Theta-logistic Model Research Articles

Estimation of the present status of the species based on the theoretical bounds of environmental noise intensity: An illustration through a big abundance data and simulation

Sibly et al. (2005) described that most species have a fundamental characteristic to follow the theta-logistic growth trait with the convex downward trend. The fundamental yardstick of this research work builds under the deterministic setup, whereas the involvement of the external noise in any growth system is inevitable. But, the involvement of external affairs in any species growth can not be well judged only through its density dependence; it requires a further assessment. So, we frame the theta-logistic model with the stochastic analog in two directions, i.e., the discrete and continuous setup. The analysis of the discrete stochastic model is manifested by the bifurcation analysis, which shows that the attainment of the chaotic regime enhances with the increase in noise intensity level. Although the role of chaos in species extinction is debatable, a literature survey suggests that chaos with stochasticity accelerates the extinction of species. Similarly, in the case of the continuous version, we performed a theoretical study on the stochastic theta-logistic model to provide a critical value of the noise intensity parameter. This threshold magnitude acts as the sustainability criterion of any species environmental tolerability. In this connection, we use the data of four major taxonomic groups, i.e., Bird, Insect, Mammal, and Fish, from the GPDD database and classifies the species based on environmental sensitivity. The highly sensitive species have a low tolerance level, associated with the small magnitude of environmental noise intensity parameter. Moreover, the simulation prediction model on these four taxonomic classes also shows that the overall extinction probability of the considered birds in our research work is almost negligible for the current time window.

On estimating the parameters of generalized logistic model from census data: Drawback of classical approach and reliable inference using Bayesian framework

In ecology, a nonconstant functional relationship between per capita growth rate and population size is referred as density dependence and mathematical models are utilized to detect its presence in natural populations. The theta-logistic model (parameterized by rm: intrinsic growth rate; θ: shape parameter; and K: carrying capacity) has been extensively discussed through various generalizations in the literature due to its flexibility and sound ecological interpretations which can be generalized for many natural populations. In this article, we show that nonlinear least squares approach is not an appropriate choice for estimating the model using real data. Using simulation, we show that the unknown parameters are better estimated under a Bayesian framework. We utilize the Gibbs algorithm for simulating samples from the posterior density, which is approximated by grid approximation and Bayesian credible intervals are obtained. Reliability of the estimation process is shown by using simulated data sets. Rules for choosing the prior distributions are discussed. We also apply the proposed strategy to estimate parameters using real data sets from the global population dynamics database. The robustness of the method with respect to prior distribution of the parameters is investigated by taking different choice of priors. We also establish its effectiveness in estimating parameters in a predator-prey system. The discussed method is computationally intensive, but its simple implementation will be useful for fitting complex models to study growth patterns of natural populations.

An extended stochastic Allee model with harvesting and the risk of extinction of the herring population

Overexploitation of commercially beneficial fish is a serious ecological problem around the world. The growth profiles of most of the species are likely to follow density regulated theta-logistic model irrespective of any taxonomy group [Sibly et al., Science, 2005]. Rapid depletion of population size may cause reduced fitness, and the species is exposed to Allee phenomena. Here sustainability is addressed by modelling the herring population as a stochastic process and computing the probability of extinction and expected time to extinction. The models incorporate an Allee effect, crowding effect which reduce birth and death rates at large populations, and two possible choices of harvesting models viz. linear harvesting and nonlinear harvesting. A seminal attempt is made by Saha [Saha et al., Ecol. Model, 2013] for this economically beneficial fish, but ignored the vital phenomena of harvesting. Moreover, in this model, the demographic stochasticity is introduced through the white-noise term, which has certain limitations when harvesting is introduced into the system. White noise is appropriate for such a system where immigration and emigration are allowed, but a harvesting model is rational for a closed system. The demographic stochasticity is introduced by replacing an ordinary differential equation model with a stochastic differential equation model, where the instantaneous variance in the SDE is derived directly from the birth and death rates of a birth-death process. The modelling parameters are fit to data of the herring populations collected from Global Population Dynamics Database (GPDD), and the risk of extinction of each population is computed under different harvesting protocols. A threshold for handling times is computed beneath which the risk of extinction is high. This is proposed as a recommendation to management for sustainable harvesting.

The effect of harvesting on the spatial synchrony of population fluctuations

Harvesting in space affects, in general, the spatial scale of the synchrony in the population fluctuations, which determines the size of the areas subjected to simultaneous quasi-extinction risk. Here we show that harvesting reduces the population synchrony scale if it depends more strongly on population fluctuations than the density dependence of the growth rate in the absence of harvesting. We show that constant and proportional harvesting always increases the spatial scale, using a theta-logistic model for density regulation. We also provide exact scaling results under harvesting for the Beverton–Holt and the Ricker stock-recruitment models that are commonly applied, e.g. in fisheries. Our results indicate that harvest in areas with large abundances should be encouraged to avoid increase of the spatial scale of synchrony in the population fluctuations that can lead to unexpected quasi-extinction of populations over large areas. Our results quantify this harvesting impact giving the resulting scales of spatial synchrony of population fluctuations. This emphasizes the importance of estimating the form of density dependence as well as the dependency of harvest upon population density of exploited populations, in order to get reliable predictions of the size of areas that can undergo simultaneous quasi-extinction.

On the relationship between the encounter rate and population density: Are classical models of population dynamics justified?

A stochastic model of migrations on a lattice and with discrete time is considered. It is assumed that space is homogenous with respect to its properties and during one time step every individual (independently of local population numbers) can migrate to nearest nodes of lattice with equal probabilities. It is also assumed that population size remains constant during certain time interval of computer experiments. The following variants of estimation of encounter rate between individuals are considered: when for the fixed time moments every individual in every node of lattice interacts with all other individuals in the node; when individuals can stay in nodes independently, or can be involved in groups in two, three or four individuals. For each variant of interactions between individuals, average value (with respect to space and time) is computed for various values of population size. The samples obtained were compared with respective functions of classic models of isolated population dynamics: Verhulst model, Gompertz model, Svirezhev model, and theta-logistic model. Parameters of functions were calculated with least square method. Analyses of deviations were performed using Kolmogorov-Smirnov test, Lilliefors test, Shapiro-Wilk test, and other statistical tests. It is shown that from traditional point of view there are no correspondence between the encounter rate and functions describing effects of self-regulatory mechanisms on population dynamics. Best fitting of samples was obtained with Verhulst and theta-logistic models when using the dataset resulted from the situation when every individual in the node interacts with all other individuals.

A simple approximation of moments of the quasi-equilibrium distribution of an extended stochastic theta-logistic model with non-integer powers

The stochastic versions of the logistic and extended logistic growth models are applied successfully to explain many real-life population dynamics and share a central body of literature in stochastic modeling of ecological systems. To understand the randomness in the population dynamics of the underlying processes completely, it is important to have a clear idea about the quasi-equilibrium distribution and its moments. Bartlett et al. (1960) took a pioneering attempt for estimating the moments of the quasi-equilibrium distribution of the stochastic logistic model. Matis and Kiffe (1996) obtain a set of more accurate and elegant approximations for the mean, variance and skewness of the quasi-equilibrium distribution of the same model using cumulant truncation method. The method is extended for stochastic power law logistic family by the same and several other authors (Na˙sell, 2003; Singh and Hespanha, 2007). Cumulant truncation and some alternative methods e.g. saddle point approximation, derivative matching approach can be applied if the powers involved in the extended logistic set up are integers, although plenty of evidence is available for non-integer powers in many practical situations (Sibly et al., 2005). In this paper, we develop a set of new approximations for mean, variance and skewness of the quasi-equilibrium distribution under more general family of growth curves, which is applicable for both integer and non-integer powers. The deterministic counterpart of this family of models captures both monotonic and non-monotonic behavior of the per capita growth rate, of which theta-logistic is a special case. The approximations accurately estimate the first three order moments of the quasi-equilibrium distribution. The proposed method is illustrated with simulated data and real data from global population dynamics database.

Guidelines when estimating temporal changes in density dependent populations

Anthropogenic activity can cause changes in the population dynamics of species. The changes can be modelled by density dependent models with time varying parameters. The following study looks at the accuracy of model parameter estimates using primarily simulations, in addition to a real data set of Grey Heron. A key point is the amount of data required to detect deterministic changes, either step-wise or gradual, in parameters for species with different population dynamics. The theta-logistic model is used to simulate the data and fitted to realizations of step-wise change in growth rate, and a linear model is fitted to gradual or step-wise changes in carrying capacity. Bayesian analysis is applied to study the effect of different prior distributions on the strength of density regulation. The range of the data is especially important when trying to detect step-wise changes in growth rate. Detection of changes in carrying capacity depends on the dynamics of the population, e.g. it is difficult to observe change for species with long return time to equilibrium within short time frames. The estimates of change in carrying capacity can become more accurate using a strong prior on the strength of density regulation. However, the prior may give more conservative estimates, if the prior assumes a weak density regulation. The results provide ecologists and decision makers with a general idea of what to expect of analyses of time series data of populations in changing environments.

Cooperation in species: Interplay of population regulation and extinction through global population dynamics database

Throughout the history of evolutionary process cooperation has played a significant role in species sociological development and social groups are more likely to be survived than individuals in apparently unfavorable conditions. We bring in the concept of cooperation and introduce a new parameter γ in the generalized theta-logistic equation to model non-monotonic behavior of per capita growth rate. This phenomenological modeling approach highlights the interplay of two opposing forces, namely, the positive effect of cooperation and the negative density dependence (competition). Sibly et al. (2005) found that, the density dependent population growth curve seems to be concave in most animal populations, and follows the theta-logistic equation. To explore this phenomenon further, we investigated 715 time series from global population dynamics database and identified 28 cases, where, incorporating intra-species cooperation as an allometric trait into the logistic improves the description of the population growth. Nonlinear regression and bootstrap resampling technique were used to estimate the model parameters during the fitting of real data and the mean square error was used to compare the fitting performance of theta-logistic model and the model with cooperation. This study may help to identify the presence of cooperation from species growth profile obtained from census data. The numerical estimate of cooperation parameter and its threshold value can be used as a potential tool to decide the optimal management action in species conservation. The stability and bifurcation analysis are used to choose the critical value of γ above which the risk of extinction is high. The illustration has been provided by means of simulation and real data.

Why preferring parametric forecasting to nonparametric methods?

A recent series of papers by Charles T. Perretti and collaborators have shown that nonparametric forecasting methods can outperform parametric methods in noisy nonlinear systems. Such a situation can arise because of two main reasons: the instability of parametric inference procedures in chaotic systems which can lead to biased parameter estimates, and the discrepancy between the real system dynamics and the modeled one, a problem that Perretti and collaborators call "the true model myth". Should ecologists go on using the demanding parametric machinery when trying to forecast the dynamics of complex ecosystems? Or should they rely on the elegant nonparametric approach that appears so promising? It will be here argued that ecological forecasting based on parametric models presents two key comparative advantages over nonparametric approaches. First, the likelihood of parametric forecasting failure can be diagnosed thanks to simple Bayesian model checking procedures. Second, when parametric forecasting is diagnosed to be reliable, forecasting uncertainty can be estimated on virtual data generated with the fitted to data parametric model. In contrast, nonparametric techniques provide forecasts with unknown reliability. This argumentation is illustrated with the simple theta-logistic model that was previously used by Perretti and collaborators to make their point. It should convince ecologists to stick to standard parametric approaches, until methods have been developed to assess the reliability of nonparametric forecasting.

A Quantitative Genetic Model ofr- andK-Selection in a Fluctuating Population

We analyze a stochastic quantitative genetic model for the joint dynamics of population size N and evolution of a multidimensional mean phenotype (z) under density-dependent selection. This generalizes our previous theories of evolution in fluctuating environments to include density-dependent (but frequency-independent) selection on quantitative characters. We assume that appropriate constraints or trade-offs between fitness components exist to prevent unlimited increase of fitness. We also assume weak selection such that the expected rate of return to equilibrium is much slower for (z) than N. The mean phenotype evolves to a stationary distribution around an equilibrium point z(opt) that maximizes a simple function determined by ecological parameters governing the dynamics of population size. For any (z), the expected direction of phenotypic evolution is determined by the additive genetic covariance matrix G and the gradient of this function with respect to the mean phenotype. For the theta-logistic model of density dependence, evolution tends to maximize the expected value of N(θ).

Impulsive harvesting and by-catch mortality for the theta logistic model

In this paper, we investigate the population dynamics described by the theta logistic model with periodic impulsive harvesting and by-catch mortality. We examine the existence and stability of two positive periodic solutions by using qualitative methods and cobwebs. Then the sufficient conditions under which the unique positive periodic solution exists and is semi-stable are established, and qualifications for the solutions approach zero are also obtained. Further, choosing the maximum sustainable yield as the management objective, we investigate the optimal harvesting policy for the theta logistic model with periodic impulsive harvesting. Moreover the corresponding theta logistic difference equation is considered subject to the impulsive perturbation, and the dynamics which is parallel to that for the differential equation is examined. The main results extend and generalize the classical results for populations described by the autonomous logistic equation in renewable resources management.

Estimation methods for nonlinear state-space models in ecology

The use of nonlinear state-space models for analyzing ecological systems is increasing. A wide range of estimation methods for such models are available to ecologists, however it is not always clear, which is the appropriate method to choose. To this end, three approaches to estimation in the theta logistic model for population dynamics were benchmarked by Wang (2007). Similarly, we examine and compare the estimation performance of three alternative methods using simulated data. The first approach is to partition the state-space into a finite number of states and formulate the problem as a hidden Markov model (HMM). The second method uses the mixed effects modeling and fast numerical integration framework of the AD Model Builder (ADMB) open-source software. The third alternative is to use the popular Bayesian framework of BUGS. The study showed that state and parameter estimation performance for all three methods was largely identical, however with BUGS providing overall wider credible intervals for parameters than HMM and ADMB confidence intervals.

An Analysis of Eurasian Badger (Meles meles) Population Dynamics: Implications for Regulatory Mechanisms

Based on 15 years of observations of a badger (Meles meles) population at Wytham Woods in the central United Kingdom, we investigated which parameters governed demographic changes, using data recording the life histories of 868 individuals. We modeled the population in terms of a stage-classified matrix, giving an exponential population growth rate of r 5 0.063 (l 5 1.065). Population elasticity values, derived from this matrix, produced a relative order of importance of regulatory parameters governing this population growth rate, of: adult survival (Pa) . fertility (F) . juvenile survival (Pj) . age at 1st reproduction (a) . age at last reproduction (v). Thus, changes in Pa and F have the greatest potential to influence population growth rate. The population underwent a dramatic change in the last 5 years of the study, with a decline, then stabilization in population growth. However, this change was not related to any change in elasticity patterns between the 2 periods. Comparing the latter period to the preceding period of marked population growth revealed that fertility rate had little actual influence, whereas adult and juvenile survival rates were far more influential demographic variables. These findings prompted a 2nd, retrospective, analytical approach, a life-table response experiment (LTRE), revealing that the importance of Pj in the LTRE contrasted with its lesser prospective importance in the elasticity analysis. Change in fertility (DF) apparently had little environmental or genetic scope (according to the results of either technique) to influence regulation in this population. We also tested for delayed density dependence using the theta logistic model. Because territorial animals are expected to respond quickly to the effects of density dependence, corresponding to theta , 1, our rejection of this hypothesis indicated that the restrictions of territorial sociospatial regulation are relaxed in this badger population. These results are used to highlight population vulnerabilities.

A note on density dependence in population models

One of the most studied phenomena in ecology is density dependent regulation. The model most frequently used to study this behaviour is the theta-logistic model. However, disagreement has developed within the ecology community pertaining to the interpretation of this model’s parameters, and thus as to appropriate values for the parameters to assume. In particular, the parameter θ has been allowed to take negative values, resulting in the ‘growth rate parameter’ estimated to be negative for species which are extant and exhibit no signs of becoming extinct in the short-term. Here we explain this phenomenon by formulating the theta-logistic model in the manner in which the original logistic model was formulated by Verhulst (1838), in doing so providing a simple interpretation of model parameters and thus restrictions on values the parameters may assume. We conclude that θ should (almost always) be restricted to values greater than − 1 . This has implications for studies assessing the form of density dependence from data. Additionally, another model appearing in the literature is presented which provides a more flexible model of density dependence at the expense of only one additional parameter.

Determining the Functional Form of Density Dependence: Deductive Approaches for Consumer‐Resource Systems Having a Single Resource

Consumer-resource models are used to deduce the functional form of density dependence in the consumer population. A general approach to determining the form of consumer density dependence is proposed; this involves determining the equilibrium (or average) population size for a series of different harvest rates. The relationship between a consumer's mortality and its equilibrium population size is explored for several one-consumer/one-resource models. The shape of density dependence in the resource and the shape of the numerical and functional responses all tend to be "inherited" by the consumer's density dependence. Consumer-resource models suggest that density dependence will very often have both concave and convex segments, something that is impossible under the commonly used theta-logistic model. A range of consumer-resource models predicts that consumer population size often declines at a decelerating rate with mortality at low mortality rates, is insensitive to or increases with mortality over a wide range of intermediate mortalities, and declines at a rapidly accelerating rate with increased mortality when mortality is high. This has important implications for management and conservation of natural populations.

Fixation probability of beneficial mutations in a fluctuating population

We compute an accurate approximation to the probability of fixation for a beneficial mutation in a population fluctuating with a stationary distribution of population size. The population dynamics are described by the theta-logistic model with environmental variance, assuming that the population size is large enough to ignore demographic variance. We show that stochastic fluctuations of population size reduce the probability of fixation. However, it is not the magnitude of the population fluctuations per se that creates this reduction. Only the environmental variance has a substantial effect on the probability of fixation. The strength of density dependence (or expected return time to equilibrium) and the functional form of density-regulation, given by the parameter theta in the theta-logistic model, have little effect on the fixation probability. Effective population size based on harmonic mean population size will therefore underestimate the expected fixation rate of beneficial mutations in fluctuating populations.

The effects of environmental perturbation and measurement error on estimates of the shape parameter in the theta-logistic model of population regulation

The theta-logistic is a widely used generalisation of the logistic model of regulated biological processes which is used in particular to model population regulation. Then the parameter theta gives the shape of the relationship between per-capita population growth rate and population size. Estimation of theta from population counts is however subject to bias, particularly when there are measurement errors. Here we identify factors disposing towards accurate estimation of theta by simulation of populations regulated according to the theta-logistic model. Factors investigated were measurement error, environmental perturbation and length of time series. Large measurement errors bias estimates of theta towards zero. Where estimated theta is close to zero, the estimated annual return rate may help resolve whether this is due to bias. Environmental perturbations help yield unbiased estimates of theta. Where environmental perturbations are large, estimates of theta are likely to be reliable even when measurement errors are also large. By contrast where the environment is relatively constant, unbiased estimates of theta can only be obtained if populations are counted precisely. Our results have practical conclusions for the design of long-term population surveys. Estimation of the precision of population counts would be valuable, and could be achieved in practice by repeating counts in at least some years. Increasing the length of time series beyond ten or 20 years yields only small benefits. If populations are measured with appropriate accuracy, given the level of environmental perturbation, unbiased estimates can be obtained from relatively short censuses. These conclusions are optimistic for estimation of theta.

A New BEAST: Bayesian Software Tools for Ecological Trend Analysis

Many software tools available to biologists are either case-specific or require the ability to write computer programs. The learning curve for newer Bayesian approaches is much steeper than traditional approaches, often requiring an investment of several years. This article introduces a new free software package, BEAST (Bayesian Ecological Analysis of Statistical Trends), for Bayesian estimation of population trends, which requires the user to do no programming. A wide range of single-species population dynamics models are supported, including single-stage population models, stage-structured population models, and age-structured population models. In addition to Bayesian parameter estimation, this software allows users to do population projections, and may be used as a teaching tool by agencies and universities. I provide an example of a theta-logistic population model of stock pigeon (Columba oenas) breeding pairs. The degree of density dependence for this population was found to be stronger when compared to maximum likelihood parameter estimates available from the literature. Short-term population projections indicate that the number of breeding pairs is relatively stable, indicating that the population is at (or near) carrying capacity. This result is in agreement with the most recent data available, which indicate the breeding counts have remained relatively constant.

On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.

TESTING THE IRRUPTIVE PARADIGM OF LARGE-HERBIVORE DYNAMICS

A dominant paradigm in understanding and managing large herbivores is that, after introduction to new range or release from harvesting, the herbivore population increases to peak abundance, crashes to a lower abundance, and then increases to a carrying capacity lower than peak abundance. However, support for the paradigm has been largely anecdotal. We first developed two mathematical models to better describe irruptive dynamics. The models differed in the form of the postcrash growth toward carrying capacity: the "Caughley model" included a time lag that generated dampening oscillations, and the "Leopold model" did not. We then evaluated which of four models (theta-logistic, delayed-logistic, Leopold, and Caughley) best described the dynamics of seven ungulate populations either introduced to new range (n = 5 populations) or released from harvesting (n = 2). The dynamics of six of the populations were best described by irruptive models (two by the Leopold, one by the Caughley, and three by the delayed-logistic), and one of the populations did not display irruptive dynamics (theta-logistic model). The limited data thus support the widespread existence of irruptive dynamics, and we encourage the consideration of irruptive models in studies of large-herbivore dynamics.