Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion

In this paper, we first propose a prey-predator model with prey-stage structure and diffusion. Then we discuss the following three problems: (1) stability of non-negative constant steady states for the reduced ODE system and the corresponding reaction diffusion system with homogeneous Neumann boundary conditions; (2) Hopf bifurcation for the ODE system; (3) Hopf bifurcation created by diffusion.     

A generalized diffusion model for growth and dispersal in a population

A reaction-diffusion model is presented in which spatial structure is maintained by means of a diffusive mechanism more general than classical Fickian diffusion. This generalized diffusion takes into account the diffusive gradient (or gradient energy) necessary to maintain a pattern even in a single diffusing species. The approach is based on a Landau-Ginzburg free energy model. A problem involving simple logistic kinetics is fully analyzed, and a nonlinear stability analysis based on a multi-scale perturbation method shows bifurcation to non-uniform states.Part of this work was done while at the Mathematical Institute, Oxford University as a Senior Visiting Fellow supported by the Science Research Council of Great Britain under grant GR/B31378     

A diffusion model for the fate of tandem gene duplicates in diploids

Suppose one chromosome in one member of a population somehow acquires a duplicate copy of the gene, fully linked to the original gene's locus. Preservation is the event that eventually every chromosome in the population is a descendant of the one which initially carried the duplicate. For a haploid population in which the absence of all copies of the gene is lethal, the probability of preservation has recently been estimated via a diffusion approximation. That approximation is shown to carry over to the case of diploids and arbitrary strong selection against the absence of the gene. The techniques used lead to some new results. In the large population limit, it is shown that the relative probability that descendants of a small number of individuals carrying multiple copies of the gene fix in the population is proportional to the number of copies carried. The probability of preservation is approximated when chromosomes carrying two copies of the gene are subject to additional, fully non-functionalizing mutations, thereby modelling either an additional cost of replicating a longer genome, or a partial duplication of the gene. In the latter case the preservation probability depends only on the mutation rate to null for the duplicated portion of the gene.     

The relationship between endophyte transition and plant species coexistence

The survival possibilities of terrestrial plant species are determined by their competitive abilities. One factor that affects competitive ability is the community of microorganisms that lives in association with the plants. Microorganisms affect the competitive dominance among plants by means of their metabolites. In this paper, we study the multiple plant species coexistence mediated by interactions with endophytes (fungi). The population dynamics are described by a revised lottery competition model for multiple plant species, each of which is divided into two classes: plants with endophytes (denoted EP) and plants without endophytes (NEP). The model includes the transition of seeds from EP to NEP. We show multiple species of plants cannot coexist in a steady state if this transition is density independent, but can coexist in a steady state if this transition is an increasing function of population density.     

森林种群径阶转移模型中转移概率的估算方法

基于统计分析理论和微分方程理论,给出了森林种群径阶转移模型中估算转移概率的方法：第一种是在有两次样地观测数据,不考虑林分环境因子等因素的条件下估算转移概率;第二种是在已知林分环境因子条件下,不需要对样地有两次观测数据来估算转移概率．实例验证结果表明,两种估算转移概率的方法具有计算简单和实用性强的特点,对森林经营与管理有一定的理论指导和实际应用价值．     

一种模拟昆虫种群动态的改进的变维矩阵模型

提出了一种模拟昆虫种群动态的改进的变维矩阵模型,该模型以发有历期为维数,采用分解与合成的方法变维,并考虑了个体间的发育差异。经模拟检验,模型模拟结果略优于徐汝梅等（1981）变维矩阵模型的结果。     

Comparison and content of the Wright-Fisher model of random genetic drift, the diffusion approximation, and an intermediate model

We investigate the detailed connection between the Wright-Fisher model of random genetic drift and the diffusion approximation, under the assumption that selection and drift are weak and so cause small changes over a single generation. A representation of the mathematics underlying the Wright-Fisher model is introduced which allows the connection to be made with the corresponding mathematics underlying the diffusion approximation. Two ‘hybrid’ models are also introduced which lie ‘between’ the Wright-Fisher model and the diffusion approximation. In model 1 the relative allele frequency takes discrete values while time is continuous; in model 2 time is discrete and relative allele frequency is continuous. While both hybrid models appear to have a similar status and the same level of plausibility, the different nature of time and frequency in the two models leads to significant mathematical differences. Model 2 is mathematically inconsistent and has to be ruled out as being meaningful. Model 1 is used to clarify the content of Kimura's solution of the diffusion equation, which is shown to have the natural interpretation as describing only those populations where alleles are segregating. By contrast the Wright-Fisher model and the solution of the diffusion equation of McKane and Waxman cover populations of all categories, namely populations where alleles segregate, are lost, or fix.     

The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics, Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig’s result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.     

Simulation of diffusion of O 2 and CO 2 in amorphous poly(ethylene terephthalate) and related alkylene and isomeric polyesters

The diffusion of small molecules through polymers is important in many areas of polymer science, such as gas barrier and separation membrane materials, polymeric foams, and in the processing and properties of polymers. Molecular simulation techniques have been applied to study the diffusion of oxygen and carbon dioxide as small molecule penetrants in models of bulk amorphous poly(ethylene terephthalate) (PET) and related alkylene and isomeric polyesters. A bulk amorphous configuration with periodic boundary conditions made into a unit cell whose dimensions were determined for each of the simulated polyesters in the cell having the experimental density. The diffusion coefficients for O 2 and CO 2 were determined via NVE molecular dynamics simulations using the Dreiding 2.21 molecular mechanics force field over a range of temperatures (300, 500 and 600 K) using up to 3 ns simulation time. We have focussed on the influence of the temperature, polymer dynamics, number of CH 2 groups, density and free volume distribution on the diffusion properties. Correlation of diffusion coefficients with free volume and number of CH 2 groups was found.     

A parity-structured matrix model for tsetse populations

A matrix model is used to describe the dynamics of a population of female tsetse flies structured by parity (i.e., by the number of larvae laid). For typical parameter values, the intrinsic growth rate of the population is zero when the adult daily survival rate is 0.970, corresponding to an adult life expectancy of 1/0.030 = 33.3 days. This value is plausible and consistent with results found earlier by others. The intrinsic growth rate is insensitive to the variance of the interlarval period. Temperature being a function of the time of the year, a known relationship between temperature and mean pupal and interlarval times was used to produce a time-varying version of the model which was fitted to temperature and (estimated) population data. With well-chosen parameter values, the modeled population replicated at least roughly the population data. This illustrates dynamically the abiotic effect of temperature on population growth. Given that tsetse flies are the vectors of trypanosomiasis ("sleeping sickness") the model provides a framework within which future transmission models can be developed in order to study the impact of altered temperatures on the spread of this deadly disease.     

The relationship between body size and population size in bromeliad tank faunas

Relationships between body size and abundance in collections of animals from the tanks of 73 bromeliads belonging to five species were analysed. Unlike data in previously published studies on this relationship, these collections of species are not taxonomically restricted and represent complete communities over the macroscopic range of organisms. There is no overall tendency for there to be a positive or negative relationship between population abundance and body size of morphospecies. We can find no evidence that body size-abundance patterns are triangular in complete communities. However, there is weak evidence that the relationship in the aquatic subsets of those communities may have some underlying triangularity, with medium-sized species having the largest populations.     

Probability distribution of haplotype frequencies under the two-locus Wright-Fisher model by diffusion approximation

The probability distribution of haplotype frequencies in a population, and the way it is influenced by genetical forces such as recombination, selection, random drift ...is a question of fundamental interest in population genetics. For large populations, the distribution of haplotype frequencies for two linked loci under the classical Wright-Fisher model is almost impossible to compute because of numerical reasons. However the Wright-Fisher process can in such cases be approximated by a diffusion process and the transition density can then be deduced from the Kolmogorov equations. As no exact solution has been found for these equations, we developed a numerical method based on finite differences to solve them. It applies to transient states and models including selection or mutations. We show by several tests that this method is accurate for computing the conditional joint density of haplotype frequencies given that no haplotype has been lost. We also prove that it is far less time consuming than other methods such as Monte Carlo simulations.     

黄酒中酒液粒度与混浊沉淀的关系

对煎酒前后酒液品质进行了分析,并应用９０°光散射实验考查煎酒前后,储存阶段酒液平均粒度和酒液沉淀物的平均粒度,发现酒液中沉淀物的出现和酒液平均粒径的一定大小存在耦合,当酒液平均径达到一定水平,沉淀过程就逐渐开始。     

The relationship between active transport and the exchange diffusion effect

Dependences of unidirectional ionic fluxes across biological membranes on the trans concentrations of the same ion, commonly described as exchange diffusion, and the association of this phenomenon with active transport, are noted. It is suggested that this effect could arise as a result of energetic coupling between the movement of ions conveyed in each direction by the pump if the latter operates near thermodynamic equilibrium and if the rate of the energizing reactions are restricted. This hypothesis is supported by an analysis in which the transport step and the energizing reactions are separated and described according to the laws of chemical kinetics. A likely cause for such restriction of the maximum rate of energy supply is shown to lie in evolutionary optimization of the efficiency of active transport if the energizing reaction is not perfectly coupled. Similar optimization will produce gross ionic fluxes large compared with the net flux, especially if the transport step approaches perfect coupling, when restriction of the rate of energy supply will cause a large exchange diffusion effect. The range of validity of the analysis is examined with particular reference to the ionic exchanges between osmoregulating animals and their surroundings.     

A multi-scale mechanobiological model of in-stent restenosis: deciphering the role of matrix metalloproteinase and extracellular matrix changes

Since their first introduction, stents have revolutionised the treatment of atherosclerosis; however, the development of in-stent restenosis still remains the Achilles' heel of stent deployment procedures. Computational modelling can be used as a means to model the biological response of arteries to different stent designs using mechanobiological models, whereby the mechanical environment may be used to dictate the growth and remodelling of vascular cells. Changes occurring within the arterial wall due to stent-induced mechanical injury, specifically changes within the extracellular matrix, have been postulated to be a major cause of activation of vascular smooth muscle cells and the subsequent development of in-stent restenosis. In this study, a mechanistic multi-scale mechanobiological model of in-stent restenosis using finite element models and agent-based modelling is presented, which allows quantitative evaluation of the collagen matrix turnover following stent-induced arterial injury and the subsequent development of in-stent restenosis. The model is specifically used to study the influence of stent deployment diameter and stent strut thickness on the level of in-stent restenosis. The model demonstrates that there exists a direct correlation between the stent deployment diameter and the level of in-stent restenosis. In addition, investigating the influence of stent strut thickness using the mechanobiological model reveals that thicker strut stents induce a higher level of in-stent restenosis due to a higher extent of arterial injury. The presented mechanobiological modelling framework provides a robust platform for testing hypotheses on the mechanisms underlying the development of in-stent restenosis and lends itself for use as a tool for optimisation of the mechanical parameters involved in stent design.     

Exact solution of the multi-allelic diffusion model

We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus, this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.     

Phospholipids diffusion on the surface of model lipid droplets

Lipid droplets (LD) are organelles localized in the membrane of the Endoplasmic Reticulum (ER) that play an important role in metabolic functions. They consist of a core of neutral lipids surrounded by a monolayer of phosphoplipids and proteins resembling an oil-in-water emulsion droplet. Many studies have focused on the biophysical properties of these LDs. However, despite numerous efforts, we are lacking information on the mobility of phospholipids on the LDs surface, although they may play a key role in the protein distribution. In this article, we developed a microfluidic setup that allows the formation of a triolein–buffer interface decorated with a phospholipid monolayer. Using this setup, we measured the motility of phospholipid molecules by performing Fluorescent Recovery After Photobleaching (FRAP) experiments for different lipidic compositions. The results of the FRAP measurements reveal that the motility of phospholipids is controlled by the monolayer packing decorating the interface.     

The reaction force and the transition region of a reaction

The reaction force F(R) and the position-dependent reaction force constant κF(R) are defined by F(R)=-∂V(R)/∂R and κ(R)=∂²V(R)/∂R², where V(R) is the potential energy of a reacting system along a coordinate R. The minima and maxima of F(R) provide a natural division of the process into several regions. Those in which F(R) is increasing are where the most dramatic changes in electronic properties take place, and where the system goes from activated reactants (at the force minimum) to activated products (at the force maximum). κ(R) is negative throughout such a region. We summarize evidence supporting the idea that a reaction should be viewed as going through a transition region rather than through a single point transition state. A similar conclusion has come out of transition state spectroscopy. We describe this region as a chemically-active, or electronically-intensive, stage of the reaction, while the ones that precede and follow it are structurally-intensive. Finally, we briefly address the time dependence of the reaction force and the reaction force constant.     

Energy expenditure and cardiorespiratory responses at the transition between walking and running

We investigated whether the spontaneous transition between walking and running during moving with increasing speed corresponds to the speed at which walking becomes less economical than running. Seven active male subjects [mean age, 23.7 (SEM 0.7) years, mean maximal oxygen uptake ( ), 57.5 (SEM 3.3) ml·kg ^–1·min ^–1, mean ventilatory threshold (VTh), 37.5 (SEM 3) ml·kg ^–1 ·min ^–1] participated in this study. Each subject performed four exercise tests separated by 1-week intervals: test 1, and VTh were determined; test 2, the speed at which the transition between walking and running spontaneously occurs (ST) during increasing speed (increases of 0.5 km·h ^–1 every 4 min from 5 km·h ^–1) was determined; test 3, the subjects were constrained to walk for 4 min at ST, at ST ± 0.5 km·h ^–1 and at ST ± 1 km·h ^–1; and test 4, the subjects were constrained to run for 4 min at ST, at ST±0.5 km·-h ^–1 and at ST±1 km·h ^–1. During exercise, oxygen uptake ( ), heart rate (HR), ventilation ( ), ventilatory equivalents for oxygen and carbon dioxide (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaca% WaaSbaaSqaaiaabweaaeqaaOGaai4laiqadAfagaGaamaaBaaaleaa% caqGYaaabeaakiaacYcacaqGGaGaaeiiaiqadAfagaGaamaaBaaale% aacaqGfbaabeaakiaac+caceWGwbGbaiaacaqGdbGaae4tamaaBaaa% leaacaaIYaaabeaaaaa!4240!\[\dot V_{\text{E}} /\dot V_{\text{2}} ,{\text{ }}\dot V_{\text{E}} /\dot V{\text{CO}}_2 \]), respiratory exchange ratio (R), stride length (SL), and stride frequency (SF) were measured. The results showed that: ST occurred at 2.16 (SEM 0.04) m·s ^–1; , HR and speed at ST were significantly lower than the values measured at VTh (P< 0.001, P< 0.001 and P< 0.05, respectively); changed significantly with speed (P< 0.001) but was greater during running than walking below ST (ST minus 1 km·h ^–1, P< 0.001; ST minus 0.5 km·h ^–1, P< 0.05) with the converse above ST (ST.plus 1 km·h ^–1, P<0.05), whereas at ST the values of were very close [23.9 (SEM 1.1) vs 23.7 (SEM 0.8) ml·kg ^–1 · min ^–1 not significant, respectively, for walking and running]; SL was significantly greater during walking than running (P<0.001) and SF lower (P<0.001); and HR and were significantly greater during running than walking below ST (ST minus 1 km·h ^–1, P<0.01; ST minus 0.5 km·h ^–1, P{<0.05) with the converse above ST (ST plus 1 km·h ^–1, P·< 0.05), whereas no difference appeared for and R between the two types of locomotion. We concluded from this study that ST corresponded to the speed at which the energy expenditure of running became lower than the energy expenditure of walking but that the mechanism of the link needed further investigation.