Extinction phase transitions in a model of ecological and evolutionary dynamics

We study the non-equilibrium phase transition between survival and extinction of spatially extended biological populations using an agent-based model. We especially focus on the effects of global temporal fluctuations of the environmental conditions, i.e., temporal disorder. Using large-scale Monte-Carlo simulations of up to 3 × 10⁷ organisms and 10⁵ generations, we find the extinction transition in time-independent environments to be in the well-known directed percolation universality class. In contrast, temporal disorder leads to a highly unusual extinction transition characterized by logarithmically slow population decay and enormous fluctuations even for large populations. The simulations provide strong evidence for this transition to be of exotic infinite-noise type, as recently predicted by a renormalization group theory. The transition is accompanied by temporal Griffiths phases featuring a power-law dependence of the life time on the population size.     

In search for the optimal strategy in population dynamics

A unification of recently proposed models describing population dynamics is presented. We study the effect of different factors, like environmental conditions, concentration of individuals in a given area and migration strategies, on population dynamics. Moreover, we show that a population occupying a smaller area is more susceptible to extinction, which is a well known biological fact. We solve the model using Monte Carlo simulations and the mean-field approach. Constructing flow diagrams we find the optimal strategy in population dynamics. Received 10 June 2001 and Received in final form 22 November 2001     

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     

Impact of time delay and a multiplicative periodic signal on stochastic resonance and steady states shift for a stochastic insect outbreak system subjected to Gaussian noises

In this paper, we aim to investigate comprehensively the steady-states characteristics, the stochastic resonance phenomenon and the mean decline time for an insect outbreak system caused by the terms of the multiplicative, additive noises and time delay,. Our results exhibit that the multiplicative noise and the time delay can both reduce the stability of the biological system and speed up the extinction process of the insect population, while the additive noise can decrease the possibility of the decline of the biological population by a wide margin and make contribution to the survival and reproduction of the insect system to some extent. On the other hand, as regards to the stochastic resonance phenomenon (SR) induced by noise terms, time delay term and a weak multiplicative periodic signal, the numerical results show that the multiplicative noise intensity Q always suppresses the SR effect in any case, while the additive noise intensity M can inhibit the SR effect in the case of a big value of Q, but excite the maximum of the SNR for the case of a small value of Q. Moreover, time delay τ exerts mainly the inhibitory effect on the SR phenomenon except that in the SNR-Q plot.     

The diffusion equation of random genetic drift – biology’s analogue of the Schrödinger equation?

This work is a self-contained introduction to some basic aspects of the dynamics that occurs in biological populations. It focusses on the proportion (or frequency) of a population that carries a particular gene. We make use of the notion of a force, in the context of genetics and evolution, to describe the dynamics of the frequency in an effectively infinite population. We then show how randomness enters into the dynamics of populations with a finite size, a randomness known as random genetic drift. We derive an equation, involving random numbers, which describes how the frequency behaves in a population of finite size. It is shown that in some situations this equation exhibits irreversible absorption phenomena. These phenomena are associated with the extinction (or loss) of the gene, or the complete takeover by the gene (termed fixation), where 100% of the population carries the gene. Taking the theory further, we show how an approximation leads to a stochastic differential equation for the frequency, where random genetic drift takes the form of an additional contribution to the force, that randomly fluctuates. The stochastic differential equation is, in turn, related to a diffusion equation, which encompasses many fundamental phenomena. Because of this, the diffusion equation plausibly has a similar status in biology to the Schrödinger equation in physics. It is notable that both the Schrödinger equation and the diffusion equation have a somewhat similar mathematical structure: they both involve first order derivatives of time and second order derivatives of space (or the analogue of space). There are, however, some significant mathematical differences. In contrast to the Schrödinger equation, the diffusion equation can routinely have solutions which are singular, in that the solutions contain Dirac delta functions. The delta functions are not, however, problematic, and have an explicit biological significance. We illustrate results with some basic calculations and computer simulations.     

Mean Extinction Time in Predator-Prey Model

A basic predator-prey (Lotka-Volterra) system exhibits marginal stability on the deterministic level. Intrinsic demographic stochasticity destroys this stability and drives the system toward extinction of one or both species. We analytically calculate the mean extinction time of such a system and investigate its scaling with the system’s parameters. This mean extinction time, measured in number of population cycles, scales as the square root of the size of the smaller population and as the minus three halves power of the size of the larger population. The analytic results are fully confirmed by Monte-Carlo simulations.     

Impact of colored cross-correlated noises on stochastic resonance and mean extinction rate for a metapopulation system with a multiplicative periodic signal

In this paper, we aim to explore the mean extinction rate and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by a multiplicative periodic signal, colored cross-correlated multiplicative and additive Gaussian noises. By use of the fast descent method and the adiabatic approximation theory for the signal-to-noise ratio, we obtain the expression of the signal-to-noise ratio (SNR). Numerical results indicate that the various SR phenomena occur in the metapopulation system due to the variation of the noise terms and the correlation time. Specifically, the noise correlation always plays a critical role in motivating the SR phenomenon, while the multiplicative noise exerts the inhibition effect on the SR. Interestingly, the weak additive noise can stimulate the resonant peak of the SNR, while the further increase of the noise intensity will lead to the reduction of the SR effect. On the other hand, the noise correlation time τ plays antipodal roles in motivating the SR phenomenon under different circumstances. With regard to the mean extinction rate of the population from the boom state to the extinction one, by performing the numerical calculations, it is found that the additive noise always accelerate the extinction of the population, while the correlation noise will slow down the decline for the population. The role that the noise correlation time plays in the population extinction depends on the values that λ takes.     

Nontrivial Effect of Time-Varying Migration on the Three Species Prey-Predator System

Migration is ubiquitous in ecosystem and often plays an important role in biological diversity. In this work,by introducing a time-varying migration rate associated with the difference of subpopulation density into a prey, we study the Hopf bifurcation and the critical phenomenon of predator extinction of the three species prey-predator system,which consists of a predator, a prey and a mobile prey. It is found that the system with migration exhibits richer dynamic behaviors than that without migration, including two Hopf bifurcations and two limit cycles. Interestingly,the parameters of migration have a drastically influence on the critical point of predator extinction, determining the coexistence of species. Moreover, the population evolution dynamics of one-dimensional multiple prey-predator system are also discussed.     

Effects of gradual spatial variation in resources on population extinction: Analytic calculations for abrupt transitions

We study bifurcations in a spatially extended nonlinear system representing population dynamics with the help of analytic calculations. The result we obtain helps in the understanding of the onset of abrupt transitions leading to the extinction of biological populations. The result is expressed in terms of Airy functions and sheds light on the behavior of bacteria in a Petri dish as well as of large animals such as rodents moving over a landscape.     

A computational predator-prey model, pursuit-evasion behavior based on different range of vision

We studied an extended predator-prey model of three interacting species in a two-dimensional lattice. Numerous factors have been taken into account in our research such as individual mobility and pursuit-evasion abilities. Our major focus is on the stochastic character of vision distribution for predator and prey. The model we made displays the features upon the population evolving through time, the spatial distribution of population, and the cross correlation of three species. What we observed showed the increase of the predators’ pursuit ability works in a negative way on their population, although nature favors the predator with maximum ability during the evolution, and the increasing vision of predators causes the increase of the preys’ population. And the predators’ ability deficiency may lead to the extinction of their population. In addition, the results show that it is not necessary for prey to have more intelligent evasion abilities.     

Subcritical behavior in the alternating supercritical Domany-Kinzel dynamics

Cellular automata are widely used to model real-world dynamics. We show using the Domany-Kinzel probabilistic cellular automata that alternating two supercritical dynamics can result in subcritical dynamics in which the population dies out. The analysis of the original and reduced models reveals generality of this paradoxical behavior, which suggests that autonomous or man-made periodic or random environmental changes can cause extinction in otherwise safe population dynamics. Our model also realizes another scenario for the Parrondo’s paradox to occur, namely, spatial extensions.     

Characteristic periodicities of collective behavior at the foreign exchange market

The Battle of the Sexes describes asymmetric conflicts in mating behavior of males and females. Males can be philanderer or faithful, while females are either fast or coy, leading to a cyclic dynamics. The adjusted replicator equation predicts stable coexistence of all four strategies. In this situation, we consider the effects of fluctuations stemming from a finite population size. We show that they unavoidably lead to extinction of two strategies in the population. However, the typical time until extinction occurs strongly prolongs with increasing system size. In the emerging time window, a quasi-stationary probability distribution forms that is anomalously flat in the vicinity of the coexistence state. This behavior originates in a vanishing linear deterministic drift near the fixed point. We provide numerical data as well as an analytical approach to the mean extinction time and the quasi-stationary probability distribution.     

Population turnover and adaptation in heterogeneous environments

We study adaptive dynamics in a structured population model of asexual individuals which takes into account environmental heterogeneity among the subpopulations. The key purpose of the present work is to address how population turnovers, i.e. extinction events followed by recolonization, affect the rate of fixation of advantageous mutations. This model is a generalization of our previous model to address the interplay between environmental correlation and evolutionary forces on the adaptive process. The incorporation of population turnovers into the model enables us to make a direct correspondence between the model and host-parasite dynamics (epidemiological models). Strikingly, contrary to the intuitive and usual deleterious effect associated to extinction events, it is observed that population turnovers can in fact speed up adaptation as heterogeneity rises. On the other side, in nearly homogeneous population turnovers have a neutral effect on fixation rates, but a detrimental outcome is also achieved when extinction events become very common. In resume, population turnover outcomes on fixation rates of advantageous mutations are strongly influenced by the selective correlation among the subpopulations (demes).     

生物材料红外波段消光性能分析

对制备的三种消光材料真菌An0429孢子,真菌Bb0919孢子以及真菌Cx0507孢子的红外波段消光性能进行了测试分析。静态测试采用压片法得到三种生物材料的镜面反射光谱,然后根据Krames-Kronig(K-K)关系对三种生物材料红外波段的复折射率进行了计算。由Mie理论计算得到三种生物材料红外波段的静态质量消光系数,并与几种无机非金属材料进行了对比。搭建烟幕箱实验平台,对三种生物材料3~5 μm波段动态质量消光系数进行了测试分析,得到三种消光材料的动态质量消光系数分别为1.257,1.065以及1.009 m2·g-1。测试分析结果表明,三种生物材料的红外波段消光性能优于常见的无机材料,其生产周期短,生产成本低,生产过程无毒,对环境友好等优点,使得生物消光材料具有较好的应用前景。     

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.     

Biological evolution and statistical physics

This review is an introduction to theoretical models and mathematical calculations for biological evolution, aimed at physicists. The methods in the field are naturally very similar to those used in statistical physics, although the majority of publications have appeared in biology journals. The review has three parts, which can be read independently. The first part deals with evolution in fitness landscapes and includes Fisher's theorem, adaptive walks, quasispecies models, effects of finite population sizes, and neutral evolution. The second part studies models of coevolution, including evolutionary game theory, kin selection, group selection, sexual selection, speciation, and coevolution of hosts and parasites. The third part discusses models for networks of interacting species and their extinction avalanches. Throughout the review, attention is paid to giving the necessary biological information, and to pointing out the assumptions underlying the models, and their limits of validity.     

Scattering characteristics of relativistically moving concentrically layered spheres

The energy extinction cross section of a concentrically layered sphere varies with velocity as the Doppler shift moves the spectral content of the incident signal in the sphere's co-moving inertial reference frame toward or away from resonances of the sphere. Computations for hollow gold nanospheres show that the energy extinction cross section is high when the Doppler shift moves the incident signal's spectral content in the co-moving frame near the wavelength of the sphere's localized surface plasmon resonance. The energy extinction cross section of a three-layer sphere consisting of an olivine-silicate core surrounded by a porous and a magnetite layer, which is used to explain extinction caused by interstellar dust, also depends strongly on velocity. For this sphere, computations show that the energy extinction cross section is high when the Doppler shift moves the spectral content of the incident signal near either of olivine-silicate's two localized surface phonon resonances at 9.7 μm and 18 μm.     

内嵌圆饼空心方形银纳米结构的光学性质

采用离散偶极子近似方法计算了内嵌圆饼空心方形银纳米结构的消光光谱以及其近场的电场强度分布,并进一步与空心方形纳米结构的消光光谱和表面电场做比较.结果表明,在耦合作用下内嵌圆饼空心方形银纳米结构不仅产生了新的共振模式,而且新的共振模式在传统表面增强拉曼散射的激发波长范围内,进而可以弥补由于实验上运用纳米切片法所制备的空心方形纳米结构尺寸较大导致其共振吸收峰在远红外波长范围的不足.此外,可以通过改变内嵌圆饼空心方形银纳米结构的形貌参数调节其表面等离子体共振峰的共振波长,以满足在表面增强拉曼散射、生物分子或化学分子探测上的应用.     

On the trap of extinction and its elimination

We investigate the dynamics of a system of three interacting populations in presence of extinction and substitution: each population whose number of individuals drops under some threshold value becomes extinct, and it is substituted by another population with different fitness and different coefficients of interaction with the other populations. We study the influence of extinction on the system states, which in the absence of extinction can be fixed points, limit cycles or chaotic attractors of Shilnikov kind. The extinction can destabilize each of these states. We observe two possible kinds of evolution in the destabilized system: (i) it can remain forever in the trap of extinction, i.e., the extinctions and substitutions of populations continue for indefinitely long time or (ii) it can avoid the trap of extinction by means of the substitution, i.e., the fitness and the coefficients of the interactions between the species move the system attractor away from the zone of the threshold values, the extinction stops, and the system settles on a new attractor. The obtained results are discussed from the point of view of products competing for the preference of buyers that can change their opinion.