Parasitism and symbiosis in an N-person non-constant-sum continuous game

A non-constant-sum continuous game analyzed in a previous paper by one of the authors (A. Rapoport,Bull. Math. Biophysics,18, 15–30, 1956) is extended from two toN players with special emphasis onN=3. It is shown that the concept of stability becomes in this case one of “pairwise stability,” and depends on the so-called distribution matrix of rewards. The distribution matrix and the collusion structure jointly determine the end states of the game. Conditions which lead to the emergence of one, two, or no “parasites” are derived. An apparatus is described which provides a physical analogue of the game, making possible the isolation of behavioral variables under the prescribed conditions of the game.     

The law of payoff consistency: Games with continuous differences on resource values

We analyzed general continuous games in which two players compete for a resource with a different value. There is a pure ESS function in which an increase in the resource value increases the cost which players are willing to pay to obtain the resource. We also found that the expected payoff of the player does not change at all, even if the rules of the game were changed. If the rules of a contest become more costly and more vain, the cost equilibrium of the contest would become lower and perfectly cancel the effect of the change in the rules. The payoff is affected only by the distribution of resource value. This implies that however costly and however vain a contest in nature appears, it would not immediately relate with the individual’s loss and species benefit. We name this phenomenon “the law of payoff consistency”.     

Chaos in abstract kinetics: Two prototypes

“Spiral-type chaos” and “screw-type chaos” constitute two simple types of nonperiodic oscillatory flow in 3-variable continuous systems. The former type is exhibited, for example, by auniversal system in the switching mode, when the regimens of flow on the two stable branches of the slow manifold in state space are made to differ in an appropriate manner. Screw-type chaos occurs in ahysteresis oscillator between two stable limit cycles, if the rotation gain is positive. For either case, an analogous 2-dimensional “branched papersheet flow” exists. Both flows are determined by a single-variable discrete dynamical system of the Lorenz-Li-Yorke type (as a cross-section), as well as by an equivalent new map. Numerical simulations of two abstract reaction systems giving rise to non-idealized (that is, truly 3-dimensional) flows of either type are presented. The corresponding discrete dynamical systems (Poincaré maps) are 2-dimensional now, having the form of a flattened hairpin (“horseshoe”) in the simplest case. Thus, two actual examples for 3-dimensional flows suspended by a horseshoe diffeomorphism seem to have been found. One contains just a single functionally effective nonlinearity. Real systems of either type may be found in physics, chemistry, biochemistry, biophysics and economy.     

A game analysis of reciprocal cooperation: Sequential food-sharing and sex role alternation

The evolution of reciprocal cooperation between non-relatives is studied. Food-sharing of vampire bats studied by Wilkinson (1984) and egg-trading of simultaneous hermaphroditic fish studied by Fisher (1980) are respectively considered as Thompson's reciprocal assistance and the iterated Hero game. Those models predicted the following: (1) Reciprocal assistance between adult bats is evolutionarily stable if they repeatedly interact for a long time. However, (2) the cost for an adult to assist a juvenile is too high to be compensated by the benefit from the juvenile. (3) A particular signal, which determines the sex role of each partner, is always displayed by a “heroic” partner which releases eggs with a larger cost. (4) If alternation of sex roles is established, then the evolutionarily stable frequency of displaying a signal of egg-release increases as the frequency of divorce decreases, and (5) a “cheating” strategy, which releases sperm on every spawning bout, is less advantageous than reciprocal cooperation.     

Biomechanical analysis of movement strategies in human forward trunk bending. I. Modeling

 Two behavioral goals are achieved simultaneously during forward trunk bending in humans: the bending movement per se and equilibrium maintenance. The objective of the present study was to understand how the two goals are achieved by using a biomechanical model of this task. Since keeping the center of pressure inside the support area is a crucial condition for equilibrium maintenance during the movement, we decided to model an extreme case, called “optimal bending”, in which the movement is performed without any center of pressure displacement at all, as if standing on an extremely narrow support. The “optimal bending” is used as a reference in the analysis of experimental data in a companion paper. The study is based on a three-joint (ankle, knee, and hip) model of the human body and is performed in terms of “eigenmovements”, i.e., the movements along eigenvectors of the motion equation. They are termed “ankle”, “hip”, and “knee” eigenmovements according to the dominant joint that provides the largest contribution to the corresponding eigenmovement. The advantage of the eigenmovement approach is the presentation of the coupled system of dynamic equations in the form of three independent motion equations. Each of these equations is equivalent to the motion equation for an inverted pendulum. Optimal bending is constructed as a superposition of two (hip and ankle) eigenmovements. The hip eigenmovement contributes the most to the movement kinematics, whereas the contributions of both eigenmovements into the movement dynamics are comparable. The ankle eigenmovement moves the center of gravity forward and compensates for the backward center of gravity shift that is provoked by trunk bending as a result of dynamic interactions between body segments. An important characteristic of the optimal bending is the timing of the onset of each eigenmovement: the ankle eigenmovement onset precedes that of the hip eigenmovement. Without an earlier onset of the ankle eigenmovement, forward bending on the extremely narrow support results in falling backward. This modeling approach suggests that during trunk bending, two motion units – the hip and ankle eigenmovements – are responsible for the movement and for equilibrium maintenance, respectively. Received: 1 July 1999 / Accepted in revised form: 23 October 2000     

An innovative LC/MS approach applied to the determination of zeralenone in maize: Alternate Isotope-coded Derivatization Assay (AIDA)

Zearalenone is a mycotoxin mainly produced by severalFusarium species, which are known to colonize grains in temperate climates. The purpose of the study is to provide a reliable isotope dilution method for the quantification of this mycotoxin. A derivative of the analyte to be used as standard is obtained by reaction with acetic anhydride, which is available in two pure isotopic forms, a protonated (“light”) and a hexadeuterated (“heavy”). The derivatized standards are added to the matrix split intwo parts. Then, the derivatization procedure is repeated on both matrices derivatizing the part containing the “heavy” labelled standard with the “light” acetic anhydride and the part containing the “light” labelled standard with the “heavy” acetic anhydride. Both extracted mixtures are analyzed by LC/MS, monitoring the “light” and the “heavy” labelled analytes and using the former as standard for the latter in one case and viceversa in the other case. The method allowed to obtain very good results, without the need of IAC purification. Presented at the 27^th Mykotoxin-Workshop, Dortmund, Germany, June 13–15, 2005. Financial support: The Italian Ministry of Health     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.     

Applications of matrix algebra to communication nets

A “generic” problem amenable to matrix algebraic treatment is outlined. Several examples are given and one, a communication system, is studied in some detail. A typical structure matrix is used to describe the channels of communication and a “status” matrix is used to describe the distribution of information in the system at any time. A theorem is proved relating the status matrix at any timet to thetth power of the structure matrix. The elements of the communication system are interpreted as individuals who can send messages to each other. For the individuals attempting to solve a “group problem” certain relations are derived between the structure and status matrices and time of solution. The structure of the communication system is permitted to vary with time. A general theorem is proved relating the status matrix to the matrix product of the series of structure matrices representing the changing structure of the system. Some suggestions are made for further generalizations. In particular, it is suggested that so-called “higher order” information transmission can be similarly treated.     

From sepiamelanin to chemical ecology of opisthobranchs and diatoms: an exciting 40 years game in the field of marine natural products

This paper reports the lecture delivered on the occasion of the “Ceremony of Professional Recognition” to the author at the 6th European Conference on Marine Natural Products (Porto, Portugal, 19–23 July 2009). The most recent results obtained studying opisthobranchs and diatoms at ICB are presented after a synthetic overview of the most significant moves played during this exciting 40 years game in the field of Marine Natural Products. In a symbolic manner this career is compared to a chess game. After the “romantic” opening dedicated to the study of natural black (melanin) and red-brown (phaeomelanins) pigments in the prestigious group of Rodolfo A. Nicolaus, the research on marine organisms started in 1969 and rapidly led to the characterization of an impressive number of new compounds exhibiting very unusual structures. Substantially, the “structure hunt” ended with the publication of a series of alkaloids with absolutely unprecedented structural features: the saraines. Slowly, the scientific interest shifted “from structure to function”. Opisthobranchs were selected as models to investigate. The majority of the protective allomones possessed by opisthobranch molluscs are sequestered through the food chain from algae and invertebrates. However, opisthobranchs were also able to biosynthesize many compounds structurally related to typical molecules possessed by their prey. These aspects “from function to ecology, to biosynthesis, to evolution” are discussed in detail.     

Stability in negotiation games and the emergence of cooperation

Consider a two-player game in which each player contributes a costly resource to the common good of the pair. For such contests, the Nash equilibrium contribution, x*, is one for which neither player can increase its pay-off by unilaterally altering its contribution from x*. We study an elaboration of this game, which allows the players to exchange x-offers back and forth in a negotiation phase until they converge to a final pair of contributions, x1 and x2. A significant feature of such negotiation games, hitherto unrecognized, is the existence of a set of neutrally stable equilibrium points in negotiation phase space. To explore the long-term evolutionary outcome of such games, we simulate populations containing various mixtures of negotiation strategies and, contrary to previous results, we often find convergence to a contribution that is more cooperative than the Nash equilibrium. Mathematical analysis suggests why this might be happening, and provides a novel and robust explanation for cooperation, that negotiation can facilitate the evolution of cooperative behaviour.     

Unreasonable implications of reasonable idiotypic network assumptions

We first analyse a simple symmetric model of the idiotypic network. In the model idiotypic interactions regulate B cell proliferation. Three non-idiotypic processes are incorporated: (1) influx of newborn cells; (2) turnover of cells: (3) antigen. Antigen also regulates proliferation. A model of 2 B cell populations has 3 stable equilibria: one virgin, two immune. The twodimensional system thus remembers antigens, i.e. accounts for immunity. By contrast, if an idiotypic clone proliferates (in response to antigen), its anti-idiotypic partner is unable to control this. Symmetric idiotypic networks thus fail to account for proliferation regulation. In high-D networks we run into two problems. Firstly, if the network accounts for memory, idiotypic activation always propagates very deeply into the network. This is very unrealistic, but is an implication of the “realistic” assumption that it should be easier to activate all cells of a small virgin clone than to maintain the activation of all cells of a large (immune) clone. Secondly, graph theory teaches us that if the (random) network connectance exceeds a threshold level of one interaction per clone, most clones are interconnected. We show that this theory is also applicable to immune networks based on complementary matching idiotypes. The combination of the first “percolation” result with the “interconnectancr” result means that the first stimulation of the network with antigen should eventually affect most of the clones. We think this is unreasonable. Another threshold property of the network connectivity is the existence of a virgin state. A gradual increase in network connectance eliminates the virgin state and thus causes an abrupt change in network behaviour. In contrast to weakly connected systems, highly connected networks display autonomous activity and are unresponsive to external antigens. Similar differences between neonatal and adult networks have been described by experimentalists. The robustness of these results is tested with a network in which idiotypic inactivation of a clone occurs more generally than activation. Such “long-range inhibition” is known to promote pattern formation. However, in our model it fails to reduce the percolation, and additionally, generates semi-chaotic behaviour. In our network, the inhibition of a clone that is inhibiting can alter this clone into a clone that is activating. Hence “long-range inhibition” implies “long-range activation”, and idiotypic activation fails to remain localized. We next complicate this model by incorporating antibody production. Although this “antibody” model statically accounts for the same set of equilibrium points, it dynamically fails to account for state switching (i.e. memory). The switching behaviour is disturbed by the autonomous slow decay of the (long-lived) antibodies. After antigenic triggering the system now performs complex cyclic behaviour. Finally, it is suggested that (idiotypic) formation of antibody complexes can play only a secondary role in the network. In conclusion, our results cast doubt on the functional role of a profound idiotypic network. The network fails to account for proliferation regulation, and if it accounts for memory phenomena, it “explodes” upon the first encounter with antigen due to extensive percolation.     

Evolutionary game dynamics in diploid populations

Selection that influences behaviour can be studied using game theory if individual behavioural success depends on the frequencies of various behavioural types in the population. The evolutionarily stable strategy of J. Maynard Smith and G. R. Price (1973. Nature (London) 246, 15–18) is an equilibrium concept like the solution of a game. The dynamic model of Taylor and Jonker, studied in detail by Zeeman, goes beyond game theory using fitness to cause evolution, perhaps towards an equilibrium. A diploid version of their haploid model is considered and it is found that diploid evolution can be quite different. For example “catastrophic” bifurcations can occur between stable internal polymorphisms when the game matrix entries are changed slowly. A slight drop in food supply may cause extinction. Totally unfit altruistic genotypes can be maintained if they help the rest of the population. The relation of haploid game models to constant selection in diploids is also discussed.     

pH-Induced Molten Globule State of <Emphasis Type="Italic">Rhizopus niveus</Emphasis> Lipase is More Resistant Against Thermal and Chemical Denaturation Than Its Native State

Here, we have characterized four pH-dependent states: alkaline state, “B” (pH 9.0), native state, “N” (pH 7.4), acid-induced state, “A” (pH 2.2) and molten globule state, “MG” (pH 1.8) of Rhizopus niveus lipase (RNL) by CD, tryptophanyl fluorescence, ANS binding, DLS, and enzyme activity assay. This “MG” state lacks catalytic activity and tertiary structure but it has native-like significant secondary structure. The “R _h” of all the four states of RNL obtained from DLS study suggests that the molecular compactness of the protein increases as the pH of solution decreases. Kinetic analysis of RNL shows that it has maximum catalytic efficiency at state “B” which is 15-fold higher than state “N.” The CD and tryptophanyl fluorescence studies of RNL on GuHCl and temperature-induced unfolding reveal that the “MG” state is more stable than the other states. The DSC endotherms of RNL obtained at pH 9.0, 7.4, and 2.2 were with two transitions, while at pH 1.8 it showed only a single transition.     

An ESS for the Height of a Plant Population, or an Optimal Height for an Individual?—Rethinking Game-Theoretic Models for Plant Height

Plants only interact with neighbors over restricted distances, so local conditions are of great significance for plants. It is therefore important to consider spatial structure and neighborhood effects if we are to understand plants' strategies. We constructed a spatially-explicit, game theory model to explore optimal height growth at the individual-level. In the model, there is no ESS for height growth at the population level, because there is an “instantaneous” optimal height growth strategy for the individual plant that changes depending on the local light environment. The optimal strategy is plasticity in response to local conditions. Game-theoretic models for plant phenotypic traits should move from “mean-field approximations” towards explicit modeling of local interactions.     

Reconstruction of Some Hybrid Phylogenetic Networks with Homoplasies from Distances

Suppose G is a phylogenetic network given as a rooted acyclic directed graph. Let X be a subset of the vertex set containing the root, all leaves, and all vertices of outdegree 1. A vertex is “regular” if it has a unique parent, and “hybrid” if it has two parents. Consider the case where each gene is binary. Assume an idealized system of inheritance in which no homoplasies occur at regular vertices, but homoplasies can occur at hybrid vertices. Under our model, the distances between taxa are shown to be described using a system of numbers called “originating weights” and “homoplasy weights.” Assume that the distances are known between all members of X. Sufficient conditions are given such that the graph G and all the originating and homoplasy weights can be reconstructed from the given distances.     

Two definitions of stability of equilibria

In classical physics the stability of an equilibrium requires that any, even infinitesimal, displacement from the configuration of equilibrium results in forces which tend to restore the original equilibrium configuration. In case of several stable equilibrium configurations, the height of the threshold, which must be exceeded by the deviarion from the stable equilibrium in order to bring the configuration into another stable equilibrium is taken as a measure of stability of the first configuration. In quantum mechanics, and in the recent work of I. Baianu, S. Comorosan and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968;31, 59–70, 1969;32, 539–561, 1970) on organismic supercategories, preference is given to take, as ameasure of the degree of stability of a configuration, or of a “state”, the length of time during which the system remains in that configuration. It is shown that under rather general conditions the two criteria are equivalent.     

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.     

The evolution of “egalitarian” and “despotic” social systems among macaques

Recent studies of captive macaques have revealed considerable inter-species differences in dominance styles among females. In “egalitarian” species such as stumptail (Macaca arctoides) or tonkean macaques (M. tonkeana), social interactions are more symmetrical and less kin-biased than in “despotic” species such as Japanese (M. fuscata) or rhesus macaques (M. mulatta). Field observations of moor macaques (M. maurus), close relatives of tonkean macaques, suggest that tolerance during feeding characterizes their egalitarian dominance style in the natural habitat. Although it has been proposed that communal defense against other groups may be the main selective force in the evolution of egalitarian dominance style among females, few field data support this prediction. A game theory analysis showed that both an “egalitarian” strategy and a “despotic” strategy are possible evolutionarily stable strategies (ESS) under certain conditions. The difference in dominance styles might reflect the difference in ESS. This means that an egalitarian dominance style can emerge without strong between-group contest competition. A phylogenetic comparison among macaques suggests that despotic dominance styles very likely evolved from egalitarian dominance styles. In the future, primate socioecological studies should pay more attention to the evolutionary history of each species.     

Circumscription of Malvaceae (Malvales) as determined by a preliminary cladistic analysis of morphological, anatomical, palynological, and chemical characters

We report a phylogenetic analysis of “core” Malvales (Tiliaceae, Sterculiaceae, Bombacaceae, and Malvaceae) based on morphological, anatomical, palynological, and chemical features. The results of the analyses lead to the conclusion that Tiliaceae, Sterculiaceae, and Bombacaceae, as variously delimited, are paraphyletic; only the Malvaceae are likely monophyletic. The genera of “core” Malvales form a well-defined clade. Genera of “Tiliaceae” constitute the basal complex within “core” Malvales. The “Sterculiaceae” (most genera)+ “Bombacaceae” + Malvaceae form a clade on the basis of a monadelphous androecium; “Bombacaceae”+ Malvaceae also form a clade, which is diagnosable on the basis of monoloculate anthers. It is clear that the traditional classification, with its arbitrarily delimited evolutionary grades, is unsatisfactory, especially if one seeks to reflect phylogeny accurately. Thus, Malvaceae is redefined to refer to the most recent common ancestor of plants previously considered to be “Tiliaceae,” “Sterculiaceae,” “Bombacaceae,” and Malvaceae, and all of the descendants of that ancestor. This broadly circumscribed Malvaceae can be diagnosed by several presumed synapomorphies, but we draw special attention to the unusual floral nectaries that are composed of densely packed, multicellular, glandular hairs on the sepals (or less commonly on the petals or androgynophore).     

Conditional confined oscillatory dynamics of <Emphasis Type="Italic">Escherichia coli</Emphasis> strain K12-MG1655 in chemostat systems

A series of continuous- and sequencing-batch reactor experiments were performed to assess the growth dynamics of Escherichia coli strain K12-MG1655 in chemostat systems. Previous mathematical predictions and early experimental results had shown that confined oscillatory dynamics ensue in bioreactor populations, which relates to “group birth and death” events within the population. New results are reported here that generally verify the predictions of the model and show that confined oscillations occur under different initial conditions, but the characteristics of the oscillatory dynamics vary as a function of the hydraulic retention time (HRT). Bioreactors were operated at HRTs ranging from 2.7 to 35 h and, regardless of initial conditions or the imposition of transient operational instabilities, highly patterned oscillations developed when HRT was between ∼3 and 8 h. However, outside of this range, bioreactor populations tended to form biofilms on the reactor walls (although the majority of the cells remained suspended in the bulk solution) and stable oscillations were not seen in the bulk phase. This suggests that alternate operating “states” might exist in chemostat populations with biofilm formation and non-homogenous spatial growth influencing “system” dynamics at very low and high HRTs. Although the model accurately predicts a confined dynamic equilibrium for mid-range HRT operations, experimental data show that model predictions do not extend outside of this range, when an alternate stable-state seems to be attained.