Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

Asymptotic regime in N random interacting species

The asymptotic regime of a complex ecosystem with N random interacting species and in the presence of an external multiplicative noise is analyzed. We find the role of the external noise on the long time probability distribution of the ith density species, the extinction of species and the local field acting on the ith population. We analyze in detail the transient dynamics of this field and the cavity field, which is the field acting on the ith species when this is absent. We find that the presence or the absence of some population give different asymptotic distributions of these fields.     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Instabilities in population dynamics

Biologists have long known that the smaller the population, the more susceptible it is to extinction from various causes. Biologists define minimum viable population size (MVP), which is the critical population size, below which the population has a very small chance to survive. There are several theoretical models for predicting the probability that a small population will become extinct. But these models either embody unrealistic assumptions or lead to currently unresolved mathematical problems. In other popular models of population dynamics, like the logistic model, MVP does not exist. In this paper we find the existence of such a critical concentration in a simple model of evolution. We solve this model by a mean field theory and show, in one and two dimensions, the existence of the critical adaptation and concentration below which a population dies out. We also show that, like in the logistic model, above the critical value a population reaches its carrying capacity. Moreover, in the two-dimensional case we find - the so common in biological models - periodic solutions and their biffurcations. Received 15 February 2000     

Susceptible-infected-recovered epidemics in populations with heterogeneous contact rates

Heterogeneity of contact patterns is recognized as an important feature for realistic modeling of many epidemics. During an outbreak, the frequency of contacts can vary a great deal from person to person and period to period. Contact heterogeneity has been shown to have a large impact on epidemic thresholds and the final size of epidemics. We develop and apply a model which incorporates an arbitrary distribution of contact rates. The model consists of a low-dimensional system of ordinary differential equations which incorporates arbitrary heterogeneity by making use of generating functions of the contact rate distribution. We show further how this model can be applied to the study of simple intervention strategies, such as quarantine of public venues with probability proportional to size. The dynamic model allows us to investigate the effects of gradually implementing such strategies in response to an ongoing epidemic, and we investigate these strategies using data on the contact patterns within a large US city.     

Coexistence and Extinction Pattern of Asymmetric Cyclic Game Species in a Square Lattice

The co-evolutionary dynamics of a cyclic game system is investigated in a two-dimensional square lattice with the asymmetrical rates for three species. Different with the well-mixed system, coexistence and extinction emerge alternately in the system, where a ＂zero-one＂ behavior is robust for a small population size, whereas, the system is predominated by coexistence for a big population one. We study in detail the influence about the fluctuation to the change of the state, and find that the difference between the maximal amplitude about the fluctuation and the average intensity determines which state the system is ultimately. In addition, we introduce Ports energy to explain the reason of the ＂zero-one＂ behavior. It is shown that the average Ports energy per site is the distance to the ＂zero-one＂ behavior in the model.     

Transient selection in multicellular immune networks

We analyze the dynamics of a multi-clonotype naive T-cell population competing for survival signals from antigen-presenting cells. We find that this competition provides with an efficacious selection of clonotypes, making the less able and more repetitive get extinct. We uncover the scaling principles for large systems the extinction rate obeys and calibrate the model parameters to their experimental counterparts. For the first time, we estimate the physiological values of the T-cell receptor-antigen presentation profile recognition probability and T-cell clonotypes niche overlap. We demonstrate that, while the ultimate state is a stable fixed point, sequential transients dominate the dynamics over large timescales that may span over years, if not decades, in real time. We argue that what is currently viewed as “homeostasis” is a complex sequential transient process, while being quasi-stationary in the total number of T-cells only. The discovered type of sequential transient dynamics in large random networks is a novel alternative to the stable heteroclinic channel mechanism.     

Stochastic fluctuation induced the competition between extinction and recurrence in a model of tumor growth

We investigate the phenomenon that stochastic fluctuation induced the competition between tumor extinction and recurrence in the model of tumor growth derived from the catalytic Michaelis–Menten reaction. We analyze the probability transitions between the extinction state and the state of the stable tumor by the Mean First Extinction Time (MFET) and Mean First Return Time (MFRT). It is found that the positional fluctuations hinder the transition, but the environmental fluctuations, to a certain level, facilitate the tumor extinction. The observed behavior could be used as prior information for the treatment of cancer.     

Coexistence in the two-dimensional May-Leonard model with random rates

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-) steady state in two-dimensional stochastic May-Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May-Leonard system (for small system sizes): (1) as the mobility rate exceeds a threshold that separates a species coexistence (quasi-) steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ∼ e ^cN /N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.     

On the structure of competitive societies

We model the dynamics of social structure by a simple interacting particle system. The social standing of an individual agent is represented by an integer-valued fitness that changes via two offsetting processes. When two agents interact one advances: the fitter with probability p and the less fit with probability 1-p. The fitness of an agent may also decline with rate r. From a scaling analysis of the underlying master equations for the fitness distribution of the population, we find four distinct social structures as a function of the governing parameters p and r. These include: (i) a static lower-class society where all agents have finite fitness; (ii) an upwardly-mobile middle-class society; (iii) a hierarchical society where a finite fraction of the population belongs to a middle class and a complementary fraction to the lower class; (iv) an egalitarian society where all agents are upwardly mobile and have nearly the same fitness. We determine the basic features of the fitness distributions in these four phases.     

Minimal agent based model for financial markets I

We introduce a minimal agent based model for financial markets to understand the nature and self-organization of the stylized facts. The model is minimal in the sense that we try to identify the essential ingredients to reproduce the most important deviations of price time series from a random walk behavior. We focus on four essential ingredients: fundamentalist agents which tend to stabilize the market; chartist agents which induce destabilization; analysis of price behavior for the two strategies; herding behavior which governs the possibility of changing strategy. Bubbles and crashes correspond to situations dominated by chartists, while fundamentalists provide a long time stability (on average). The stylized facts are shown to correspond to an intermittent behavior which occurs only for a finite value of the number of agents N. Therefore they correspond to finite size effects which, however, can occur at different time scales. We propose a new mechanism for the self-organization of this state which is linked to the existence of a threshold for the agents to be active or not active. The feedback between price fluctuations and number of active agents represents a crucial element for this state of self-organized intermittency. The model can be easily generalized to consider more realistic variants.     

Type-II intermittency characteristics in the presence of noise

We consider a type of intermittent behavior that occurs as the result of the interplay between dynamical mechanisms giving rise to type-II intermittency and random dynamics. We analytically deduce the law for the distribution of the laminar phases, which has never been obtained hitherto. The already known dependence of the mean length of the laminar phases on the criticality parameter [Phys. Rev. E 68, 036203 (2003)] follows as a corollary of the carried out research. We also prove that this dependence obtained earlier under the assumption of the fixed form of the reinjection probability does not depend on the relaminarization properties, and, correspondingly, the obtained expression of the mean length of the laminar phases on the criticality parameter remains correct for different types of the reinjection probability.     

Loops in one-dimensional random walks

Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate numerically and analytically loops in several types of RWs, including RWs with continuous step-length distribution. We show that for long walks the probability density of the longest loop becomes independent of the details of the walks and definition of the loops. We investigate crossovers and convergence of probability densities to the limiting behavior, and obtain some of the analytical properties of the universal probability density. Received 8 January 1999     

The power of choice in growing trees

The “power of choice” has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of random tree growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting tree can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the tree with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k ≫ 1 to see a power law over a wide range of degrees.     

Effects of migration on the evolutionary game dynamics in finite populations with community structures

We investigate the impacts of migration on the evolutionary game dynamics in finite populations with community structures in the framework of evolutionary game theory. In contrast to deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Based on the analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that it is most likely to find the population in the community where all individuals have the lower migration rate. Furthermore, we show that reducing the difference between the migration rates of distinct communities can increase the first hitting time to the homogeneous absorbing state and can prolong the coexistence time of different species, promoting the conservation of biodiversity.     

Electron-phonon relaxation and excited electron distribution in zinc oxide and anatase

We propose a first-principles method for evaluations of the time-dependent electron distribution function of excited electrons in the conduction band of semiconductors. The method takes into account the excitations of electrons by an external source and the relaxation to the bottom of the conduction band via electron-phonon coupling. The methods permit calculations of the non-equilibrium electron distribution function, the quasi-stationary distribution function with a steady-in-time source of light, the time of setting of the quasi-stationary distribution and the time of energy loss via relaxation to the bottom of the conduction band. The actual calculations have been performed for titanium dioxide in the anatase structure and zinc oxide in the wurtzite structure. We find that the quasi-stationary electron distribution function has a peak near the bottom of the conduction band and a tail whose maximum energy rises linearly with increasing energy of excitation. The calculations demonstrate that the relaxation of excited electrons and the setting of the quasi-stationary distribution occur within a time of no more than 500?fs for ZnO and 100?fs for anatase. We also discuss the applicability of the effective phonon model to energy-independent electron-phonon transition probability. We find that the model only reproduces the trends in the change of the characteristic times whereas the precision of such calculations is not high. The rate of energy transfer to phonons at the quasi-stationary electron distribution also have been evaluated and the effect of this transfer on the photocatalysis has been discussed. We found that for ZnO this rate is about five times less than in anatase.     

Self-Similarity and Power-Like Tails in Nonconservative Kinetic Models

In this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann–Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time-monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.     

Optical extinction spectroscopy of single silver nanoparticles

The extinction spectrum of single silver nanoparticles with size ranging from 20 to 80 nm is investigated with the spatial modulation spectroscopy technique using either a tunable laser or a white lamp as the broadband source. Results are in good agreement with the prediction of the Mie theory, permitting to extract the nanoparticle size from the measured absolute value of the optical extinction cross-section. In contrast, the deduced refractive index of the nanoparticle environment and the reduction of the electron mean free path show a large dependence on the precise value of the bulk silver dielectric function.     

Survival and extinction in cyclic and neutral three-species systems

We study the ABC model ( A + B↦2B, B + C↦2C, C + A↦2A), and its counterpart: the three-component neutral drift model ( A + B↦2A or 2B, B + C↦2B or 2C, C + A↦2C or 2A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates. Received 14 August 2002 and Received in final form 14 February 2003 / Published online: 1 April 2003 RID="a" ID="a"e-mail: ita@physics.ubc.ca     

Correlated noises in an absorptive optical bistable model

We study the steady state properties of an absorptive optical bistable model in the presence of correlated noises. Based on the corresponding Fokker-Planck equation the steady state solution of the probability distribution and the average value of the transmitted light have been investigated. We have found that fluctuations of the input light amplitude improve the transmitted light and an optimized value exists for the fluctuations of the population difference at which the transmitted light takes its maximum value. The correlation between the two noises reduce the transmitted light and the noises in the model can induce a phase transition.