An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect

We consider a single-species model of population dynamics allowing for migrations and the Allee effect. Two types of migration are taken into account: one caused by environmental factors (e.g., a passive transport with the wind or water current) and the other associated with biological mechanisms. While the first type is apparently density-independent, the speed of migration in the second one can depend on the population density. Mathematically, this model consists of a non-linear partial differential equation of advection-diffusion-reaction type. Using an appropriate change of variables, we obtain an exact solution of the equation describing propagation of travelling population fronts. We show that, depending on parameter values and thus on the relative intensity of density-dependent and density-independent factors, the direction of the propagation can be different thus describing either species invasion or species retreat.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.     

Do Triplets Have Enough Information to Construct the Multi-Labeled Phylogenetic Tree?

The evolutionary history of certain species such as polyploids are modeled by a generalization of phylogenetic trees called multi-labeled phylogenetic trees, or MUL trees for short. One problem that relates to inferring a MUL tree is how to construct the smallest possible MUL tree that is consistent with a given set of rooted triplets, or SMRT problem for short. This problem is NP-hard. There is one algorithm for the SMRT problem which is exact and runs in time, where is the number of taxa. In this paper, we show that the SMRT does not seem to be an appropriate solution from the biological point of view. Indeed, we present a heuristic algorithm named MTRT for this problem and execute it on some real and simulated datasets. The results of MTRT show that triplets alone cannot provide enough information to infer the true MUL tree. So, it is inappropriate to infer a MUL tree using triplet information alone and considering the minimum number of duplications. Finally, we introduce some new problems which are more suitable from the biological point of view.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows

A key problem in environmental flow assessment is the explicit linking of the flow regime with ecological dynamics. We present a hybrid modeling approach to couple hydrodynamic and biological processes, focusing on the combined impact of spatial heterogeneity and temporal variability on population dynamics. Studying periodically alternating pool-riffle rivers that are subjected to seasonally varying flows, we obtain an invasion ratchet mechanism. We analyze the ratchet process for a caricature model and a hybrid physical–biological model. The water depth and current are derived from a hydrodynamic equation for variable stream bed water flows and these quantities feed into a reaction-diffusion-advection model that governs population dynamics of a river species. We establish the existence of spreading speeds and the invasion ratchet phenomenon, using a mixture of mathematical approximations and numerical computations. Finally, we illustrate the invasion ratchet phenomenon in a spatially two-dimensional hydraulic simulation model of a meandering river structure. Our hybrid modeling approach strengthens the ecological component of stream hydraulics and allows us to gain a mechanistic understanding as to how flow patterns affect population survival.     

Coexistence and Spread of Competitors in Heterogeneous Landscapes

Competition between species is ubiquitous in nature and therefore widely studied in ecology through experiment and theory. One of the central questions is under which conditions a (rare) invader can establish itself in a landscape dominated by a resident species at carrying capacity. Applying the same question with the roles of the invader and resident reversed leads to the principle that “mutual invasibility implies coexistence.” A related but different question is how fast a locally introduced invader spreads into a landscape (with or without competing resident), provided it can invade. We explore some aspects of these questions in a deterministic, spatially explicit model for two competing species with discrete non-overlapping generations in a patchy periodic environment. We obtain threshold values for fragmentation levels and dispersal distances that allow for mutual invasion and coexistence even if the non-spatial competition model predicts competitive exclusion. We obtain exact results when dispersal is governed by a Laplace kernel. Using the average dispersal success, we develop a mathematical framework to obtain approximate results that are independent of the exact dispersal patterns, and we show numerically that these approximations are very accurate.     

Anticooperativity in diffusion-controlled reactions with pairs of anisotropic domains: a model for the antigen–antibody encounter

The encounter between anisotropic agents in diffusion-controlled reactions is a topic of very general relevance in chemistry and biology. Here we introduce a simplified model of encounter of an isotropic molecule with a pair of partially reacting agents and apply it to the encounter reaction between an antibody and its antigen. We reduce the problem to the solution of dual series relations, which can be solved iteratively, yielding the exact solution for the encounter rate constant at any desired order of accuracy. We quantify the encounter effectiveness by means of a simple indicator and show that the two binding centers systematically behave in an anti-cooperative fashion. However, we demonstrate that a reduction of the binding active sites allows the composite molecule to recover binding effectiveness, in spite of the overall reduction of the rate constant. In addition, we provide a simple formula that enables one to calculate the anti-cooperativity as a function of the size of the binding site for any values of the separation between the two active lobes and of the antigen size. Finally, some biological implications of our results are discussed.     

Picking alignments from (Steiner) trees.

The application of Needleman-Wunsch alignment techniques to biological sequences is complicated by two serious problems when the sequences are long: the running time, which scales as the product of the lengths of sequences, and the difficulty in obtaining suitable parameters that produce meaningful alignments. The running time problem is often corrected by reducing the search space, using techniques such as banding, or chaining of high-scoring pairs. The parameter problem is more difficult to fix, partly because the probabilistic model, which Needleman-Wunsch is equivalent to, does not capture a key feature of biological sequence alignments, namely the alternation of conserved blocks and seemingly unrelated nonconserved segments. We present a solution to the problem of designing efficient search spaces for pair hidden Markov models that align biological sequences by taking advantage of their associated features. Our approach leads to an optimization problem, for which we obtain a 2-approximation algorithm, and that is based on the construction of Manhattan networks, which are close relatives of Steiner trees. We describe the underlying theory and show how our methods can be applied to alignment of DNA sequences in practice, successfully reducing the Viterbi algorithm search space of alignment PHMMs by three orders of magnitude.     

Influences of size and sex on invasive species aggression and native species vulnerability: a case for modern regression techniques

Animal behaviour is of fundamental importance but is often overlooked in biological invasion research. A problem with such studies is that they may add pressure to already threatened species and subject vulnerable individuals to increased risk. One solution is to obtain the maximum possible information from the generated data using a variety of statistical techniques, instead of solely using simple versions of linear regression or generalized linear models as is customary. Here, we exemplify and compare the use of modern regression techniques which have very different conceptual backgrounds and aims (negative binomial models, zero-inflated regression, and expectile regression), and which have rarely been applied to behavioural data in biological invasion studies. We show that our data display overdispersion, which is frequent in ecological and behavioural data, and that conventional statistical methods such as Poisson generalized linear models are inadequate in this case. Expectile regression is similar to quantile regression and allows the estimation of functional relationships between variables for all portions of a probability distribution and is thus well suited for modelling boundaries in polygonal relationships or cases with heterogeneous variances which are frequent in behavioural data. We applied various statistical techniques to aggression in invasive mosquitofish, Gambusia holbrooki, and the concomitant vulnerability of native toothcarp, Aphanius iberus, in relation to individual size and sex. We found that medium sized male G. holbrooki carry out the majority of aggressive acts and that smaller and medium size A. iberus are most vulnerable. Of the regression techniques used, only negative binomial models and zero-inflated and expectile Poisson regressions revealed these relationships.     

A Geometric Buildup Algorithm for the Solution of the Distance Geometry Problem Using Least-Squares Approximation

We propose a new geometric buildup algorithm for the solution of the distance geometry problem in protein modeling, which can prevent the accumulation of the rounding errors in the buildup calculations successfully and also tolerate small errors in given distances. In this algorithm, we use all instead of a subset of available distances for the determination of each unknown atom and obtain the position of the atom by using a least-squares approximation instead of an exact solution to the system of distance equations. We show that the least-squares approximation can be obtained by using a special singular value decomposition method, which not only tolerates and minimizes small distance errors, but also prevents the rounding errors from propagation effectively, especially when the distance data is sparse. We describe the least-squares formulations and their solution methods, and present the test results from applying the new algorithm for the determination of a set of protein structures with varying degrees of availability and accuracy of the distances. We show that the new development of the algorithm increases the modeling ability, and improves stability and robustness of the geometric buildup approach significantly from both theoretical and practical points of view.     

Opera: reconstructing optimal genomic scaffolds with high-throughput paired-end sequences

Scaffolding, the problem of ordering and orienting contigs, typically using paired-end reads, is a crucial step in the assembly of high-quality draft genomes. Even as sequencing technologies and mate-pair protocols have improved significantly, scaffolding programs still rely on heuristics, with no guarantees on the quality of the solution. In this work, we explored the feasibility of an exact solution for scaffolding and present a first tractable solution for this problem (Opera). We also describe a graph contraction procedure that allows the solution to scale to large scaffolding problems and demonstrate this by scaffolding several large real and synthetic datasets. In comparisons with existing scaffolders, Opera simultaneously produced longer and more accurate scaffolds demonstrating the utility of an exact approach. Opera also incorporates an exact quadratic programming formulation to precisely compute gap sizes (Availability: http://sourceforge.net/projects/operasf/ ).     

Effect of pooling samples on the efficiency of comparative studies using microarrays

MOTIVATION: Many biomedical experiments are carried out by pooling individual biological samples. However, pooling samples can potentially hide biological variance and give false confidence concerning the data significance. In the context of microarray experiments for detecting differentially expressed genes, recent publications have addressed the problem of the efficiency of sample pooling, and some approximate formulas were provided for the power and sample size calculations. It is desirable to have exact formulas for these calculations and have the approximate results checked against the exact ones. We show that the difference between the approximate and the exact results can be large. RESULTS: In this study, we have characterized quantitatively the effect of pooling samples on the efficiency of microarray experiments for the detection of differential gene expression between two classes. We present exact formulas for calculating the power of microarray experimental designs involving sample pooling and technical replications. The formulas can be used to determine the total number of arrays and biological subjects required in an experiment to achieve the desired power at a given significance level. The conditions under which pooled design becomes preferable to non-pooled design can then be derived given the unit cost associated with a microarray and that with a biological subject. This paper thus serves to provide guidance on sample pooling and cost-effectiveness. The formulation in this paper is outlined in the context of performing microarray comparative studies, but its applicability is not limited to microarray experiments. It is also applicable to a wide range of biomedical comparative studies where sample pooling may be involved.     

Diffusion exchange between a membrane-bounded sphere and its surrounding

The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.     

Exact sample sizes needed to detect dependence in 2 x 3 tables

Many medical and biological studies entail classifying a number of observations according to two factors, where one has two and the other three possible categories. This is the case of, for example, genetic association studies of complex traits with single-nucleotide polymorphisms (SNPs), where the a priori statistical planning, analysis, and interpretation of results are of critical importance. Here, we present methodology to determine the minimum sample size required to detect dependence in 2 x 3 tables based on Fisher's exact test, assuming that neither of the two margins is fixed and only the grand total N is known in advance. We provide the numerical tools necessary to determine these sample sizes for desired power, significance level, and effect size, where only the computational time can be a limitation for extreme parameter values. These programs can be accessed at . This solution of the sample size problem for an exact test will permit experimentalists to plan efficient sampling designs, determine the extent of statistical support for their hypotheses, and gain insight into the repeatability of their results. We apply this solution to the sample size problem to three empirical studies, and discuss the results with specified power and nominal significance levels.     

Dichotomous noise with feedback and charge-conformational interactions

The states of the charge transfer chain fragment depending on the charge localization are modelled by interaction with the two-level system as a source of dichotomous noise whose features depend themselves on the fragment states. Basing on the exact solution of the stochastic problem, we describe the noise-induced phase transitions of the (mono-bi)stability type which are possible even in the simplest case of the fragment represented by an overdamped harmonic oscillator. The results are discussed in the context of the functional nonlinearities in biological charge transport processes.     

模拟酸雨对入侵植物豚草与伴生种鬼针草竞争关系的影响

植物种间相互作用对物种的生存至关重要。对于入侵植物而言, 更强的种间竞争能力是其成功入侵的重要机制。然而, 环境条件的变化可能会改变种间关系并影响植物入侵的最终结果。该研究主要探索当今严重的环境问题之一——酸雨对中国入侵植物豚草(Ambrosia artemisiifolia)和伴生本地种鬼针草(Bidens bipinnata)种间相互作用的影响, 进一步了解环境扰动对生物入侵的影响。该研究于2021年3月在北京师范大学室外条件下对豚草和鬼针草开展de Wit替代竞争实验, 并施加不同浓度的酸雨模拟溶液(pH = 3、4、5、7), 记录植株在第24、34、45天的株高以及生长季结束时的株高和地上生物量, 通过计算相对邻株效应指数(RNE)并绘制取代系列图表来评估两者的竞争关系。主要结果如下: 单一种植时, 中浓度酸雨(pH = 4)促进了豚草和鬼针草的早期生长, 而高浓度酸雨(pH = 3)显著抑制了两者的早期生长, 但不影响生长季结束时的株高。混合种植时, 高浓度酸雨(pH = 3)显著降低了豚草在第34、45天时的株高, 但在生长季结束时对豚草无影响, 而鬼针草在生长季结束时株高显著降低。鬼针草相对豚草的RNE在任何酸浓度下都不显著。而豚草相对鬼针草的RNE在无酸雨条件下显著, 酸雨处理时在鬼针草比例较低时显著。取代系列实验图表显示, 低浓度酸雨(pH = 5)增加了在低豚草比例下的鬼针草的竞争优势, 而高浓度酸雨增强了豚草的竞争优势。该研究表明酸雨显著影响了豚草和鬼针草的生长及种间竞争关系, 高浓度酸雨增加了入侵植物豚草的竞争优势。     

Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

Unbiased bootstrap error estimation for linear discriminant analysis

Convex bootstrap error estimation is a popular tool for classifier error estimation in gene expression studies. A basic question is how to determine the weight for the convex combination between the basic bootstrap estimator and the resubstitution estimator such that the resulting estimator is unbiased at finite sample sizes. The well-known 0.632 bootstrap error estimator uses asymptotic arguments to propose a fixed 0.632 weight, whereas the more recent 0.632+ bootstrap error estimator attempts to set the weight adaptively. In this paper, we study the finite sample problem in the case of linear discriminant analysis under Gaussian populations. We derive exact expressions for the weight that guarantee unbiasedness of the convex bootstrap error estimator in the univariate and multivariate cases, without making asymptotic simplifications. Using exact computation in the univariate case and an accurate approximation in the multivariate case, we obtain the required weight and show that it can deviate significantly from the constant 0.632 weight, depending on the sample size and Bayes error for the problem. The methodology is illustrated by application on data from a well-known cancer classification study.     

Species Invasiveness in Biological Invasions: A Modelling Approach

The study of invasiveness, the traits that enable a species to invade a habitat, and invasibility, the habitat characteristics that determine its susceptibility to the establishment and spread of an invasive species, provide a useful conceptual framework to formulate the biological invasion problem in a modelling context. Another important aspect is the complex interaction emerging among the invader species, the noninvader species already present in the habitat, and the habitat itself. Following a modelling approach to the biological invasion problem, we present a spatially explicit cellular automaton model (Interacting Multiple Cellular Automata (IMCA)). We use field parameters from the invader Gleditsia triacanthos and the native Lithraea ternifolia in montane forests of central Argentina as a case study to compare outputs and performance of different models. We use field parameters from another invader, Ligustrum lucidum, and the native Fagara coco from the same system to run the cellular automaton model. We compare model predictions with invasion values from aerial photographs. We discuss in detail the importance of factors affecting species invasiveness, and give some insights into habitat invasibility and the role of interactions between them. Finally, we discuss the relevance of mathematical modelling for studying and predicting biological invasions. The IMCA model provided a suitable context for integrating invasiveness, invasibility, and the interactions. In the invasion system studied, the presence of an invader's juvenile bank not only accelerated the rate of invasion but was essential to ensure invasion. Using the IMCA model, we were able to determine that not only adult survival but particularly longevity of the native species influenced the spread velocity of the invader, at least when a juvenile bank is present. Other factors determining velocity of invasion detected by the IMCA model were seed dispersal distance and age of reproductive maturity. We derived relationships between species' adult survival, fecundity and longevity of both theoretical and applied relevance for biological invasions. Invasion velocities calculated from the aerial photographs agreed well with predictions of the IMCA model.     

A note on a more general solution of Eigen's rate equation for selection

In this note we examine Eigen's nonlinear rate equations for the relation of coupled biomacromolecules. We obtain an exact solution to the equations for constant overall population densities and constant rate parameters. We conclude that there is only one stationary solution where all molecular species coexist when they are coupled by mutation.