Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II

This is a continuation of our paper [Liu, M., Wang, K., 2010. Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol. 264, 934-944]. Taking both white noise and colored noise into account, a stochastic single-species model under regime switching in a polluted environment is studied. Sufficient conditions for extinction, stochastic nonpersistence in the mean, stochastic weak persistence and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is obtained. The results show that a different type of noise has a different effect on the survival results.     

Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle

Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.     

The effect of phenotypic variation on metapopulation persistence

Demographic stochasticity (due to the probabilistic nature of the birth–death process) and demographic heterogeneity (between-individual differences in demographic parameters) have long been seen as factors affecting extinction risk. While demographic stochasticity can be independent of underlying species traits, demographic heterogeneity may strongly depend on phenotypic variation. However, how phenotypic variation can affect extinction risk is largely unknown. Here, I develop a stochastic metapopulation model that takes into account the effects of demographic stochasticity and phenotypic variation in the traits controlling colonization rates to assess what the effect of phenotypic variation may be on the persistence of the metapopulation. Although phenotypic variation can lead to a decrease in metapopulation persistence under some conditions, it also may lead to an increase in persistence whenever phenotypic mismatch—or the distance between the optimal trait value and the population mean—is large. This mismatch can in turn arise from a variety of ecological and evolutionary reasons, including weak selection or a recent history of invasion. Last, the effect of phenotypic variation has a deterministic component on colonization rates, and a stochastic component on persistence through colonization rates, but both are important to understand the overall effect. These results have important implications for the conservation of threatened species and management practices that may historically have overlooked phenotypic variation as unimportant noise around mean values of interest.     

Inhibition of rhythmic spiking by colored noise in neural systems

We study the effect of colored noise on the rhythmic spiking activity of neural systems in this paper. The phenomenon of the so-called inverse stochastic resonance , that is, noise with appropriate intensity suppresses the spiking activity in neural systems, is clearly observed in a special parameter regime. We find that the inhibition effect of colored noise is stronger than that of Gaussian white noise. Furthermore, our simulation results show that the inhibition effect of colored noise provides a useful mechanism for the generation of synchronized burst in type-2 mixed-feed-forward-feedback loop neuronal network motif, which indicates that such inhibition effect might have some biological implications.     

The effect of colored noise on spatiotemporal dynamics of biological invasion in a diffusive predator-prey system

Spatiotemporal dynamics of a predator-prey system is considered under the assumption that the predator is sensitive to colored noise. Mathematically, the model consists of two coupled diffusion-reactions. By means of extensive numerical simulations, the complex invasion pattern formations of the system are identified. The results show that a geographical invasion emerges without regional persistence when the intensity of colored noise is small. Remarkably, as the noise intensity increases, the species spreads via a patchy invasion only when the system is affected by red noise. Meanwhile, the relationship between local stability and global invasion is also considered. The predator, which becomes extinct in the system without diffusion, could invade locally when the system is affected by white noise. However, the local invasion is not followed by geographical spread.     

Optimizing Metapopulation Sustainability through a Checkerboard Strategy

The persistence of a spatially structured population is determined by the rate of dispersal among habitat patches. If the local dynamic at the subpopulation level is extinction-prone, the system viability is maximal at intermediate connectivity where recolonization is allowed, but full synchronization that enables correlated extinction is forbidden. Here we developed and used an algorithm for agent-based simulations in order to study the persistence of a stochastic metapopulation. The effect of noise is shown to be dramatic, and the dynamics of the spatial population differs substantially from the predictions of deterministic models. This has been validated for the stochastic versions of the logistic map, the Ricker map and the Nicholson-Bailey host-parasitoid system. To analyze the possibility of extinction, previous studies were focused on the attractiveness (Lyapunov exponent) of stable solutions and the structure of their basin of attraction (dependence on initial population size). Our results suggest that these features are of secondary importance in the presence of stochasticity. Instead, optimal sustainability is achieved when decoherence is maximal. Individual-based simulations of metapopulations of different sizes, dimensions and noise types, show that the system''s lifetime peaks when it displays checkerboard spatial patterns. This conclusion is supported by the results of a recently published Drosophila experiment. The checkerboard strategy provides a technique for the manipulation of migration rates (e.g., by constructing corridors) in order to affect the persistence of a metapopulation. It may be used in order to minimize the risk of extinction of an endangered species, or to maximize the efficiency of an eradication campaign.     

生物钟体系中色噪音诱导的日夜节律振荡和内信号随机共振

利用脉孢菌生物钟体系,研究了色噪音对其进行诱导所产生的日夜节律振荡信号及其内信号随机共振的行为．结果表明,色噪音的相关时间对该体系内信号随机共振的强弱起较大的影响作用．当无外信号存在时,色噪音的相关时间对体系内信号随机共振强度起抑制的作用,且随相关时间的增大,抑制作用增强．当外信号加到体系中时,由于相关时间和外信号的协同作用,相关时间不仅对其内信号随机共振强度起抑制的作用,而且还影响内信号随机共振峰的数目,即随相关时间的增大,可使单峰随机共振变为随机双共振．存在最佳的外信号频率使体系的内信号随机共振强度得到最大的增强,而其他频率的外信号却起抑制作用．色内噪音和色外噪音相比,前者对该体系进行诱导所得的内信号随机共振强度比后者的更强,而且体系对前者更敏感．另外,存在极限的噪音强度使白噪音和色噪音对该体系内信号随机共振的影响差异得以消失．所得结果可为治疗生物钟紊乱综合症提供理论依据,同时可更好地理解其他节奏机理,如心脏搏动节奏、呼吸节奏以及荷尔蒙水平的波动节奏等．     

Signal transduction and amplification in a circadian oscillator: Interaction between two colored noises

The signal transduction and amplification in a Neurospora circadian clock system is studied by using the mechanism of internal signal stochastic resonance (ISSR). Two cases have been investigated: the case of no correlations between multiplicative and additive colored noises and the case of correlations between two noises. The results show that, in both cases, the noise-induced circadian oscillations can be transduced with the phenomenon of internal signal stochastic resonance (ISSR). However, the correlation time and intensity of an additive colored noise play different roles for the ISSR, driven by multiplicative colored noise, while the correlation time and intensity of multiplicative colored noise hardly influence the ISSR driven by additive colored noise. In addition, the ISSR can be amplified or suppressed at an appropriate range of the correlation intensity between two colored noises. The fundamental frequency of noise-induced circadian oscillations is hardly shifted with the increment of the intensity and correlation time of colored noises, which implies that the Neurospora system could be resistant to colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms.     

Interspike interval correlations in neuron models with adaptation and correlated noise

The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel’s time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability.     

The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations

Analysing the persistence and viability of small populations is a key issue in extinction theory and population viability analysis. However, there is still no consensus on how to quantify persistence and viability. We present an approach to evaluate any simulation model concerned with extinction. The approach is devised from general Markov models of stochastic population dynamics. From these models, we distil insights into the general mathematical structure of the risk of extinction by time t, P₀(t). From this mathematical structure, we devise a simple but effective protocol – the ln(1−P₀)-plot – which is applicable for situations including environmental noise or catastrophes. This plot delivers two quantities which are fundamental to the assessment of persistence and viability: the intrinsic mean time to extinction, T_m, and the probability c₁ of the population reaching the established phase. The established phase is characterized by typical fluctuations of the population's state variable which can be described by quasi-stationary probability distributions. The risk of extinction in the established phase is constant and given by 1/T_m. We show that T_m is the basic currency for the assessment of persistence and viability because T_m is independent of initial conditions and allows the risk of extinction to be calculated for any time horizon. For situations where initial conditions are important, additionally c₁ has to be considered.     

Pushed beyond the brink: Allee effects,environmental stochasticity,and extinction

To understand the interplay between environmental stochasticity and Allee effects, we analyse persistence, asymptotic extinction, and conditional persistence for stochastic difference equations. Our analysis reveals that persistence requires that the geometric mean of fitness at low densities is greater than one. When this geometric mean is less than one, asymptotic extinction occurs with high probability for low initial population densities. Additionally, if the population only experiences positive density-dependent feedbacks, conditional persistence occurs provided the geometric mean of fitness at high population densities is greater than one. However, if the population experiences both positive and negative density-dependent feedbacks, conditional persistence only occurs if environmental fluctuations are sufficiently small. We illustrate counter-intuitively that environmental fluctuations can increase the probability of persistence when populations are initially at low densities, and can cause asymptotic extinction of populations experiencing intermediate predation rates despite conditional persistence occurring at higher predation rates.     

Aperiodic stochastic resonance in neural information processing with Gaussian colored noise

The aim of this paper is to explore the phenomenon of aperiodic stochastic resonance in neural systems with colored noise. For nonlinear dynamical systems driven by Gaussian colored noise, we prove that the stochastic sample trajectory can converge to the corresponding deterministic trajectory as noise intensity tends to zero in mean square, under global and local Lipschitz conditions, respectively. Then, following forbidden interval theorem we predict the phenomenon of aperiodic stochastic resonance in bistable and excitable neural systems. Two neuron models are further used to verify the theoretical prediction. Moreover, we disclose the phenomenon of aperiodic stochastic resonance induced by correlation time and this finding suggests that adjusting noise correlation might be a biologically more plausible mechanism in neural signal processing.     

The Interplay of Intrinsic and Extrinsic Bounded Noises in Biomolecular Networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, a model of enzymatic futile cycle and a genetic toggle switch. In and we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.     

Stochastic Sensitivity Analysis of Noise-Induced Extinction in the Ricker Model with Delay and Allee Effect

A susceptibility of population systems to the random noise is studied on the base of the conceptual Ricker-type model taking into account the delay and Allee effect. This two-dimensional discrete model exhibits the persistence in the form of equilibria, discrete cycles, closed invariant curves, and chaotic attractors. It is shown how the Allee effect constrains the persistence zones with borders defined by crisis bifurcations. We study the role of random noise on the contraction and destruction of these zones. This phenomenon of the noise-induced extinction is investigated with the help of direct numerical simulations and semi-analytical approach based on the stochastic sensitivity functions. Stochastic transitions from the persistence regimes to the extinction are studied by the analysis of the mutual arrangement of the basins of attraction and confidence domains.     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

污染环境中的Smith系统生存分析

研究了环境污染对Smith系统中种群生存的长期影响,考虑到种群数量的变化对种群个体体内毒素浓度和环境中毒素浓度的影响,对传统的Smith系统进行了修正,并且给出了一些种群弱平均持续生存和绝灭的充分条件．在一定条件下得到了弱平均持续生存与绝灭的阈值．     

Fisher information as a metric of locally optimal processing and stochastic resonance

The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.     

污染环境下单种群模型生存阈值

本论文研究了污染环境下毒素对单种群生存的影响。在环境容纳量较小的假设下建立了生物种群模型,在该模型中不但考虑了环境毒素浓度对生物个体生存的影响,还考虑了生物个体从食物链中吸收的毒素对其影响。通过研究得到种群一致持续生存和若平均持续生存的充分条件,同时得到种群持续生存依赖于模型参数和生物个体体内毒素净化率的某些充分条件．     

Complex dynamics of a stochastic uni-directional consumer-resource mutualism system

Like predation and competition, mutualism is recognized as a consumer-resource interaction, which includes bi-directional and uni-directional mutualisms. In this paper, we firstly propose a stochastic uni-directional consumer-resource system of two species in which the consumer has both positive and negative effects on the resource, while the resource has only a positive effect on the consumer. We then mathematically analyze the system, to demonstrate the existence, uniqueness, asymptotic pathwise behavior and stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the global attractivity and the existence of ergodic stationary distribution of the system. We also establish sufficient conditions for the extinction and persistence in mean of the resource, the consumer or the entire system. Numerical simulations are carried out to demonstrate the analytical results.     

Mean and quasideterministic equivalence for linear stochastic dynamics

In linear, stochastic dynamics it is shown that the quasideterministic population size is equivalent to the mean population size. The quasideterministic dynamics are defined by the conditional infinitesimal mean of the process. The stochastic component of the dynamics includes both Gaussian and Poisson white noise, with amplitude coefficients proportional to the population size. Generalizations are given for nonautonomous coefficients and for distributed Poisson jump amplitudes. A counter example--an exactly integrable nonlinear jump model--shows that the equivalence result does not hold for nonlinear stochastic dynamics.