on the level of objects and mental or physical means). The “component” of a psychological system is defined as any “something” that in some sense belongs to or is included in human interaction with the world. A component that belongs to the system is called an “element” of it, but a component that is included in the functioning and development of the system is called a “part” of it. The article presents the mathematical and psychological foundation of these definitions. It identifies and discusses the substantial (independently existing) components of psychological systems and their attributes (properties and conditions). It describes the relationships between them using the bipolar theoretical constructs “part-element” and “substantial-attributive” component of a system.     

Correlations between Planar Random Structures,with an Ecological Application

This paper suggests some characteristics which may help to detect and quantify “correlations” between planar geometrical structures such as point patterns/point processes, fibre systems/fibre processes and region patterns/random sets. Typical examples of “correlations” are mutual attraction or inhibition. Some of the characteristics are related to moment-measures, the definition of the others is similar to that of quantities in the mathematical morphology of MATHERON and SERRA . For all quantities unbiased estimators are given. Also the detection of orientations in random structures is discussed. An ecological example demonstrates the application of the characteristics.     

II. Model Building: An Electrical Theory of Control of Growth and Development in Animals,Prompted by Studies of Exogenous Magnetic Field Effects (Paper I), and Evidence of DNA Current Conduction,In Vitro

A theory of control of cellular proliferation and differentiation in the early development of metazoan systems, postulating a system of electrical controls “parallel” to the processes of molecular biochemistry, is presented. It is argued that the processes of molecular biochemistry alone cannot explain how a developing organism defies a stochastic universe.

The demonstration of current flow (charge transfer) along the long axis of DNA through the base-pairs (the “π-way) in vitro raises the question of whether nature may employ such current flows for biological purposes. Such currents might be too small to be accessible to direct measurement in vivo but conduction has been measured in vitro, and the methods might well be extended to living systems. This has not been done because there is no reasonable model which could stimulate experimentation. We suggest several related, but detachable or independent, models for the biological utility of charge transfer, whose scope admittedly outruns current concepts of thinking about organization, growth, and development in eukaryotic, metazoan systems. The ideas are related to explanations proposed to explain the effects demonstrated on tumors and normal tissues described in Article I (this issue).

Microscopic and mesoscopic potential fields and currents are well known at sub-cellular, cellular, and organ systems levels. Not only are such phenomena associated with internal cellular membranes in bioenergetics and information flow, but remarkable long-range fields over tissue interfaces and organs appear to play a role in embryonic development (Nuccitelli, ). The origin of the fields remains unclear and is the subject of active investigation. We are proposing that similar processes could play a vital role at a “sub-microscopic level,” at the level of the chromosomes themselves, and could play a role in organizing and directing fundamental processes of growth and development, in parallel with the more discernible fields and currents described.     

Hierarchical modelling of acclimatory processes

The term acclimation has been used with several connotations in the field of acclimatory physiology. An attempt has been made, in this paper, to define precisely the term “acclimation” for effective modelling of acclimatory processes. Acclimation is defined with respect to a specific variable, as cumulative experience gained by the organism when subjected to a step change in the environment. Experimental observations on a large number of variables in animals exposed to sustained stress, show that after initial deviation from the basal value (defined as “growth”), the variables tend to return to basal levels (defined as “decay”). This forms the basis for modelling biological responses in terms of their growth and decay.Hierarchical systems theory as presented by Mesarovic, Macko & Takahara (1970) facilitates modelling of complex and partially characterized systems. This theory, in conjunction with “growth-decay” analysis of biological variables, is used to model temperature regulating system in animals exposed to cold. This approach appears to be applicable at all levels of biological organization. Regulation of hormonal activity which forms a part of the temperature regulating system, and the relationship of the latter with the “energy” system of the animal of which it forms a part, are also effectively modelled by this approach. It is believed that this systematic approach would eliminate much of the current circular thinking in the area of acclimatory physiology.     

Complexity and stability revisited

Since Robert May's work on random community matrices it has been known that stability tends to decrease with complexity. Recently, it was shown that this is not necessarily true in competitive ecosystems. We investigated the stability of random ecosystems and found that it can largely be predicted by simple matrix statistics such as the mean and the variance of the interaction coefficients. We use this to explain why stability can increase as well as decrease with complexity in ecological communities. We argue that the variance, and to a lesser extent the mean, of the interaction coefficients go a long way in explaining patterns in the stability of ecosystems.     

Reply to Rosenberg

I respond to two of the main arguments in Rosenberg’s commentary on “Mind, Matter, and Metabolism.” Rosenberg’s claim that metabolic activities are “modularized” in a way that sets them apart from cognitive processes is not true given the broad sense of the “metabolic” employed in my paper, and contemporary neuroscience, including the work on navigation cited by Rosenberg, has begun to yield an understanding of subjectivity and “point of view.”     

Evolution of transfer RNA

Evolution by gene duplication and subsequent divergence is indicated by similarities common to 43 different transfer RNAs. Pairwise comparisons of these tRNAs reveal additional similarity, greatest for certain pairs of tRNAs for the same amino acid in the same organism, and also occurring in certain pairs of tRNAs for different amino acids in the same organism. Although tRNAs functionally interact with several other molecules, there have been surprisingly few restrictions on the divergence of their primary structures. This divergence has proceeded so far that clear phylogenetic separations are absent in most cases: it it impossible to construct a coherent phylogeny for most of the 43. Selection and stochastic processes have both been active in the evolution of tRNA. Selection has favored moderate change more than expected and has reduced radical change below that expected from stochastic processes alone. Two obvious effects of selection are nine invariant loci, another five that are always purines and five others that are always pyrimidines, in the tRNAs involved in protein synthesis. In addition to these constraints in the primary nucleotide sequence, the method of “identical site equivalents”, introduced here, demonstrates that further constraints exist equivalent to about 12 additional invariant loci. These “invisible” restraints reflect disperse chemical forces maintaining the tertiary structure and reducing evolutionary divergence to an extent quantitatively comparable to that of the nine observable invariant loci. The average divergence (49·4%) for pairs of tRNAs for different amino acids involved in protein synthesis represents an equilibrium between natural selection and stochastic processes. These tRNAs have had time to diverge nearly to the 75% maximum expected from stochastic process alone; this is shown by comparing the two glycine tRNAs involved in peptidoglycan synthesis with tRNAs for different amino acids participating in polypeptide synthesis. The rates of nucleotide replacements in genes coding for the tRNAs and the cytochromes c are about the same: 2 × 10 ^?10 replacements per nucleotide site per year.     

Was heißt “lebendig”?

“Life” means “being alive” of special entities, which we call “organisms”. From a physical point of view, living entities are open systems, which exchange matter as well as energy with their surroundings. Against disruptive influences permanently present, they maintain actively and autonomously a steady state far from the thermodynamic equilibrium. This dynamic state of living beings represents a functional order, an internal “organization”. That means that the involved processes one and all must be correlated in such a way that they in sum prevent the breakdown of the living state. Organization implies functionality, which in turn requires structural relationships, and structures require information for their specification. Information in turn presupposes a source, which is constituted in living systems by the nucleic acids. Organisms are unique in having a capacity to use information, which is stored in the nucleic acid and yields the basis for their specific internal organization in its perpetuation: Living beings, and only they, show a self‐maintained organization.     

Free radicals in aging: Causal complexity and its biomedical implications

Superoxide generated adventitiously by the mitochondrial respiratory chain can give rise to much more reactive radicals, resulting in random oxidation of all classes of macromolecules. Harman's 1956 suggestion that this process might drive aging has been a leading strand of biogerontological thinking since the discovery of superoxide dismutase. However, it has become apparent that the many downstream consequences of free radical damage can also be caused by processes not involving oxidation. Moreover, free radicals have been put to use by evolution to such an extent that their wholesale elimination would certainly be fatal. This multiplicity of parallel pathways and side-effects illustrates why attempts to postpone aging by “cleaning up” metabolism will surely fail for the foreseeable future: we simply understand metabolism too poorly. This has led me to pursue the alternative, “repair and maintenance” approach that sidesteps our ignorance of metabolism and may be feasible relatively soon.     

Statistical Description of Food Microstructure. Extraction of Some Correlation Functions From 2D Images

Two algorithms able to compute lineal-path distribution function L(z) and two-point correlation function, S ₂ (r ₁ ,r ₂ ) from 2D digital images of foods were developed and validated. Particularly, “Altamura” and “White” breads as well as “Napoli” sausages were studied. Lineal-path function was modeled by considering the foods as two-phase random systems composed from polydisperse overlapping disks. Instead, two-point correlation function was modeled by using a Debye’s equation. In all cases a good agreement between experimental and theoretical trend was obtained stating the high accuracy of the algorithms in the extraction of correlation functions as well as the possibility to model breads and sausages as two-phase random systems. Also, important morphological properties such as pores distribution, pore size homogeneity and their spatial distribution, were obtained. For instance, although the estimated radius of the pores was lower for “Altamura” bread rather that for “White”, its slowest decay of L(z) highlighted that the pores showed a high dimensional inhomogeneity.     

Univariate Community Assembly Analysis (UniCAA): Combining hierarchical models with null models to test the influence of spatially restricted dispersal,environmental filtering,and stochasticity on community assembly

Identifying the influence of stochastic processes and of deterministic processes, such as dispersal of individuals of different species and trait‐based environmental filtering, has long been a challenge in studies of community assembly. Here, we present the Univariate Community Assembly Analysis (UniCAA) and test its ability to address three hypotheses: species occurrences within communities are (a) limited by spatially restricted dispersal; (b) environmentally filtered; or (c) the outcome of stochasticity—so that as community size decreases—species that are common outside a local community have a disproportionately higher probability of occurrence than rare species. The comparison with a null model allows assessing if the influence of each of the three processes differs from what one would expect under a purely stochastic distribution of species. We tested the framework by simulating “empirical” metacommunities under 15 scenarios that differed with respect to the strengths of spatially restricted dispersal (restricted vs. not restricted); habitat isolation (low, intermediate, and high immigration rates); and environmental filtering (strong, intermediate, and no filtering). Through these tests, we found that UniCAA rarely produced false positives for the influence of the three processes, yielding a type‐I error rate ≤5%. The type‐II error rate, that is, production of false negatives, was also acceptable and within the typical cutoff (20%). We demonstrate that the UniCAA provides a flexible framework for retrieving the processes behind community assembly and propose avenues for future developments of the framework.     

The terminology and classification of steles: Historical perspective and the outlines of a system

Discussion focuses on the history of classification of the steles of vascular plants, from Jeffrey (1898) to the present. Recent stelar classifications are largely based (1) on Jeffrey’s system (protostele vs. siphonostele, the latter with ectophloic vs. amphiphloic types), (2) on the system developed by the early British morphologists, notably Brebner (1902), Gwynne-Vaughan (1901), Tansley (1907-08), and Tansley and Lulham (1902) (protostele vs. medullated protostele vs. solenostele vs. dictyostele vs. eustele, with various subtypes), or (3) on a combination of the preceding two systems, the British usually superimposed upon the Jeffreyian, although generally erratically so. An extended classification of three main stelar types and 25 stelar subtypes derived from Brebner (1902), Gwynne-Vaughan (1901), Jeffrey (1898), Tansley (1907-08), and others is presented in outline form with accompanying etymology, synonymy, examples, and with detailed notes on controversial taxa, classificatory rationale, and historical aspects. Recognized are (1) “protostele” (divided into “ectophloic,” “endophloic,” and “amphiphloic,” with 7, 1, and 1 subtypes respectively), (2) “siphonostele” (divided into “ectophloic” with 2 subtypes and “amphiphloic,” the latter divided into “solenostele” and “dictyostele” with 3 subtypes each, plus “siphonostele incertae sedis”), and (3) “eustele” (divided into “eustele sensu stricto” with 4 subtypes, “pseudosiphonostele,” “reduced eustele,” and “polycyclic eustele”).