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We show that when selection is extreme-the fittest strategy always reproduces or is imitated-the unequivalence between the possible evolutionary game scenarios in finite and infinite populations resolves, in the sense that the three generic outcomes-dominance, coexistence, and mutual exclusion-emerge in well-mixed populations of any size. We consider the simplest setting of a 2-player-2-strategy symmetric game and the two most common microscopic definitions of strategy spreading-the frequency-dependent Moran process and the imitation process by pairwise comparison-both in the case allowing any intensity of selection. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations-fixation being more or less likely to occur and rapid compared to the neutral game-the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and fast fixation of one strategy; (2) mutual invasion and slow fixation of one strategy; (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection 2 becomes mutual invasion and no fixation, a case not possible for finite intensity of selection that better corresponds to the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes. [ABSTRACT FROM AUTHOR]

Eco-genetic models describe the contemporary evolution of quantitative genetic traits by integrating the key ecological aspects with the modes of inheritance. Because of the flexibility in incorporating physiological, ecological, and genetic detail, eco-genetic models are typically individual-based and stochastic. Here we propose a deterministic eco-genetic population model to study the evolution of maturation schedules in exploited fish stocks. It is the first deterministic model addressing the evolution of the probabilistic maturation reaction norm (PMRN)—the genetic trait recognized to control maturation in fish species. The PMRN is the probability of interseasonal maturation as a function of age and size of juvenile individuals. Being independent of the growth trajectory followed by the individual, it can be considered a non-plastic phenotype, under limited change in temperature and climate. We consider the continuous age and size structures of the fish stock and explicitly represent foraging on a single food resource and mating under 1:1 sex ratio. Inheritance follows a single-locus-two-alleles Mendelian rule to limit the number of genotypes. We adapt the standard escalator boxcar train method for the simulation of physiologically structured population models to the case of two-dimensional age distributions—mature individuals being also distributed according to the age at maturation. The model simulations qualitatively reproduce empirical trends, specifically the genotypic redistribution toward early-maturing types under increasing exploitation and the eventual collapse of the stock. Interestingly, increased effort targeted at large sizes seems to erode the genetic diversity, an effect missed by stochastic models that might serve as warning to fishery managers and policy makers. [ABSTRACT FROM AUTHOR]

We propose a model to investigate the dynamics of fashion traits purely driven by social interactions. We assume that people adapt their style to maximize social success, and we describe the interaction as a repeated group game in which the payoffs reflect the social norms dictated by fashion. On one hand, the tendency to imitate the trendy stereotypes opposed to the tendency to diverge from them to proclaim identity; on the other hand, the exploitation of sex appeal for dating success opposed to the moral principles of the society. These opposing forces promote diversity in fashion traits, as predicted by the modeling framework of adaptive dynamics. Our results link the so-called horizontal dynamics-- the primary driver of fashion evolution, compared with the vertical dynamics accounting for interclass and economic drivers--to style variety. [ABSTRACT FROM AUTHOR]

We investigate the competition between two groups of similar agents in the restricted, but classical context of unstructured populations varying in continuous time in an isolated, homogeneous, and constant abiotic environment. Individual behavioral and phenotypic traits are quantified by one-dimensional strategies and intra- as well as inter-specific interactions are described in the vicinity of a stationary regime. Some known results are revisited: invasion by a new strategy generically implies the substitution of the former resident; and resident–invader coexistence is possible close to singular strategies—the stationary points of the invasion fitness—and is generically protected—each of the two competing groups can invade the other. An (almost known) old conjecture is shown true: competition close to a singular strategy is “essentially Lotka–Volterra”—dominance of one strategy, protected coexistence at an intermediate equilibrium, and mutual exclusion are the generic outcomes. And the unfolding of the competition scenarios is completed with the analysis of three degenerate singular strategies—characterized by vanishing second-order fitness derivatives—near which resident–invader coexistence can be unprotected. Our approach is based on the series expansion of a generic demographic model, w.r.t. the small strategy difference between the two competing groups, and on known results on time-scale separation and bifurcation theories. The analysis is carried out up to third order and is extendable to any order. For each order, explicit genericity conditions under which higher orders can be neglected are derived and, interestingly, they are known prior to invasion. An important result is that degeneracies up to third-order are required to have more than one stable way of coexistence. Such degeneracies can be due to particular symmetries in the model formulation, and breaking the genericity conditions provides a direct way to draw biological interpretations. The developed body of theory is exemplified on a model for the evolution of cannibalism and on Lotka–Volterra competition models. [ABSTRACT FROM AUTHOR]

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits - phenotypes or strategies - of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present - resident - types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits - the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching - the joint effect of resident-mutant coexistence and evolutionary instability - have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime. [ABSTRACT FROM AUTHOR]

Love stories are dynamic processes that begin, develop, and often stay for a relatively long time in a stationary or fluctuating regime, before possibly fading. Although they are, undoubtedly, the most important dynamic process in our life, they have only recently been cast in the formal frame of dynamical systems theory. In particular, why it is so difficult to predict the evolution of sentimental relationships continues to be largely unexplained. A common reason for this is that love stories reflect the turbulence of the surrounding social environment. But we can also imagine that the interplay of the characters involved contributes to make the story unpredictable--that is, chaotic. In other words, we conjecture that sentimental chaos can have a relevant endogenous origin. To support this intriguing conjecture, we mimic a real and well-documented love story with a mathematical model in which the environment is kept constant, and show that the model is chaotic. The case we analyze is the triangle described in Jules et Jim, an autobiographic novel by Henri-Pierre Roché that became famous worldwide after the success of the homonymous film directed by Francois Truffaut. [ABSTRACT FROM AUTHOR]

We show in this paper how simulations of ODEs and continuations of systems of algebraic equations can be combined to study the evolution of biodiversity in multispecies systems where phenotypic traits are genetically transmitted. We follow the adaptive dynamics (AD) approach, which provides a deterministic approximation of the evolutionary dynamics of stationary coexisting populations in terms of an ODE system, the so-called AD canonical equation. AD also provides explicit conditions to test whether a stable evolutionary equilibrium of the canonical equation is a branching point--resident and mutant morphs coexist and further differentiate, thus increasing biodiversity--or not. We analyze a standard parameterized family of prey-predator communities, described by the most standard ecological model, and propose an iterative procedure to obtain the branching portrait, explaining the dependence of branching scenarios on two (demographic, environmental, or control) parameters. Among a number of interesting results, in line with field studies and known ecological principles, we find that prey branching is induced by the predation pressure, and is favored when prey intraspecific competition is highly sensitive to the resident-mutant phenotypic mismatch, while predator branching is not possible when prey and predators are present in an equal number of morphs. This results in alternate prey-predator branching sequences with possible phases of prey unilateral branching. The guidelines for deriving a general method for analyzing the evolution of biodiversity are also discussed. The indications that can be obtained typically have a qualitative nature, but can be of help for the long-term conservation and management of biodiversity. [ABSTRACT FROM AUTHOR]

Abstract: The fixation of cooperation among unrelated individuals is one of the fundamental problems in biology and social sciences. It is investigated by means of public goods games, the generalization of the prisoner''s dilemma to more than two players. In compulsory public goods games, defect is the dominant strategy, while voluntary participation overcomes the social dilemma by allowing a cyclic coexistence of cooperators, defectors, and non-participants. Experimental and theoretical research has shown how the combination of voluntary participation and altruistic punishment—punishing antisocial behaviors at a personal cost—provides a solution to the problem, as long as antisocial punishment—the punishing of cooperators—is not allowed. Altruistic punishment can invade at low participation and pave the way to the fixation of cooperation. Specifically, defectors are overpunished, in the sense that their payoff is reduced by a sanction proportional to the number of punishers in the game. Here we show that qualitatively equivalent results can be achieved with a milder punishing mechanism, where defectors only risk a fixed penalty per round—as in many real situations—and the cost of punishment is shared among the punishers. The payoffs for the four strategies—cooperate, defect, abstain, and cooperate-&-punish—are derived and the corresponding replicator dynamics analyzed in full detail. [Copyright &y& Elsevier]

Abstract: We analyze a codimension-two bifurcation in generic -dimensional Filippov (autonomous, piecewise smooth, discontinuous) systems, when a hyperbolic equilibrium collides with the discontinuity boundary. At generic (codim 1) boundary equilibria (BE), two different scenarios can occur: the ‘persistence’ of the stationary solution, that changes from a standard equilibrium on one side of the discontinuity boundary into a pseudo-equilibrium, and the ‘nonsmooth-fold’ between the standard equilibrium and a coexisting pseudo-equilibrium. Generalized (codim 2) boundary equilibria (GBE) separate persistence and nonsmooth-fold scenarios and are here shown to occur together with a fold bifurcation between pseudo-equilibria (PLP). In a two-parameter plane, the PLP bifurcation curve emanates from the GBE point tangentially to the BE curve. The asymptotic of the PLP curve locally to the GBE point is also derived and tested on a 4-dimensional ecological example. [Copyright &y& Elsevier]

We contribute to the analysis of codimension-two bifurcations in discontinuous systems by studying all equilibrium bifurcations of 2D Filippov systems that involve a sliding limit cycle. There are only two such local bifurcations: (1) a degenerate boundary focus, which we call the homoclinic boundary focus (HBF), and (2) the boundary Hopf (BH).We prove that-besides local bifurcations of equilibria and pseudoequilibria-the universal unfolding of the HBF singularity includes a codimension-one global bifurcation at which a sliding homoclinic orbit to a pseudosaddle exists, while that of the BH singularity has a codimension-one bifurcation curve along which a cycle grazing occurs. We define two canonical forms, one for each singularity, to which a generic 2D Filippov system can be locally reduced by smooth changes of variables and parameters and time reparametrization. Explicit genericity conditions are also provided, as well as the asymptotics of the bifurcation curves in the two-parameter space. We show that both studied codimension-two bifurcations occur in a known 2D Filippov system modeling an ecosystem subject to on-off harvesting control, and we provide two Mathematica scripts that automatize all computations. [ABSTRACT FROM AUTHOR]

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka-Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics. [ABSTRACT FROM AUTHOR]

The analysis of a simple model shows that exploitation of fish stocks can entrain in the long run the substantial decline or even the collapse of the stock, as well as difficulties in stock recovery, loss of fishery resilience, and reduction of the mean fish size. The results are in agreement with numerous observations, even though they are obtained with a simple model in which the harvesting fleet and the fish stock are considered as unstructured predator and prey. The study is carried out for the typical case of fleet dimension not too sensitive to the year-to-year fluctuations of the stock and assuming that the sole cause of evolution is technological innovation. The analysis is performed by means of Adaptive Dynamics, an approach born in theoretical biology which is used here in the context of technological change. Although the results are qualitatively consistent with those obtained long ago through the principles of bioeconomics, it is fair to stress that the underlying assumptions are different. In fact, in the bioeconomic approach fleet technology does not evolve and fishing effort varies to produce economic optimization, while in the Adaptive Dynamics approach technological innovation is the key driver. The paper is purely theoretical and the proposed model can hardly be tuned on any real fishery. No practical guidelines for managers can therefore be drawn, if not the general conclusion that long-term sustainability of exploited fish stocks can only be achieved if strategic parameters influencing technological change are kept under strict control. [Copyright &y& Elsevier]

We analyze three codimension-two bifurcations occurring in nonsmooth systems, when a nonhyperbolic cycle (fold, flip, and Neimark-Sacker cases, in both continuous and discrete time) interacts with one of the discontinuity boundaries characterizing the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proved here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported. [ABSTRACT FROM AUTHOR]

BPCONT, a driver for the software package AUTO for the numerical continuation of simple branch points of algebraic and boundary-value problems, is described in detail. Simple branch points are points in the continuation space where two solution branches intersect transversally. Detection and accurate location of branch points along the continuation of a solution branch and switching to the continuation of the other branch are implemented in AUTO and other software packages, but branch point continuation has not been fully supported. BPCONT fills this gap. It handles both generic problems, where branch point continuation is performed in two extra parameters, and nongeneric cases, where one extra parameter is typically involved due to problem-specific symmetries. This paper presents the extended algebraic and boundary-value problems which define branch point continuation, discusses their initialization at branch point detection and the symmetry-breaking technique used to automatically handle nongeneric cases, and describes the BPCONT implementation. Several examples, including generic and nongeneric algebraic problems, generic and nongeneric periodic boundary-value problems, and a nongeneric nonperiodic boundary-value problem, are also presented. [ABSTRACT FROM AUTHOR]

The problem of synchronization of metacommunities is investigated in this article with reference to a rather general model composed of a chaotic environmental compartment driving a biological compartment. Synchronization in the absence of dispersal (i.e., the so-called Moran effect) is first discussed and shown to occur only when there is no biochaos. In other words, if the biological compartment is reinforcing environmental chaos, dispersal must be strictly above a specified threshold in order to synchronize population dynamics. Moreover, this threshold can be easily determined from the model by computing a special Lyapunov exponent. The application to prey-predator metacommunities points out the influence of frequency and coherence of the environmental noise on synchronization and agrees with all experimental studies performed on the subject. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations. [Copyright &y& Elsevier]

The paper considers the evolution of a particular class of networks of identical chaotic oscillators, namely that of ecological networks. In these networks, nodes represent patches where a certain number of plant and animal populations interact on ecological timescale, arcs represent migration flows due to dispersal, and Darwinian evolution is responsible for variations, on a longer evolutionary timescale, of the demographic parameters characterizing the populations. Up to now, this problem has been mainly studied with reference to single-population patches described by one-dimensional discrete-time models and by considering only the dispersal rates of migrating populations as an evolving trait. Here, we propose a method of investigation which allows to study multipopulation patches described by continuous-time models with evolving traits influencing various demographic parameters (including or not dispersal). The method is casted within the frame of the so-called master stability function approach for the analysis of synchronization of coupled systems, and the results obtained in a first and very simple application support the conjecture that evolution drives ecological networks toward weak forms of synchronization. [ABSTRACT FROM AUTHOR]

Abstract: The behavior of the simplest forest fire model is studied in this paper through bifurcation analysis. The model is a second-order continuous-time impact model where vegetational growth is described as a continuous and slow dynamic process, while fires are modeled as instantaneous and disruptive events. The transition from Mediterranean forests (characterized by wild chaotic fire regimes) to savannas and boreal forests (where fires are almost periodic) is recognized to be a catastrophic transition known as border collision bifurcation in the context of discrete-tine systems. In the present case such a bifurcation can be easily detected numerically and then continued by solving a standard boundary-value problem. The result of the analysis complements previous simulation studies and are consistent with biological intuition. [Copyright &y& Elsevier]

The paper describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation. [ABSTRACT FROM AUTHOR]

SLIDECONT, an Auro97 driver for sliding bifurcation analysis of discontinuous piecewise-smooth autonomous systems, known as Filippov systems, is described in detail. Sliding bifurcations are those in which some sliding on the discontinuity boundary is critically involved. The software allows for detection and continuation of codimension-1 sliding bifurcations as well as detection of some codimension-2 singularities, with special attention to planar systems (n = 2). Some bifurcations are also supported for n-dimensional systems. This article gives a brief introduction to Filippov systems, describes the structure of SLIDECONT and all computations supported by SLIDECONT 2.0. Several examples, which are distributed together with the source code of SLIDECONT, are presented. [ABSTRACT FROM AUTHOR]

The problem of determining the horizon of a terrain viewed from a given point is addressed. A triangulated surface model of the terrain is often used to this aim, because such a model is simple and flexible. In all contributions where the problem is explicitly discussed, it is assumed that the horizon can be determined by computing the upper envelope of a set of segments in the plane. This conjecture is shown here to be wrong. One of the major consequences of this error concerns the determination of the visible portion of a terrain. Indeed, the algorithms available for this purpose process the terrain triangles in distance order from the viewpoint and are based on the updating of the horizon associated with the already visited triangles, that can cast shadow on the not yet visited ones. The consequence of a wrong computation of the horizon is that some invisible portions of the terrain can be classified as visible. Moreover, the extension of the wrongly classified area depends upon the configuration of the terrain and can be arbitrarily large even for an arbitrarily small error in horizon computation. [Copyright &y& Elsevier]

We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model [The American Naturalist, 97 (1963), pp. 209-223] and, following the so-called adaptive dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trait. Then, we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters. Surprisingly, the evolutionary dynamics turn out to be much richer than the population dynamics. Up to three evolutionary attractors can be present, and the bifurcation diagrams contain numerous global bifurcations and codimension-2 bifurcation points. Interesting biological properties can be extracted from these bifurcation diagrams. In particular, one can conclude that evolution of the traits can be cyclic and easily promote prey species diversity. [ABSTRACT FROM AUTHOR]

The evolution of cannibalistic traits in consumer populations is studied in this paper with the approach of adaptive dynamics theory. The model is kept at its minimum complexity by eliminating some environmental characteristics, like heterogeneity and seasonalities, and by hiding the size-structure of the population. Evolutionary dynamics are identified through numerical bifurcation analysis, applied both to the ecological (resident–mutant) model and to the canonical equation of adaptive dynamics. The result is a rich catalog of evolutionary scenarios involving evolutionary stable strategies and branching points both in the monomorphic and dimorphic dynamics. The possibility of evolutionary extinction of highly cannibalistic populations is also ascertained. This allows one to explain why cannibalism can be a transient stage of evolution. [Copyright &y& Elsevier]

Presents a study that demonstrated ecological bistability and evolutionary reversals under asymmetrical competition in a given population. Analysis of a single species whose population dynamics are determined by a set of fixed ecological parameters; Description of the ecological dynamics of the population of a single competition phenotype; Discussion on the application of the mutant-resident model; Ecological parameters of the occurrence of evolutionary reversals.

Summary: The aim of this study was to assess, with numerical simulations, if the complex mechanism of two (or more) interacting spinal/supraspinal structures generating periodic leg movements can be modelled with a single‐generator approach. For this, we have developed the first phenomenological model to generate periodic leg movements in‐silico. We defined the onset of a movement in one leg as the firing of a neuron integrating excitatory and inhibitory inputs from the central nervous system, while the duration of the movement was defined in accordance to statistical evidence. For this study, polysomnographic leg movement data from 32 subjects without periodic leg movements and 65 subjects with periodic leg movements were used. The proportion of single‐leg and double‐leg inputs, as well as their strength and frequency, were calibrated on the without periodic leg movements dataset. For periodic leg movements subjects, we added a periodic excitatory input common to both legs, and the distributions of the generator period and intensity were fitted to their dataset. Besides the many simplifying assumptions – the strongest being the stationarity of the generator processes during sleep – the model‐simulated data did not differ significantly, to a large extent, from the real polysomnographic data. This represents convincing preliminary support for the validity of our single‐generator model for periodic leg movements. Future model extensions will pursue the ambitious project of a supportive diagnostic and therapeutic tool, helping the specialist with realistic forecasting, and with cross‐correlations and clustering with other patient meta‐data. [ABSTRACT FROM AUTHOR]

Understanding cooperative behavior in biological and social systems constitutes a scientific challenge, being the object of intense research over the past decades. Many mechanisms have been proposed to explain the presence and persistence of cooperation in those systems, showing that there is no unique explanation, as different scenarios have different possible driving forces. In this paper, we propose a model to study situations in which voluntary participation involves an access cost to the cooperative interaction, besides the cost associated with cooperation. The proposed costly-access prisoner’s dilemma (PD), a symmetric donation game with voluntary and costly participation, breaks the symmetry between abstainers and participants of the voluntary PD. A mean-field approach shows that, in well-mixed populations, the dynamic always leads the system to abstention. However, depending on the return parameter, numerical simulations in structured populations display an alternating behavior between mono-strategic, multi-stable, and coexistence phases. This behavior is fully explained through a theoretical analysis of the strategic motifs, the transitions being determined by the change in stability of those motifs. [ABSTRACT FROM AUTHOR]

Network reciprocity has been successfully put forward (since M. A. Nowak and R. May's, 1992, influential paper) as the simplest mechanism—requiring no strategical complexity—supporting the evolution of cooperation in biological and socioeconomic systems. The mechanism is actually the network, which makes agents' interactions localized, while network reciprocity is the property of the underlying evolutionary process to favor cooperation in sparse rather than dense networks. In theoretical models, the property holds under imitative evolutionary processes, whereas cooperation disappears in any network if imitation is replaced by the more rational best-response rule of strategy update. In social experiments, network reciprocity has been observed, although the imitative behavior did not emerge. What did emerge is a form of conditional cooperation based on direct reciprocity—the propensity to cooperate with neighbors who previously cooperated. To resolve this inconsistency, network reciprocity has been recently shown in a model that rationally confronts the two main behaviors emerging in experiments—reciprocal cooperation and unconditional defection—with rationality introduced by extending the best-response rule to a multi-step predictive horizon. However, direct reciprocity was implemented in a non-standard way, by allowing cooperative agents to temporarily cut the interaction with defecting neighbors. Here, we make this result robust to the way cooperators reciprocate, by implementing direct reciprocity with the standard tit-for-tat strategy and deriving similar results. [ABSTRACT FROM AUTHOR]

Since M. A. Nowak & R. May's (1992) influential paper, limiting each agent's interactions to a few neighbors in a network of contacts has been proposed as the simplest mechanism to support the evolution of cooperation in biological and socio-economic systems. The network allows cooperative agents to self-assort into clusters, within which they reciprocate cooperation. This (induced) network reciprocity has been observed in several theoreticalmodels and shown to predict the fixation of cooperation under a simple rule: the benefit produced by an act of cooperation must outweigh the cost of cooperating with all neighbors. However, the experimental evidence among humans is controversial: though the rule seems to be confirmed, the underlying modeling assumptions are not. Specifically, models assume that agents update their strategies by imitating better performing neighbors, even though imitation lacks rationality when interactions are far from all-to-all. Indeed, imitation did not emerge in experiments. What did emerge is that humans are conditioned by their own mood and that, when in a cooperative mood, they reciprocate cooperation. To help resolve the controversy, we design a model in which we rationally confront the two main behaviors emerging from experiments—reciprocal cooperation and unconditional defection—in a networked prisoner's dilemma. Rationality is introduced by means of a predictive rule for strategy update and is bounded by the assumed model society. We show that both reciprocity and a multi-step predictive horizon are necessary to stabilize cooperation, and sufficient for its fixation, provided the game benefit-to-cost ratio is larger than a measure of network connectivity. We hence rediscover the rule of network reciprocity, underpinned however by a different evolutionary mechanism. [ABSTRACT FROM AUTHOR]

We study a mathematical model based on ordinary differential equations to describe the dynamic interaction in the market of two types of energy called standard and innovative. The model consists of an adaptation of the generalized Lotka-Volterra system in which the parameters are assumed to depend on a quantitative and continuous attribute characteristic of energy generation. Using the analysis of the model the fitness function for the innovative energy is determined, from which conditions of invasion can be established in a market dominated by the conventional power. The canonical equation of the adaptive dynamics is studied to know the long-term behavior of the characteristic attribute and its impact on the market. Then we establish conditions under which evolutionary ramifications occur, that is to say, the requirements of coexistence and divergence of the characteristic attributes, whose occurrence leads to the origin of diversity in the energy market. [ABSTRACT FROM AUTHOR]

Simple Summary: Neuroblastoma is a rare type of cancer that usually affects children. The high-risk patients' expected survival rate is less than 50%. One reason is the lack of precision in the standard treatment protocol: a one-size-fits-all multi-modal therapy. The study presented in this paper was designed to address this deficit by optimising the use of two chemotherapeutic agents—vincristine and cyclophosphamide—during induction chemotherapy—the part of the protocol that shrinks the primary tumour before surgical removal. We combined a mathematical model and an optimisation algorithm to identify the best chemotherapy schedules for a cohort of virtual patients with different initial tumour compositions. Our results reveal novel strategies to exploit a pair of drugs with different levels of efficacy, provide a platform on which to individualise induction chemotherapy, and lay the foundation for a personalised therapy that leverages targeted therapies, multi-region sequencing, liquid biopsies, and modern computational methods to improve today's multi-modal therapy. Neuroblastoma is the most common extra-cranial solid tumour in children. Despite multi-modal therapy, over half of the high-risk patients will succumb. One contributing factor is the one-size-fits-all nature of multi-modal therapy. For example, during the first step (induction chemotherapy), the standard regimen (rapid COJEC) administers fixed doses of chemotherapeutic agents in eight two-week cycles. Perhaps because of differences in resistance, this standard regimen results in highly heterogeneous outcomes in different tumours. In this study, we formulated a mathematical model comprising ordinary differential equations. The equations describe the clonal evolution within a neuroblastoma tumour being treated with vincristine and cyclophosphamide, which are used in the rapid COJEC regimen, including genetically conferred and phenotypic drug resistance. The equations also describe the agents' pharmacokinetics. We devised an optimisation algorithm to find the best chemotherapy schedules for tumours with different pre-treatment clonal compositions. The optimised chemotherapy schedules exploit the cytotoxic difference between the two drugs and intra-tumoural clonal competition to shrink the tumours as much as possible during induction chemotherapy and before surgical removal. They indicate that induction chemotherapy can be improved by finding and using personalised schedules. More broadly, we propose that the overall multi-modal therapy can be enhanced by employing targeted therapies against the mutations and oncogenic pathways enriched and activated by the chemotherapeutic agents. To translate the proposed personalised multi-modal therapy into clinical use, patient-specific model calibration and treatment optimisation are necessary. This entails a decision support system informed by emerging medical technologies such as multi-region sequencing and liquid biopsies. The results and tools presented in this paper could be the foundation of this decision support system. [ABSTRACT FROM AUTHOR]

In discontinuous systems, the fundamental solution matrix of the linearized dynamics about a reference trajectory can be noninvertible. This feature can be exploited, for instance, to design robust control algorithms, to synchronize a network, or to stabilize otherwise unstable or chaotic dynamics. In this paper we classify all the phenomena that cause rank defect in the fundamental solution matrix of a generic discontinuous system. We relate these phenomena to simple geometric conditions at a point of vector field switching, sliding, or impact, and we derive necessary and sufficient conditions for the rank defect. This constitutes a valuable tool to detect flow noninvertibility or to purposefully include it in the design of a system. In terms of Lyapunov exponents, the singularity of the fundamental solution matrix means that an infinitesimal sphere of initial perturbations is mapped onto a lower-dimensional ellipsoid. This consequently reduces the number of finite exponents and, most remarkably, makes them depend on the full history of the reference trajectory used for the computation. We introduce a numerical procedure which allows the computation of the Lyapunov exponents by resorting to a properly reduced variational system. The validity of the approach is verified in two examples. [ABSTRACT FROM AUTHOR]

Through the analysis of a mathematical model we construct in this paper the full catalog of possible love stories among standard (i.e. secure and non-synergic) individuals. For this, we first distinguish between robust and fragile couples, i.e. couples which have one or two stable equilibria. We also distinguish between high and low quality relationships by looking at the sign of the feelings among the partners. Finally, we split fragile couples into two distinct classes, namely those that start from the state of indifference to evolve toward their most desirable equilibrium, as opposed to those that have an unfavorable evolution. The boundaries separating the various classes in any two-dimensional parameter space are either bifurcation curves or special curves characterized by the annihilation of a suitably defined indicator associated to the equilibria or to their basins of attraction. The analysis is performed in part analytically and in part numerically (through continuation methods), and is mainly focused on the role of the appeals of the individuals. The results are in agreement with common wisdom and observations and show, in particular, that love stories can evolve positively even if one of the two individuals has negative feelings in the initial phase of the relationship. [ABSTRACT FROM AUTHOR]