In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI. The relevant mathematical tools are: (1) Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black-Sholes models). (2) The probability distributions known as b-lognormals, i.e. lognormals starting at a certain positive instant b>0 rather than at the origin. In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics. (3) The (Shannon) entropy of such b-lognormals is then seen to represent the 'degree of progress' reached by each living organism or by each big set of living organisms, like historic human civilizations. (4) All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy. (5) The well-known 'Molecular Clock of Evolution', namely the 'constant rate of Evolution at the molecular level' as shown by Kimura's Neutral Theory of Molecular Evolution, identifies with growth rate of the entropy of our Evo-SETI model. (6) Furthermore, we apply our Evo-SETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past. (7) Finally, we show that the Markov & Korotayev model for Darwinian Evolution identifies with an Evo-SETI model.
The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our "Peak-Locus Theorem" translates cladistics: each species created by evolution is a b-lognormal w…
Theoretical ecologists have long sought to understand how the persistence of populations depends on biotic and abiotic factors. Classical work showed that demographic stochasticity causes the mean time to extinction to increase exponentially with population size, whereas variation in environmental conditions can lead to a power law scaling. Recent work has focused especially on the influence of the autocorrelation structure ("color") of environmental noise. In theoretical physics, there is a burst of research activity in analyzing large fluctuations in stochastic population dynamics. This rese…
In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to the current population size. Using the direct Lyapunov method, we established the global properties of this stochastic differential equation. In particular, we found that solutions of the equation oscillate around an interval, and explicitly found the end points of this interval. Moreover, we found that, if the magnitude of the noise exceeds a certain critical …
In this paper we propose a new mathematical model capable of merging Darwinian Evolution, Human History and SETI into a single mathematical scheme: 1) Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution. 2) In 2008 this author provided the statistical generalization of the Drake equation yielding the numb…
The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. We prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapuno…
We consider a stochastic population model where the intrinsic or demographic noise causes cycling between states before the population eventually goes extinct. A master equation approach coupled with a WKB (Wentzel-Kramers-Brillouin) approximation is used to construct the optimal path to extinction. In addition, a probabilistic argument is used to understand the pre-extinction dynamics and approximate the mean time to extinction. Analytical results agree well with numerical Monte Carlo simulations. A control method is implemented to decrease the mean time to extinction. Analytical results quan…
We consider the problem of extinction processes on random networks with a given structure. For sufficiently large well-mixed populations, the process of extinction of one or more state variable components occurs in the tail of the quasi-stationary probability distribution, thereby making it a rare event. Here we show how to extend the theory of large deviations to random networks to predict extinction times. In particular, we use the theory to find the most probable path leading to extinction. We apply the methodology to epidemic models and discover how mean extinction times scale with epidemi…
Species extinction is a core process that affects the diversity of life on Earth. Competition between species in a population is considered by ecological niche-based theories as a key factor leading to different severity of species extinctions. There are population dynamics models that describe a simple and easily understandable mechanism for resource competition. However, these models can not efficiently characterize and quantify new emergent extinctions in a large population appearing due to environmental forcing. To address this issue we develop a stochastic physics-inspired approach to ana…
We present an explicit unified stochastic model of fluctuations in population size due to random birth, death, density-dependent competition and environmental fluctuations. Stochastic dynamics provide insight into small populations, including processes such as extinction, that cannot be correctly treated by deterministic methods. We present exact analytical and simulation-based results for extinction times of our stochastic model and compare the different effects of environmental stochasticity and intrinsic demographic stochasticity. We use both the discrete master equation approach and an exa…
Invading activity fronts or near-critical clusters may stabilize vulnerable excitable systems against stochastic trajectories terminating in an absorbing state
Virginia Tech（弗吉尼亚理工大学）；Faculty of Health Sciences, Virginia Tech（弗吉尼亚理工大学健康科学学院）
Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasi-stationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover -- renewal and removal -- is much slower than all other processes. In this case there is a time scale separation in the system which enables one to introduce a short-time quasi-stationary extinction rate W_1 and a long-time quasi-stationary extinction rate W_2, and develop a time-dependent theory of the transition bet…
We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the rec…
Community ecology has traditionally relied on the competitive exclusion principle, a piece of common wisdom in conceptual frameworks developed to describe species assemblages. Key concepts in community ecology, such as limiting similarity and niche partitioning, are based on competitive exclusion. However, this classical paradigm in ecology relies on implications derived from simple, deterministic models. Here we show how the predictions of a symmetric, deterministic model about the way extinctions proceed can be utterly different from the results derived from the same model when ecological dr…
In this study, spatial stochastic and logistic model (SSLM) describing dynamics of a population of a certain species was analysed. The behaviour of the extinction threshold as a function of model parameters was studied. More specifically, we studied how the critical values for the model parameters that separate the cases of extinction and persistence depend on the spatial scales of the competition and dispersal kernels. We compared the simulations and analytical results to examine if and how the mathematical approximations break down at the vicinity of the extinction threshold, and found a fun…
We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as es…
We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dyna…
The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number $R^{*}_{0}$, the expected survival probability of the plasmodium $E(e^{-(μ_{v}T_{1}+μT_{2})})$, and other threshold values are calculated. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local…
Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multi-population systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first causing a rapid extinction of the predators. A…
How high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population extinct? Is there any relation between the population size distributions with and without immigration? Under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch, and treat a patch simply as either occupied or empty? We address these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the Verhulst model: a pr…
Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. In this paper, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact o…
We study the role of the noise in the dynamics of two competing species. We consider generalized Lotka-Volterra equations in the presence of a multiplicative noise, which models the interaction between the species and the environment. The interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence of a periodic driving term, which accounts for the environment temperature variation. We find noise-induced periodic oscillations of the species concentrations and stochastic resonance phenomenon. W…
We investigate stochastic extinction in an epidemic model and the impact of random vaccinations in large populations formulated in terms of an optimal escape path. We find that different random vaccination strategies can have widely different results in decreasing expected time till extinction, for the same total amount of vaccines used. Vaccination strategies are considered in terms of two parameters: average frequency of vaccinations, given by $γ$, and the amplitude of the vaccinations, $ε$, where $ε\ll 1$ refers to the proportion of the population being vaccinated at some particular instant…
We study the combined impact of a colored environmental noise and demographic noise on the extinction risk of a long-lived and well-mixed isolated stochastic population which exhibits the Allee effect. The environmental noise modulates the population birth and death rates. Assuming that the Allee effect is strong, and the environmental noise is positively correlated and Gaussian, we derive a Fokker-Planck equation for the joint probability distribution of the population sizes and environmental fluctuations. In WKB approximation this equation reduces to an effective two-dimensional Hamiltonian …
We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment, focusing on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global in-all-patch extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and its stochastic analog. The deterministic model has either two or four attractors. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low…
Understanding the mechanisms governing population extinctions is of key importance to many problems in ecology and evolution. Stochastic factors are known to play a central role in extinction, but the interactions between a population's demographic stochasticity and environmental noise remain poorly understood. Here, we model environmental forcing as a stochastic fluctuation between two states, one with a higher death rate than the other. We find that in general there exists a rate of fluctuations that minimizes the mean time to extinction, a phenomenon previously dubbed "resonant activation."…
In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by $n$-dimensional K…
We consider non-demographic noise in the form of uncertainty in the reaction step size, and reveal a dramatic effect this noise may have on the stability of self-regulating populations. Employing the reaction scheme mA->kA, but allowing, e.g., the product number k to be a-priori unknown and sampled from a given distribution, we show that such non-demographic noise can greatly reduce the population's extinction risk compared to the fixed k case. Our analysis is tested against numerical simulations, and by using empirical data of different species, we argue that certain distributions may be more…
We investigate extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0. In scenario B there is an intermediate repelling point n=n_1 between the attracting point n=0 and another attracting point n=n_2 in the vicinity o…
Coextinction models are useful to understand community robustness to species loss and resilience to disturbances. We simulated pollinator extinctions in pollination networks by using a hybrid model that combined a recently developed stochastic coextinction model (SCM) for plant extinctions and a topological model (TCM) for animal extinctions. Our model accounted for variation in interaction strengths and included empirical estimates of plant dependence on pollinators to set seeds. The stochastic nature of such model allowed us determining plant survival to single (and multiple) extinction even…
We analyze the long term behavior of interacting populations which can be controlled through harvesting. The dynamics is assumed to be discrete in time and stochastic due to the effect of environmental fluctuations. We present extinction and coexistence criteria when there are one or two interacting species. We then use these tools in order to see when harvesting leads to extinction or persistence of species, as well as what the optimal harvesting strategies, which maximize the expected long term yield, look like. For single species systems, we show under certain conditions that the optimal ha…
Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see e.g. B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan [Nature {\bf 418}, 171 (2002)] and B. Kirkup and M. A. Riley [Nature {\bf 428}, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely the `rock-paper-scissors' or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting i…
Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-tim…
Phenotypic plasticity and its evolution may help evolutionary rescue in a novel and stressful environment, especially if environmental novelty reveals cryptic genetic variation that enables the evolution of increased plasticity. However, the environmental stochasticity ubiquitous in natural systems may alter these predictions because high plasticity may amplify phenotype-environment mismatches. Although previous studies have highlighted this potential detrimental effect of plasticity in stochastic environments, they have not investigated how it affects extinction risk in the context of evoluti…
Disentangling the processes leading populations to extinction is a major topic in ecology and conservation biology. The difficulty to find a mate in many species is one of these processes. Here, we investigate the impact of self-incompatibility in flowering plants, where several inter-compatible classes of individuals exist but individuals of the same class cannot mate. We model pollen limitation through different relationships between mate availability and fertilization success. After deriving a general stochastic model, we focus on the simple case of distylous plant species where only two cl…
A wide range of stochastic processes that model the growth and decline of populations exhibit a curious dichotomy: with certainty either the population goes extinct or its size tends to infinity. There is a elegant and classical theorem that explains why this dichotomy must hold under certain assumptions concerning the process. In this note, I explore how these assumptions might be relaxed further in order to obtain the same, or a similar conclusion, and obtain both positive and negative results.
Motivated by modeling the dynamics of a population living in a flowing medium where the environmental factors are random in space, we have studied an asymmetric variant of the one-dimensional contact process, where the quenched random reproduction rates are systematically greater in one direction than in the opposite one. The spatial disorder turns out to be a relevant perturbation but, according to results of Monte Carlo simulations, the behavior of the model at the extinction transition is different from the (infinite randomness) critical behavior of the disordered, symmetric contact process…
Sexually reproducing populations with small number of individuals may go extinct by stochastic fluctuations in sex determination, causing all their members to become male or female in a generation. In this work we calculate the time to extinction of isolated populations with fixed number $N$ of individuals that are updated according to the Moran birth and death process. At each time step, one individual is randomly selected and replaced by its offspring resulting from mating with another individual of opposite sex; the offspring can be male or female with equal probability. A set of $N$ time s…
We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios, from single species domination to coexistence between non-interacting species. Using exact results and numerical simulations we discuss the temporal evolution of the system for different values of $N$, for different values of the reaction rates, as well as for different initial conditions. As expected, the stochastic evolution is found to closely follow the me…
We analyse a simple discrete-time stochastic process for the theoretical modeling of the evolution of protein lengths. At every step of the process a new protein is produced as a modification of one of the proteins already existing and its length is assumed to be random variable which depends only on the length of the originating protein. Thus a Random Recursive Trees (RRT) is produced over the natural integers. If (quasi) scale invariance is assumed, the length distribution in a single history tends to a lognormal form with a specific signature of the deviations from exact gaussianity. Compar…
Many species are unsustainable at small population densities (Allee Effect), i.e., below a threshold named Allee threshold, the population decreases instead of growing. In a closed local population, environmental fluctuations always lead to extinction. Here, we show how, in spatially extended habitats, dispersal can lead to a sustainable population in a region, provided the amplitude of environmental fluctuations is below an extinction threshold. We have identified two types of sustainable populations: high-density and low-density populations (through a mean-field approximation, valid in the l…
Disease control is of paramount importance in public health with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Thus intervention controls, such as vaccination of susceptible individuals and/or treatment of infectives, are typically based on a deterministic schedule, such as periodically vaccinating susceptible children based on school calendars. In reality, however, such pol…
We investigate stochastic extinction in an epidemic model and the impact of random vaccinations in large populations. We show that, in the absence of vaccinations, the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point, with an unusual critical exponent. Even a comparatively weak Poisson-distributed vaccination leads to an exponential increase in the extinction rate, with the exponent that strongly depends on the vaccination parameters.
An ecological system with multiple stable equilibria is prone to undergo catastrophic change or regime shift from one steady-state to another. It should be noted that, if one of the steady states is an extinction state, the catastrophic change may lead to extinction. A suitable manual measure may control the prevention of catastrophic changes of different species from one equilibrium to another. We consider two stochastic models with linear and nonlinear harvesting terms. We inspect either density regulation or Allee type density regulated models [Saha et al., Ecological Modelling, 2013], whic…
RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simul…
We analyse metapopulation dynamics in terms of an individual-based, stochastic model of a finite metapopulation. We suggest a new approach, using the number of patches in the population as a large parameter. This approach does not require that the number of individuals per patch is large, neither is it necessary to assume a time-scale separation between local population dynamics and migration. Our approach makes it possible to accurately describe the dynamics of metapopulations consisting of many small patches. We focus on metapopulations on the brink of extinction. We estimate the time to ext…
The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection and the Moran model with frequency-dependent selection (equivalently, a game-theoretic formulation of the Moran model). We demonstrate how these mappings …
Recently, a first step was made by the authors towards a systematic investigation of the effect of reaction-step-size noise - uncertainty in the step size of the reaction - on the dynamics of stochastic populations. This was done by investigating the effect of bursty influx on the switching dynamics of stochastic populations. Here we extend this formalism to account for bursty reproduction processes, and improve the accuracy of the formalism to include subleading-order corrections. Bursty reproduction appears in various contexts, where notable examples include bursty viral production from infe…
We employ Monte Carlo simulations to study a stochastic Lotka-Volterra model on a two-dimensional square lattice with periodic boundary conditions. If the (local) prey carrying capacity is finite, there exists an extinction threshold for the predator population that separates a stable active two-species coexistence phase from an inactive state wherein only prey survive. Holding all other rates fixed, we investigate the non-equilibrium relaxation of the predator density in the vicinity of the critical predation rate. As expected, we observe critical slowing-down, i.e., a power law dependence of…
