Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters

Mathematical modelling of the directed movement of animals, microorganisms and cells is of great relevance in the fields of biology and medicine. Simple diffusive models of movement assume a random walk in the position, while more realistic models include the direction of movement by assuming a random walk in the velocity. These velocity jump processes, although more realistic, are much harder to analyse and an equation that describes the underlying spatial distribution only exists in one dimension. In this communication we set up a realistic reorientation model in two dimensions, where the mean turning angle is dependent on the previous direction of movement and bias is implicitly introduced in the probability distribution for the direction of movement. This model, and the associated reorientation parameters, is based on data from experiments on swimming microorganisms. Assuming a transport equation to describe the motion of a population of random walkers using a velocity jump process, together with this realistic reorientation model, we use a moment closure method to derive and solve a system of equations for the spatial statistics. These asymptotic equations are a very good match to simulated random walks for realistic parameter values.     

Statistical model of variable allometric growth: otolith growth in Micropogonias furnieri(Actinopterygii, Sciaenidae)

The main objective of this study was to develop a statistical model for accurate estimates of relative growth. The method was based on identifying patterns of the residuals obtained from the Huxley's allometric equation. Three different approaches were applied: (1) growth with variable proportionality and constant allometry coefficient, (2) growth with constant proportionality and variable allometry coefficient and (3) distinct growth phases in which proportionality and allometry coefficients remained constant. The proposed statistical models were applied to the relationship of the otolith size and fish size of whitemouth croaker Micropogonias furnieri . The best fit was obtained when using approach (3). A change in the growth parameters was associated with the attainment of sexual maturity.     

Kinetic theory of the positive column of a low-pressure discharge in a transverse magnetic field

The influence of a transverse magnetic field on the characteristics of the positive column of a planar low-pressure discharge is studied theoretically. The motion of magnetized electrons is described in the framework of a continuous-medium model, while the ion motion in the ambipolar electric field is described by means of a kinetic equation. Using mathematical transformations, the problem is reduced to a secondorder ordinary differential equation, from which the spatial distribution of the potential is found in an analytic form. The spatial distributions of the plasma density, mean plasma velocity, and electric potential are calculated, the ion velocity distribution function at the plasma boundary is found, and the electron energy as a function of the magnetic field is determined. It is shown that, as the magnetic field rises, the electron energy increases, the distributions of the plasma density and mean plasma velocity become asymmetric, the maximum of the plasma density is displaced in the direction of the Ampère force, and the ion flux in this direction becomes substantially larger than the counter-directed ion flux.     

Effects of dissipation and external fields on the dynamics of conformational distortions in DNA

The dynamics of conformational distortions in DNA has been studied in the framework of the simple mathematical model based on the sine-Gordon equation with two additional terms. The first of the terms describes the effects of dissipation, and the second takes the action of external field into account. With the help of the energetic method, an analytical expression for the velocity of local conformational distortion as a function of time has been found, and conditions have been determined under which the influence of dissipation and constant external force are in a balance, providing the distortion to move with a constant velocity along the DNA. The graphs of changes in the velocity of movement of local conformational distortions in different homogeneous polynucleotide chains have been constructed using the model values of the parameters v0 = 189 (m/s), beta(DNA) = 4.25 x 10(-34) (J x s) and F0(DNA) = 3.12 x 10(-22) (J), which determine the initial velocity of the distortion, the coefficient of dissipation, and the value of the external generalized force, respectively. The accordance of the model values used with the experimental values available from the literature is discussed.     

Effects of perceptual and movement ranges on joint predator–prey distributions

We developed a spatially‐explicit individual‐based model to study how limited perceptual and movement ranges affect spatial predator–prey interactions. Earlier models of ‘predator–prey space games’ were often developed by modifying ideal free distribution models, which are spatially‐implicit and also assume that individuals are omniscient, although some more recent models have relaxed these assumptions. We found that under some conditions, the spatially‐explicit model generated similar predictions to previous models. However, the model showed that limited range in a spatially‐explicit context generated different predictions when 1) predator density and range are both small, and 2) when the predator movement range varied while the prey range was small. The model suggests that the differences were the result of 1) movement range changing the value of information sources and thus changing the behavior of individual predators and prey and 2) movement range limiting the ability of individuals to exploit the environment.     

The random walk of Azospirillum brasilense

The bacterium Azospirillum brasilense has been frequently studied in laboratory experiments. It performs movements in space where long forward and backward runs on a straight line occur simultaneously with slow changes of direction of the line. A model is presented in which a correlated random walk on a line is joined to diffusion on a sphere of directions. For this transport system, a hierarchy of moment approximations is derived, ranging from a hyperbolic system with four dependent variables to a scalar damped wave equation (telegraph equation) and then to a single diffusion equation for particle density. The original parameters are compounded in the diffusion quotient. The effects of these parameters, such as particle speed or turning rate, on the diffusion coefficient are discussed in detail.     

Modeling Vortex Swarming In Daphnia

Based on experimental observations in Daphnia, we introduce an agent-based model for the motion of single and swarms of animals. Each agent is described by a stochastic equation that also considers the conditions for active biological motion. An environmental potential further reflects local conditions for Daphnia, such as attraction to light sources. This model is sufficient to describe the observed cycling behavior of single Daphnia. To simulate vortex swarming of many Daphnia, i.e. the collective rotation of the swarm in one direction, we extend the model by considering avoidance of collisions. Two different ansatzes to model such a behavior are developed and compared. By means of computer simulations of a multi-agent system we show that local avoidance—as a special form of asymmetric repulsion between animals—leads to the emergence of a vortex swarm. The transition from uncorrelated rotation of single agents to the vortex swarming as a function of the swarm size is investigated. Eventually, some evidence of avoidance behavior in Daphnia is provided by comparing experimental and simulation results for two animals.     

Growth Pattern and Movement of Epidermal Cells within Leaves of Asphodelus tenuifolius Cav.

The pattern of growth (velocity field) in the intercalary growthzones of monocotyledon leaves can be determined from patternsof cell number density (number per unit length of cell file)and leaf elongation rates using theory based on a cell numberconservation equation. The case where elongation rate is non-steadywhile the pattern of cell number density is steady is discussedand a method for extending calculations into the meristem usingobservations of numbers of mitotic cells is outlined. Applicationof these methods is illustrated using data for epidermal cellsin the first leaf of Asphodelus tenuifolius Cav. During earlyleaf development, leaf elongation rate increased exponentiallybut cell number density and mitotic number density were steady.Cells 0.1 mm from the base of the leaf when leaves were 3.2mm long took 8.3 d to move through the growth zone. In leavesthat were 4 d older, similar cells took 5.1 d to traverse thegrowth zone. Increases in the rates of leaf elongation and ofcell movement appeared to be associated mainly with increasesin total rates of cell production in the epidermal meristem. Asphodelus tenuifolius Cav., Asphodelus fistulosus L., velocity field, meristem, mitotic cell number density, extension-only zone     

Periodic dynamics in a two-stage Allee effect model are driven by tension between stage equilibria

Single species difference population models can show complex dynamics such as periodicity and chaos under certain circumstances, but usually only when rates of intrinsic population growth or other life history parameter are unrealistically high. Single species models with Allee effects (positive density dependence at low density) have also been shown to exhibit complex dynamics when combined with over-compensatory density dependence or a narrow fertility window. Here we present a simple two-stage model with Allee effects which shows large amplitude periodic fluctuations for some initial conditions, without these requirements. Periodicity arises out of a tension between the critical equilibrium of each stage, i.e. when the initial population vector is such that the adult stage is above the critical value, while the juvenile stage is below the critical value. Within this area of parameter space, the range of initial conditions giving rise to periodic dynamics is driven mainly by adult mortality rates. Periodic dynamics become more important as adult mortality increases up to a certain point, after which periodic dynamics are replaced by extinction. This model has more realistic life history parameter values than most 'chaotic' models. Conditions for periodic dynamics might arise in some marine species which are exploited (high adult mortality) leading to recruitment limitation (low juvenile density) and might be an additional source of extinction risk.     

Modeling reflex asymmetries with implicit delay differential equations

Neuromuscular reflexes with time-delayed negative feedback, such as the pupil light reflex, have different rates depending on the direction of movement. This asymmetry is modeled by an implicit first-order delay differential equation in which the value of the rate constant depends on the direction of movement. Stability analyses are presented for the cases when the rate is: (1) an increasing and (2) a decreasing function of the direction of movement. It is shown that the stability of equilibria in these dynamical systems depends on whether the rate constant is a decreasing or increasing function. In particular, when the asymmetry has the shape of an increasing step function, it is possible to have stability which is independent of the value of the time delay or the steepness (i.e., gain) of the negative feedback.     

齐口裂腹鱼上溯过程中“冲刺-滑行”行为对水动力的响应

研究以西南山区特有鱼种齐口裂腹鱼(Schizothorax prenanti)为研究对象, 对其游泳行为模式进行量化解译, 寻找其偏好的水动力学条件, 构建水流条件与生态行为的纽带。运用具有流速梯度的水槽创造非均匀流场条件, 得到齐口裂腹鱼在室内试验水槽内上溯的视频图像。运用图像识别技术, 计算上溯全过程的游泳动力学指标摆尾角度与摆尾频率, 在此基础上实现生态学与水动力学的耦合研究。研究表明, 齐口裂腹鱼在上溯过程中喜好在具有流速梯度处通过改变摆尾角度和摆尾频率等来适应非均匀流场, 其喜好摆尾角度为25°—35°, 喜好摆尾频率为2.5—3.5次/s, 偏好流速为0.20—0.40 m/s。随着水流速度的增大, 摆尾角度呈现逐渐减小的趋势, 且齐口裂腹鱼偏好选择在流速由大变小的区域, 进行摆尾冲刺加速, 且更趋向于摆尾角度变化为“弱强弱”的摆尾模式。滑行阶段引入滑行流速系数, 量化表示摆尾角度、滑行距离和流速三者间的耦合关系, 通过计算滑行距离对水流负方向上位移的贡献率, 得到滑行方向与水流负方向夹角。研究表明, 滑行流速系数为1.0—3.0时具有代表性, 齐口裂腹鱼对滑行方向与水流负方向夹角的偏好为40°—60°。研究利用多指标量化评价的方法, 以复杂流场为背景条件, 进一步满足过鱼设施建设需求。     

Prediction of Substrate Removal Rates of Attached Microorganisms and of Relative Contributions of Attached and Suspended Communities at Field Sites

A mathematical model composed of a direct proportionality relationship between bulk water velocities and field-determined second-order microbial transformation rate coefficients, and the relative rate coefficient of a benchmark chemical, was developed for estimating the substrate removal rates of rapidly degraded chemicals by attached organisms in shallow (<1 m deep) aquatic ecosystems. Data from 31 field experiments involving the addition of 2,4-dichlorophenoxyacetic acid methyl ester (2,4-DME) in nine field areas were used to determine a field-derived second-order rate coefficient for microbial transformation of the ester. By using 2,4-DME as a benchmark chemical, the model was used to predict microbial transformation rates of the butoxyethyl ester of 2,4-dichlorophenoxyacetic acid (2,4-DBE) at five other field sites. The predicted half-lives of 2,4-DBE varied 1,500-fold and were within about a threefold range or less of the measured half-lives. Under conditions of mass transport limitation, the contributions of attached microorganisms relative to total microbial activities at various field sites were related to the ratio of water velocity, U, and depth, D, showing that historical definitions of ecosystems according to flow and depth characteristics are also valid for describing the process-related structure of ecosystems. An equation was developed for predicting the relative contributions of attached and suspended communities with values of U and D for lotic and lentic ecosystems. On the basis of this equation, attached microorganisms were expected to be insignificant in deep lentic ecosystems and suspended microorganisms were expected to be insignificant in shallow lotic systems for the same process carried out by both populations. Neglecting epiphytic microorganisms, both suspended and attached organisms were expected to be significant in wetlands.     

Efficiency of a visual system under conditions of uncertainty of the appearance of a moving object

The effect of uncertainty of a moving object appearance in the noise field upon the coefficient efficiency, we studied. At short durations of presentation (40-80 msec) and high level of external noise, this effect was maximal: magnified 100 times. The efficiency coefficient dependence on the duration of a moving object presentation was shown to be characterized by two maximums. The position of minimum situated between these two maximums was found to be independent of either presence or absence of uncertainty of a number of parameters: such as initial position of the object the image, time of its appearance, noise level, velocity and direction of the movement, and has a latency approximately 120 sec. A functional model of the observed phenomena, has been proposed.     

Responses of class R3 retinal ganglion cells of the frog to moving configurational bars: effect of the stimulus velocity

Discrimination of 'prey' (bars elongated in the direction of movement; W- or H-orientation) and 'non-prey' (bars perpendicular to the direction of movement; A- or V-orientation) stimuli in freely moving amphibians is velocity-invariant. Whether or not this phenomenon is present in cells belonging to a general decision making neuronal process remains questionable. Present investigations report the effect of the angular velocity of the stimulus on the discrimination function of class R3 (transient ON-OFF) retinal ganglion cells. The main conclusions of this work are the following: (1) irrespective of the angular velocity, class R3 neurons always prefer vertically (A-) to horizontally (W-) oriented stripes as long as the stimulus length remains inferior to the receptive field size; (2) this preference for small A-stimuli is best expressed when stimuli are moved at V = 7.6 degrees/s; (3) a preference reversal is induced by stripes longer than the receptive field via a dual process involving both spatial and temporal mechanisms; (4) this preference reversal is velocity-dependent: the longer the bar, the faster the velocity should be.     

A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration,Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands

Bacterial chemotaxis can enhance the bioremediation of contaminants in aqueous and subsurface environments if the contaminant is a chemoattractant that the bacteria degrade. The process can be promoted by traveling bands of chemotactic bacteria that form due to metabolism-generated gradients in chemoattractant concentration. We developed a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to model chemotaxis, because LBMs are well suited to model reactive transport in the complex geometries that are typical for subsurface porous media. This MRT-LBM can attain a better numerical stability than its corresponding single-relaxation-time LBM. We performed simulations to investigate the effects of substrate diffusion, initial bacterial concentration, and hydrodynamic dispersion on the formation, shape, and propagation of bacterial bands. Band formation requires a sufficiently high initial number of bacteria and a small substrate diffusion coefficient. Uniform flow does not affect the bands while shear flow does. Bacterial bands can move both upstream and downstream when the flow velocity is small. However, the bands disappear once the velocity becomes too large due to hydrodynamic dispersion. Generally bands can only be observed if the dimensionless ratio between the chemotactic sensitivity coefficient and the effective diffusion coefficient of the bacteria exceeds a critical value, that is, when the biased movement due to chemotaxis overcomes the diffusion-like movement due to the random motility and hydrodynamic dispersion.     

Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators

We synthesize previous theory on ideal free habitat selection to develop a model of predator movement mechanisms, when both predators and prey are mobile. We consider a continuous environment with an arbitrary distribution of resources, randomly diffusing prey that consume the resources, and predators that consume the prey. Our model introduces a very general class of movement rules in which the overall direction of a predator's movement is determined by a variable combination of (i) random diffusion, (ii) movement in the direction of higher prey density, and/or (iii) movement in the direction of higher density of the prey's resource. With this model, we apply an adaptive dynamics approach to two main questions. First, can it be adaptive for predators to base their movement solely on the density of the prey's resource (which the predators do not consume)? Second, should predator movements be exclusively biased toward higher densities of prey/resources, or is there an optimal balance between random and biased movements? We find that, for some resource distributions, predators that track the gradient of the prey's resource have an advantage compared to predators that track the gradient of prey directly. Additionally, we show that matching (consumers distributed in proportion to resources), overmatching (consumers strongly aggregated in areas of high resource density), and undermatching (consumers distributed more uniformly than resources) distributions can all be explained by the same general habitat selection mechanism. Our results provide important groundwork for future investigations of predator-prey dynamics.     

The mechanics of stridulation of the cricketGryllus campestris

Summary Details in the stridulatory movement ofGryllus campestris were investigated using an improved high resolution miniature angle measurement system. The following results were obtained: During the closing (sound producing) stroke, the speed of the plectrum always has the same value (within measuring accuracy) at a given position. Plectrum speed is directly proportional to tooth spacing, which is known to vary along the file. The only exception to this rule were occasions when closing velocities of precisely 2 times the standard value were found. In between values were never recorded. While temperature has a large effect on the opening speed and duration, the closing speed has a very smallQ ₁₀ (0.07) which is equal to theQ ₁₀ of the resonance frequency of the harp. When the harps are removed, the proportionality between tooth spacing and scraper velocity is lost; the velocity is much increased (up to 3-fold) and the variance of the speed is enhanced 5-fold.These results are discussed with respect to 3 hypothetical models explaining the function of the sound generator system. The model describing the cricket sound generator as a clockwork with an escapement system is capable of accommodating all experimental data without any extra assumptions.     

How do genetic correlations affect species range shifts in a changing environment?

Species may be able to respond to changing environments by a combination of adaptation and migration. We study how adaptation affects range shifts when it involves multiple quantitative traits evolving in response to local selection pressures and gene flow. All traits develop clines shifting in space, some of which may be in a direction opposite to univariate predictions, and the species tracks its environmental optimum with a constant lag. We provide analytical expressions for the local density and average trait values. A species can sustain faster environmental shifts, develop a wider range and greater local adaptation when spatial environmental variation is low (generating low migration load) and multitrait adaptive potential is high. These conditions are favoured when nonlinear (stabilising) selection is weak in the phenotypic direction of the change in optimum, and genetic variation is high in the phenotypic direction of the selection gradient.