Stability in a system subject to noise with regulated periodicity

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

Integrated random pulse process with positive and negative periodicity

A study of nonstationary processes that are integrals of stationary random sequences of delta pulses is presented. An integrated renewal process can be represented as the sum of a deterministic linear function of time and a Wiener process of the corresponding intensity. This intensity is determined by the mean value and variance of the waiting times of the pulse process and is greater for super-Poisson processes than for sub-Poisson ones. Linear growth over time of all cumulants is proved. An integrated random process with fixed time intervals can be replaced by the sum of a deterministic linear function and a random process with bounded variance. The analytical results are in good agreement with the numerical ones.

Viscoelasticity and Noise Properties Reveal the Formation of Biomemory in Cells

Living cells are neither perfectly elastic nor liquid and return a viscoelastic response to external stimuli. Nanoindentation provides force–distance curves, allowing the investigation of cell mechanical properties, and yet, these curves can differ from point to point on the cell surface, revealing its inhomogeneous character. In the present work, we propose a mathematical method to estimate both viscoelastic and noise properties of cells as these are depicted on the values of the scaling exponents of relaxation function and power spectral density, respectively. The method uses as input the time derivative of the response force in a nanoindentation experiment. Generalized moments method and/or rescaled range analysis is used to study the resulting time series depending on their nonstationary or stationary nature. We conducted experiments in living Ulocladium chartarum spores. We found that spores in the approaching phase present a viscoelastic behavior with the corresponding scaling exponent in the range 0.25–0.52 and in the retracting phase present a liquid-like behavior with exponents in the range 0.67–0.85. This substantial difference of the scaling exponents in the two phases suggests the formation of biomemory as a response of the spores to the indenting AFM mechanical stimulus. The retracting phase may be described as a process driven by bluish noises, while the approaching one is driven by persistent noise.

Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton

In recent years, some researchers paid their attention to the interaction between toxic phytoplankton and zooplankton. Their studies showed that the mechanism of food selection in zooplankton is still immature and when different algae of the same species (toxic and nontoxic) coexist, some zooplankton may not be able to distinguish between toxic and nontoxic algae, and even show a slight preference for toxic strains. Thus, in this article, a fractional order mathematical model with time delay is constructed to describe the interaction of nutrient-phytoplankton-toxic phytoplankton-zooplankton. The main purpose of this paper is to study the influence of fractional order and time delay on the ecosystem. The sufficient conditions for the existence and local stability of each equilibrium are obtained by using fractional order stability theory. By choosing time delay as the bifurcation parameter, we find that Hopf bifurcation occurs when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to support the analytic results. At last we make some conclusion and point out some possible future work.

Neurohybrid Memristive CMOS-Integrated Systems for Biosensors and Neuroprosthetics

Here we provide a perspective concept of neurohybrid memristive chip based on the combination of living neural networks cultivated in microfluidic/microelectrode system, metal-oxide memristive devices or arrays integrated with mixed-signal CMOS layer to control the analog memristive circuits, process the decoded information, and arrange a feedback stimulation of biological culture as parts of a bidirectional neurointerface. Our main focus is on the state-of-the-art approaches for cultivation and spatial ordering of the network of dissociated hippocampal neuron cells, fabrication of a large-scale cross-bar array of memristive devices tailored using device engineering, resistive state programming, or non-linear dynamics, as well as hardware implementation of spiking neural networks (SNNs) based on the arrays of memristive devices and integrated CMOS electronics. The concept represents an example of a brain-on-chip system belonging to a more general class of memristive neurohybrid systems for a new-generation robotics, artificial intelligence, and personalized medicine, discussed in the framework of the proposed roadmap for the next decade period.

Transitions in the cell-fate induction induced by colored noise associated with the inductive stimulus

The cell-fate induction based on the saddle-node bifurcation is undoubtedly a very important concept in developmental biology, which provides a possible mechanism to explain the intrinsic irreversibility in the developmental process. In this paper, the effect of a colored noise, which is associated with the inductive stimulus, on the saddle-node landscape of cell-fate induction is investigated, especially, the effect of the change of correlation time of colored noise on cell-fate induction. The main results show clearly that the change of correlation time of colored noise could induce the transitions of the system. This implies that the colored noise associated with inductive stimulus may have a profound effect on the saddle-node bifurcation landscape of cell-fate induction. This will also help us to understand more deeply the role of cell-fate induction in developmental biology.

Effect of multiplicative noise on stationary stochastic process

An open system that can be analyzed using the Langevin equation with multiplicative noise is considered. The stationary state of the system results from a balance of deterministic damping and random pumping simulated as noise with controlled periodicity. The dependence of statistical moments of the variable that characterizes the system on parameters of the problem is studied. A nontrivial decrease in the mean value of the main variable with an increase in noise stochasticity is revealed. Applications of the results in several physical, chemical, biological, and technical problems of natural and humanitarian sciences are discussed.

Evolutionary stability concepts in a stochastic environment

Over the past 30 years, evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behaviors, but also widely used in economics and social sciences. Nonetheless, the stochastic dynamical properties of evolutionary games in randomly fluctuating environments are still unclear. In this study, we investigate conditions for stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model with random payoffs following pairwise interactions. Based on this model, we develop the concepts of stochastic evolutionary stability (SES) and stochastic convergence stability (SCS). We show that the condition for a pure strategy to be SES and SCS is more stringent than in a constant environment, while the condition for a constant mixed strategy to be SES is less stringent than the condition to be SCS, which is less stringent than the condition in a constant environment.

Delayed bet-hedging resilience strategies under environmental fluctuations

Many biological populations, such as bacterial colonies, have developed through evolution a protection mechanism, called bet hedging, to increase their probability of survival under stressful environmental fluctuation. In this context, the concept of preadaptation refers to a common type of bet-hedging protection strategy in which a relatively small number of individuals in a population stochastically switch their phenotypes to a dormant metabolic state in which they increase their probability of survival against potential environmental shocks. Hence, if an environmental shock took place at some point in time, preadapted organisms would be better adapted to survive and proliferate once the shock is over. In many biological populations, the mechanisms of preadaptation and proliferation present delays whose influence in the fitness of the population are not well understood. In this paper, we propose a rigorous mathematical framework to analyze the role of delays in both preadaptation and proliferation mechanisms in the survival of biological populations, with an emphasis on bacterial colonies. Our theoretical framework allows us to analytically quantify the average growth rate of a bet-hedging bacterial colony with stochastically delayed reactions with arbitrary precision. We verify the accuracy of the proposed method by numerical simulations and conclude that the growth rate of a bet-hedging population shows a nontrivial dependency on their preadaptation and proliferation delays. Contrary to the current belief, our results show that faster reactions do not, in general, increase the overall fitness of a biological population.

Probability characteristics of nonlinear dynamical systems driven by δ-pulse noise

For a nonlinear dynamical system described by the first-order differential equation with Poisson white noise having exponentially distributed amplitudes of δ pulses, some exact results for the stationary probability density function are derived from the Kolmogorov-Feller equation using the inverse differential operator. Specifically, we examine the "effect of normalization" of non-Gaussian noise by a linear system and the steady-state probability density function of particle velocity in the medium with Coulomb friction. Next, the general formulas for the probability distribution of the system perturbed by a non-Poisson δ-pulse train are derived using an analysis of system trajectories between stimuli. As an example, overdamped particle motion in the bistable quadratic-cubic potential under the action of the periodic δ-pulse train is analyzed in detail. The probability density function and the mean value of the particle position together with average characteristics of the first switching time from one stable state to another are found in the framework of the fast relaxation approximation.

Relaxation dynamics in the presence of pulse multiplicative noise sources with different correlation properties

The relaxation dynamics of a system described by a Langevin equation with pulse multiplicative noise sources with different correlation properties is considered. The solution of the corresponding Fokker-Planck equation is derived for Gaussian white noise. Moreover, two pulse processes with regulated periodicity are considered as a noise source: the dead-time-distorted Poisson process and the process with fixed time intervals, which is characterized by an infinite correlation time. We find that the steady state of the system is dependent on the correlation properties of the pulse noise. An increase of the noise correlation causes the decrease of the mean value of the solution at the steady state. The analytical results are in good agreement with the numerical ones.

How diffusivity, thermocline and incident light intensity modulate the dynamics of deep chlorophyll maximum in Tyrrhenian Sea

During the last few years theoretical works have shed new light and proposed new hypotheses on the mechanisms which regulate the spatio-temporal behaviour of phytoplankton communities in marine pelagic ecosystems. Despite this, relevant physical and biological issues, such as effects of the time-dependent mixing in the upper layer, competition between groups, and dynamics of non-stationary deep chlorophyll maxima, are still open questions. In this work, we analyze the spatio-temporal behaviour of five phytoplankton populations in a real marine ecosystem by using a one-dimensional reaction-diffusion-taxis model. The study is performed, taking into account the seasonal variations of environmental variables, such as light intensity, thickness of upper mixed layer and profiles of vertical turbulent diffusivity, obtained starting from experimental findings. Theoretical distributions of phytoplankton cell concentration was converted in chlorophyll concentration, and compared with the experimental profiles measured in a site of the Tyrrhenian Sea at four different times (seasons) of the year, during four different oceanographic cruises. As a result we find a good agreement between theoretical and experimental distributions of chlorophyll concentration. In particular, theoretical results reveal that the seasonal changes of environmental variables play a key role in the phytoplankton distribution and determine the properties of the deep chlorophyll maximum. This study could be extended to other marine ecosystems to predict future changes in the phytoplankton biomass due to global warming, in view of devising strategies to prevent the decline of the primary production and the consequent decrease of fish species.

Effect of time delay on pattern dynamics in a spatial epidemic model

Time delay, accounting for constant incubation period or sojourn times in an infective state, widely exists in most biological systems like epidemiological models. However, the effect of time delay on spatial epidemic models is not well understood. In this paper, spatial pattern of an epidemic model with both nonlinear incidence rate and time delay is investigated. In particular, we mainly focus on the effect of time delay on the formation of spatial pattern. Through mathematical analysis, we gain the conditions for Hopf bifurcation and Turing bifurcation, and find exact Turing space in parameter space. Furthermore, numerical results show that time delay has a significant effect on pattern formation. The simulation results may enrich the finding of patterns and may well capture some key features in the epidemic models.

Dynamics of two picophytoplankton groups in mediterranean sea: analysis of the deep chlorophyll maximum by a stochastic advection-reaction-diffusion model

A stochastic advection-reaction-diffusion model with terms of multiplicative white Gaussian noise, valid for weakly mixed waters, is studied to obtain the vertical stationary spatial distributions of two groups of picophytoplankton, i.e., picoeukaryotes and Prochlorococcus, which account about for 60% of total chlorophyll on average in Mediterranean Sea. By numerically solving the equations of the model, we analyze the one-dimensional spatio-temporal dynamics of the total picophytoplankton biomass and nutrient concentration along the water column at different depths. In particular, we integrate the equations over a time interval long enough, obtaining the steady spatial distributions for the cell concentrations of the two picophytoplankton groups. The results are converted into chlorophyll a and divinil chlorophyll a concentrations and compared with experimental data collected in two different sites of the Sicily Channel (southern Mediterranean Sea). The comparison shows that real distributions are well reproduced by theoretical profiles. Specifically, position, shape and magnitude of the theoretical deep chlorophyll maximum exhibit a good agreement with the experimental values.

Velocity distribution for quasistable acceleration in the presence of multiplicative noise

Processes that are far both from equilibrium and from phase transition are studied. It is shown that a process with mean velocity that exhibits power-law growth in time can be analyzed using the Langevin equation with multiplicative noise. The solution to the corresponding Fokker-Planck equation is derived. Results of the numerical solution of the Langevin equation and simulation of the motion of particles in a billiard system with a time-dependent boundary are presented.