One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.     

Global stability of population models

Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.     

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.     

Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense simple, and that this simplicity causes local and global stability to coincide.     

Stability analysis of within-host parasite models with delays

We provide a global analysis of systems of within-host parasitic infections. The systems studied have parallel classes of different length of latently infected target cells. These systems can also be thought as systems arising from within-host parasitic systems with distributed continuous delays. We compute the basic reproduction ratio R0 for the systems under consideration. If R0< or =1 the parasite is cleared, if R0>1 and if a sufficient condition is satisfied we conclude to the global asymptotic stability (GAS) of the endemic equilibrium. For some generic class of models this condition reduces to R0>1. These results make possible to revisit some parasitic models including intracellular delays and to study their global stability.     

A general approach for two-stage analysis of multilevel clustered non-Gaussian data

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.     

Global stability in discrete population models with delayed-density dependence

We address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker's and Pielou's equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model.     

再论生物多样性与生态系统的稳定性

本文在简述生物多样性与生态系统稳定性研究动态的基础上,从生物多样性和稳定性的概念出发,指出忽视多样性和稳定性的生物组织层次可能是造成观点纷争的根源之一。特定生物组织层次的稳定性可能更多地与该层次的多样性特征相关,探讨多样性和稳定性的关系应从不同的生物组织层次上进行,抗动是生态系统多样性与稳定性关系悖论中的重要因子,如果根据扰动的性质,把生态系统（或其他组织层次）区分为受非正常外力干扰和受环境因子时间异质性波动干扰2类系统,稳定性的4个内涵可以理解为：对于受非正常外力干扰的系统而言,抵抗力和恢复力是稳定性适宜的测度指标;对于受环境因子时间异质性波动干扰和系统而言。利用持久性和变异性衡量系统的稳定性则更具实际意义。结合对群落和种群层次多样性与稳定性相关机制的初步讨论,本文认为;在特定的前提下,多样性可以导致稳定性。     

A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks

Reaction networks are systems in which the populations of a finite number of species evolve through predefined interactions. Such networks are found as modeling tools in many biological disciplines such as biochemistry, ecology, epidemiology, immunology, systems biology and synthetic biology. It is now well-established that, for small population sizes, stochastic models for biochemical reaction networks are necessary to capture randomness in the interactions. The tools for analyzing such models, however, still lag far behind their deterministic counterparts. In this paper, we bridge this gap by developing a constructive framework for examining the long-term behavior and stability properties of the reaction dynamics in a stochastic setting. In particular, we address the problems of determining ergodicity of the reaction dynamics, which is analogous to having a globally attracting fixed point for deterministic dynamics. We also examine when the statistical moments of the underlying process remain bounded with time and when they converge to their steady state values. The framework we develop relies on a blend of ideas from probability theory, linear algebra and optimization theory. We demonstrate that the stability properties of a wide class of biological networks can be assessed from our sufficient theoretical conditions that can be recast as efficient and scalable linear programs, well-known for their tractability. It is notably shown that the computational complexity is often linear in the number of species. We illustrate the validity, the efficiency and the wide applicability of our results on several reaction networks arising in biochemistry, systems biology, epidemiology and ecology. The biological implications of the results as well as an example of a non-ergodic biological network are also discussed.     

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.     

Population Dynamics Modelling in an Hierarchical Arborescent River Network: An Attempt with Salmo trutta

The balance between births and deaths in an age-structured population is strongly influenced by the spatial distribution of sub-populations. Our aim was to describe the demographic process of a fish population in an hierarchical dendritic river network, by taking into account the possible movements of individuals. We tried also to quantify the effect of river network changes (damming or channelling) on the global fish population dynamics. The Salmo trutta life pattern was taken as an example for.We proposed a model which includes the demographic and the migration processes, considering migration fast compared to demography. The population was divided into three age-classes and subdivided into fifteen spatial patches, thus having 45 state variables. Both processes were described by means of constant transfer coefficients, so we were dealing with a linear system of difference equations. The discrete case of the variable aggregation method allowed the study of the system through the dominant elements of a much simpler linear system with only three global variables: the total number of individuals in each age-class.From biological hypothesis on demographic and migratory parameters, we showed that the global population dynamics of fishes is well characterized in the reference river network, and that dams could have stronger effects on the global dynamics than channelling.     

Optimal control of hybrid variable-order fractional coronavirus (2019-nCov) mathematical model; numerical treatments

A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt.     

Parameter inference for an individual based model of chytridiomycosis in frogs

Individual based models (IBMs) and Agent based models (ABMs) have become widely used tools to understand complex biological systems. However, general methods of parameter inference for IBMs are not available. In this paper we show that it is possible to address this problem with a traditional likelihood-based approach, using an example of an IBM developed to describe the spread of chytridiomycosis in a population of frogs as a case study. We show that if the IBM satisfies certain criteria we can find the likelihood (or posterior) analytically, and use standard computational techniques, such as MCMC, for parameter inference.     

THE WELL-POSEDNESS OF SIZE STRUCTURED POPULATION MODELS WITH DENSITY DEPENDENT PARAMETERS

Thewell-posednessofnonlinearsizestructuredpopulationmodelsisstudied.Thenonlinearitiesareintroducedbyassumingthevitalparameters(thebirthrate,thedeathrate,andthegrowthrate)tobedensitydependent.TheidealadoptedhereisbasedonthemethodofGurtinandMacCamy[4]usedfornonlinearage-dependentpopulationmodels.Thenetreproductivenumberisintroducedandusedtodeterminethelocalandglobalstabilityoftrivialequilibrium.Thestabilityconditionsoftrivialequilibriumareobtained.     

A statistical approach to quasi-extinction forecasting

Forecasting population decline to a certain critical threshold (the quasi-extinction risk) is one of the central objectives of population viability analysis (PVA), and such predictions figure prominently in the decisions of major conservation organizations. In this paper, we argue that accurate forecasting of a population's quasi-extinction risk does not necessarily require knowledge of the underlying biological mechanisms. Because of the stochastic and multiplicative nature of population growth, the ensemble behaviour of population trajectories converges to common statistical forms across a wide variety of stochastic population processes. This paper provides a theoretical basis for this argument. We show that the quasi-extinction surfaces of a variety of complex stochastic population processes (including age-structured, density-dependent and spatially structured populations) can be modelled by a simple stochastic approximation: the stochastic exponential growth process overlaid with Gaussian errors. Using simulated and real data, we show that this model can be estimated with 20-30 years of data and can provide relatively unbiased quasi-extinction risk with confidence intervals considerably smaller than (0,1). This was found to be true even for simulated data derived from some of the noisiest population processes (density-dependent feedback, species interactions and strong age-structure cycling). A key advantage of statistical models is that their parameters and the uncertainty of those parameters can be estimated from time series data using standard statistical methods. In contrast for most species of conservation concern, biologically realistic models must often be specified rather than estimated because of the limited data available for all the various parameters. Biologically realistic models will always have a prominent place in PVA for evaluating specific management options which affect a single segment of a population, a single demographic rate, or different geographic areas. However, for forecasting quasi-extinction risk, statistical models that are based on the convergent statistical properties of population processes offer many advantages over biologically realistic models.     

Pattern formation in prey-taxis systems

In this paper, we consider spatial predator–prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.     

New stability criterion of neural networks with leakage delays and impulses: a piecewise delay method

This paper analyzes the global asymptotic stability of a class of neural networks with time delay in the leakage term and time-varying delays under impulsive perturbations. Here the time-varying delays are assumed to be piecewise. In this method, the interval of the variation is divided into two subintervals by its central point. By developing a new Lyapunov–Krasovskii functional and checking its variation in between the two subintervals, respectively, and then we present some sufficient conditions to guarantee the global asymptotic stability of the equilibrium point for the considered neural network. The proposed results which do not require the boundedness, differentiability and monotonicity of the activation functions, can be easily verified via the linear matrix inequality (LMI) control toolbox in MATLAB. Finally, a numerical example and its simulation are given to show the conditions obtained are new and less conservative than some existing ones in the literature.     

The role of seasonality in the dynamics of deer tick populations

In this paper, we formulate a nonlinear system of difference equations that models the three-stage life cycle of the deer tick over four seasons. We study the effect of seasonality on the stability and oscillatory behavior of the tick population by comparing analytically the seasonal model with a non-seasonal one. The analysis of the models reveals the existence of two equilibrium points. We discuss the necessary and sufficient conditions for local asymptotic stability of the equilibria and analyze the boundedness and oscillatory behavior of the solutions. A main result of the mathematical analysis is that seasonality in the life cycle of the deer tick can have a positive effect, in the sense that it increases the stability of the system. It is also shown that for some combination of parameters within the stability region, perturbations will result in a return to the equilibrium through transient oscillations. The models are used to explore the biological consequences of parameter variations reflecting expected environmental changes.     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

Estimating quality of template-based protein models by alignment stability

The error in protein tertiary structure prediction is unavoidable, but it is not explicitly shown in most of the current prediction algorithms. Estimated error of a predicted structure is crucial information for experimental biologists to use the prediction model for design and interpretation of experiments. Here, we propose a method to estimate errors in predicted structures based on the stability of the optimal target-template alignment when compared with a set of suboptimal alignments. The stability of the optimal alignment is quantified by an index named the SuboPtimal Alignment Diversity (SPAD). We implemented SPAD in a profile-based threading algorithm and investigated how well SPAD can indicate errors in threading models using a large benchmark dataset of 5232 alignments. SPAD shows a very good correlation not only to alignment shift errors but also structure-level errors, the root mean square deviation (RMSD) of predicted structure models to the native structures (i.e. global errors), and local errors at each residue position. We have further compared SPAD with seven other quality measures, six from sequence alignment-based measures and one atomic statistical potential, discrete optimized protein energy (DOPE), in terms of the correlation coefficient to the global and local structure-level errors. In terms of the correlation to the RMSD of structure models, when a target and a template are in the same SCOP family, the sequence identity showed a best correlation to the RMSD; in the superfamily level, SPAD was the best; and in the fold level, DOPE was best. However, in a head-to-head comparison, SPAD wins over the other measures. Next, SPAD is compared with three other measures of local errors. In this comparison, SPAD was best in all of the family, the superfamily and the fold levels. Using the discovered correlation, we have also predicted the global and local error of our predicted structures of CASP7 targets by the SPAD. Finally, we proposed a sausage representation of predicted tertiary structures which intuitively indicate the predicted structure and the estimated error range of the structure simultaneously.