Effects of long-range taxis and population pressure on the range expansion of invasive species in heterogeneous environments

We consider a new model for biological invasions in periodic patchy environments, in which long-range taxis and population pressure are incorporated in the framework of reaction-diffusion-advection equations. We assume that long-range taxis is induced by a weighted integral of stimuli within a certain sensing range. Population pressure is incorporated in the diffusion coefficient that linearly increases with population density. We first analyze the model in the absence of population pressure and demonstrate how the sensing length of long-range taxis influences the range expansion pattern of invasive species and its rate of spread. The effects of population pressure are examined for both homogeneous and periodic patchy environments. For the homogeneous environment, an exact and explicit traveling wave solution and the spreading speed are obtained. For the periodic patchy environment, we find numerically that a population starting from any localized distribution evolves to a traveling periodic wave if the null solution of the RDA equation is locally unstable, and that the traveling wave speed significantly increases with increasing population pressure. Furthermore, the population pressure and taxis intensity synergistically enhance the spreading speed when they are increased together.     

The effect of temporal variability on persistence conditions in rivers

There has been great interest in the invasion and persistence of algal and insect populations in rivers. Recent modeling approaches assume that the flow speed of the river is constant. In reality, however, flow speeds in rivers change significantly on various temporal scales due to seasonality, weather conditions, or many human activities such as hydroelectric dams. In this paper, we study persistence conditions by deriving the upstream invasion speed in simple reaction-advection-diffusion equations with coefficients chosen to be periodic step functions. The key methodological idea to determine the spreading speed is to use the exponential transform in order to obtain a moment generating function. In a temporally periodic environment, the averages of each coefficient function determine the minimal upstream and downstream propagation speeds for a single-compartment model. For a two-compartment model, the temporal variation can enhance population persistence.     

利用元胞自动机研究一类捕食食饵模型中的斑块扩散现象

利用概率元胞自动机模型对空间隐式的、食饵具Allee效应的一类捕食食饵模型进行模拟,发现随着相关参数的变化,种群的空间扩散前沿由连续的扩散波逐渐转变为一种相互隔离的斑块向外扩散,这种斑块扩散现象与以往的扩散模式有所不同。研究结果表明：(1)在斑块扩散的情况下,相关参数的微小变化会导致种群灭绝或者形成连续的扩散波,即斑块扩散发生在种群趋于灭绝和连续扩散之间;(2)当种群的空间扩散方式为斑块扩散时,种群的扩散速度会变慢,与其他扩散方式下的速度有着明显的区别。该研究结果对生物入侵控制和外来物种监测有重要的启发和指导作用。     

Patchy populations in stochastic environments: critical number of patches for persistence

We introduce a model for the dynamics of a patchy population in a stochastic environment and derive a criterion for its persistence. This criterion is based on the geometric mean (GM) through time of the spatial-arithmetic mean of growth rates. For the population to persist, the GM has to be >/=1. The GM increases with the number of patches (because the sampling error is reduced) and decreases with both the variance and the spatial covariance of growth rates. We derive analytical expressions for the minimum number of patches (and the maximum harvesting rate) required for the persistence of the population. As the magnitude of environmental fluctuations increases, the number of patches required for persistence increases, and the fraction of individuals that can be harvested decreases. The novelty of our approach is that we focus on Malthusian local population dynamics with high dispersal and strong environmental variability from year to year. Unlike previous models of patchy populations that assume an infinite number of patches, we focus specifically on the effect that the number of patches has on population persistence. Our work is therefore directly relevant to patchily distributed organisms that are restricted to a small number of habitat patches.     

Integrodifference equations in patchy landscapes

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton–Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.     

Modeling biological invasions into periodically fragmented environments

Range expansion of a single species in a regularly striped environment is studied by using an extended Fisher model, in which the rates of diffusion and reproduction periodically fluctuate between favorable and unfavorable habitats. The model is analyzed for two initial conditions: the initial population density is concentrated on a straight line or at the origin. For each case, we derive a mathematical formula which characterizes the spatio-temporal pattern of range expansion. When initial distribution starts from a straight line, it evolves to a traveling periodic wave (TPW), whose frontal speed is analytically determinable. When the range starts from the origin, it tends to expand radially at a constant average speed in each direction (ray speed) keeping its frontal envelope in a similar shape. By examining the relation between the ray speed and the TPW speed, we derive the ray speed in a parametric form, from which the envelope of the expanding range can be predicted. Thus we analyze how the pattern and speed of the range expansion are affected by the pattern and scale of fragmentation, and the qualities of favorable and unfavorable habitats. The major results include: (1). The envelope of the expanding range show a variety of patterns, nearly circular, oval-like, spindle-like, depending on parameter values; (2). All these patterns are elongated in the direction of stripes; (3). When the scale of fragmentation is enlarged without changing the relative spatial pattern, the ray speed in any direction increases, i.e., the rate of range expansion increases.     

Existence of Traveling Waves for the Generalized F–KPP Equation

Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher–Kolmogorov–Petrovskii–Piskunov (F–KPP) equation. This generalized F–KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F–KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth–decay rate in the generalized F–KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.     

The role of prey taxis in biological control: a spatial theoretical model

We study a reaction-diffusion-advection model for the dynamics of populations under biological control. A control agent is assumed to be a predator species that has the ability to perceive the heterogeneity of pest distribution. The advection term represents the predator density movement according to a basic prey taxis assumption: acceleration of predators is proportional to the prey density gradient. The prey population reproduces logistically, and the local population interactions follow the Holling Type II trophic function. On the scale of the population, our spatially explicit approach subdivides the predation process into random movement represented by diffusion, directed movement described by prey taxis, local prey encounters, and consumption modeled by the trophic function. Thus, our model allows studying the effects of large-scale predator spatial activity on population dynamics. We show under which conditions spatial patterns are generated by prey taxis and how this affects the predator ability to maintain the pest population below some economic threshold. In particular, intermediate taxis activity can stabilize predator-pest populations at a very low level of pest density, ensuring successful biological control. However, very intensive prey taxis destroys the stability, leading to chaotic dynamics with pronounced outbreaks of pest density.     

The effect of colored noise on spatiotemporal dynamics of biological invasion in a diffusive predator-prey system

Spatiotemporal dynamics of a predator-prey system is considered under the assumption that the predator is sensitive to colored noise. Mathematically, the model consists of two coupled diffusion-reactions. By means of extensive numerical simulations, the complex invasion pattern formations of the system are identified. The results show that a geographical invasion emerges without regional persistence when the intensity of colored noise is small. Remarkably, as the noise intensity increases, the species spreads via a patchy invasion only when the system is affected by red noise. Meanwhile, the relationship between local stability and global invasion is also considered. The predator, which becomes extinct in the system without diffusion, could invade locally when the system is affected by white noise. However, the local invasion is not followed by geographical spread.     

Demography and dispersal: invasion speeds and sensitivity analysis in periodic and stochastic environments

Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. In this paper, we extend these calculations to temporally varying environments. We present formulas for the invasion speed and its sensitivity and elasticity in both periodic and stochastic environments. Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. We show that the invasion speed is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. We present a new formulation of sensitivity analysis, using matrix calculus, that applies equally to constant, periodic, and stochastic environments.     

Running on a slope: A collision-based analysis to assess the optimal slope

When running, energy is lost during stance to redirect the center of mass of the body (COM) from downwards to upwards. The present study uses a collision-based approach to analyze how these energy losses change with slope and speed. Therefore, we evaluate separately the average collision angle, i.e. the angle of deviation from perpendicular relationship between the force and velocity vectors, during the absorptive and generative part of stance. Our results show that on the level, the collision angle of the absorptive phase is smaller than the collision angle of the generative phase, suggesting that the collision is generative to overcome energy losses by soft tissues. When running uphill, the collision becomes more and more generative as slope increases because the average upward vertical velocity of the COM becomes greater than on the level. When running downhill at a constant speed, the collision angle decreases during the generative phase and increases during the absorptive phase because the average downward vertical velocity of the COM becomes greater. As a result, the difference between the collision angles of the generative and absorptive phases observed on the level disappears on a shallow negative slope of ∼−6°, where the collision becomes 'pseudo-elastic' and collisional energy losses are minimized. At this 'optimal' slope, the metabolic energy consumption is minimal. On steeper negative slopes, the collision angle during the absorptive phase becomes greater than during the generative phase and the collision is absorptive. At all slopes, the collision becomes more generative when speed increases.     

Periodic travelling waves in cyclic predator–prey systems

Predation is an established cause of cycling in prey species. Here, the ability of predation to explain periodic travelling waves in prey populations, which have recently been found in a number of spatiotemporal field studies, is examined. The nature of periodic waves in these systems, and the way in which they can be generated by the invasion of predators into a prey population is discussed. A theoretical calculation that predicts, as a function of two parameter ratios, whether such an invasion will lead to a stable periodic travelling wave that would be observed in practice is presented ‐ the alternative outcome is spatiotemporal chaos. The calculation also predicts quantitative details of the periodic waves, such as speed and amplitude. The results give new insights into the types of predator‐prey systems in which one would expect to see periodic travelling waves following an invasion by predators.     

A movement criterion for running

The adjustment of the leg during running was addressed using a spring-mass model with a fixed landing angle of attack. The objective was to obtain periodic movement patterns. Spring-like running was monitored by a one-dimensional stride-to-stride mapping of the apex height to identify mechanically stable fixed points. We found that for certain angles of attack, the system becomes self-stabilized if the leg stiffness was properly adjusted and a minimum running speed was exceeded. At a given speed, running techniques fulfilling a stable movement pattern are characterized by an almost constant maximum leg force. With increasing speed, the leg adjustment becomes less critical. The techniques predicted for stable running are in agreement with experimental studies. Mechanically self-stabilized running requires a spring-like leg operation, a minimum running speed and a proper adjustment of leg stiffness and angle of attack. These conditions can be considered as a movement criterion for running.     

Theory of longitudinal plasma waves with allowance for ion motion

A study is made of the propagation of steady-state large-amplitude longitudinal plasma waves in a cold collisionless plasma with allowance for both electron and ion motion. Conditions for the existence of periodic potential waves are determined. The electric field, potential, frequency, and wavelength are obtained as functions of the wave phase velocity and ion-to-electron mass ratio. Taking into account the ion motion results in the nonmonotonic dependence of the frequency of the waves with the maximum possible amplitudes on the wave phase velocity. Specifically, at low phase velocities, the frequency is equal to the electron plasma frequency for linear waves. As the phase velocity increases, the frequency first decreases insignificantly, reaches its minimum value, and then increases. As the phase velocity increases further, the frequency continues to increase and, at relativistic phase velocities, again becomes equal to the plasma frequency. Finally, as the phase velocity approaches the speed of light, the frequency increases without bound.     

Patterns of Patchy Spread in Deterministic and Stochastic Models of Biological Invasion and Biological Control

A few spatiotemporal models of population dynamics are considered in relation to biological invasion and biological control. The patterns of spread in one and two spatial dimensions are studied by means of extensive numerical simulations. We show that, in the case that population multiplication is damped by the strong Allee effect (when the population growth rate becomes negative for small population density), in a certain parameter range the spread can take place not via the intuitively expected circular expanding population front but via motion and interaction of separate patches. Alternatively, the patchy spread can take place in a system without Allee effect as a result of strong environmental noise. We then show that the phenomenon of deterministic patchy invasion takes place ‘at the edge of extinction’ so that a small change of controlling parameters either brings the species to extinction or restores the travelling population fronts. Moreover, we show that the regime of patchy invasion in two spatial dimensions actually takes place when the species go extinct in the corresponding 1-D system.     

Advection–diffusion equations for generalized tactic searching behaviors

 Many organisms search for limiting resources by using repeated responses to local cues, which cumulatively cause movement towards more favorable parts of their environment. This paper presents a general asymptotic expression, derived under the assumption of shallow environmental gradients, for the population-level flux of organisms moving at a constant speed and reorienting at rates determined by the environmental conditions experienced since the last reorientation. The expression takes the form of an advection-diffusion equation, in which the diffusivity and advection velocity are determined by statistics of the turning algorithm that are directly comparable to empirical observations. This work provides a mechanism with which to systematically evaluate a wide variety of tactic and kinetic strategies for determining turning behaviors. The model is applied to searchers on spatially-variable, random distributions of discrete resource patches. Such algorithms are functions of the integrated resource density encountered between turns. It is shown that behaviors in which the turning time distribution is a function of integrated density cannot result in taxis. In contrast, behaviors in which the turning rate is a function of integrated density can result in taxis. These two classes of search algorithm differ in that the latter requires the searcher to “learn” about its local environment, whereas the former requires no such assessment. This suggests neural or physiological mechanisms for remembering previous encounters may be a biological requirement for searchers on discrete resource distributions. Received: 21 September 1995/Revised version: 18 July 1996     

Population spread in patchy landscapes under a strong Allee effect

Many species of invasive insects establish and spread in regions around the world, causing enormous economical and environmental damage, in particular in forests. Some of these insects are subject to an Allee effect whereby the population must surpass a certain threshold in order to establish. Recent studies have examined the possibility of exploiting an Allee effect to improve existing control strategies. Forests and most other ecosystems show natural spatial variation, and human activities frequently increase the degree of spatial heterogeneity. It is therefore imperative to understand how the interplay between this spatial variation and individual movement behavior affects the overall speed of spread of an invasion. To this end, we study an integrodifference equation model in a patchy landscape and with Allee growth dynamics. Movement behavior of individuals varies according to landscape quality. Our study focuses on how the speed of the resulting traveling periodic wave depends on the interaction between landscape fragmentation, patch-dependent dispersal, and Allee population dynamics.     

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect

Invasion of an exotic species initiated by its local introduction is considered subject to predator-prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion-reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.     

Different roles of CheY1 and CheY2 in the chemotaxis of Rhizobium meliloti

Cells of Rhizobium meliloti swim by the unidirectional, clockwise rotation of their right-handed helical flagella and respond to tactic stimuli by modulating the flagellar rotary speed. We have shown that wild-type cells respond to the addition of proline, a strong chemoattractant, by a sustained increase in free-swimming speed (chemokinesis). We have examined the role of two response regulators, CheY1 and CheY2, and of CheA autokinase in the chemotaxis and chemokinesis of R. meliloti by comparing wild-type and mutant strains that carry deletions in the corresponding genes. Swarm tests, capillary assays, and computerized motion analysis revealed that (i) CheY2 alone mediates 60 to 70% of wild-type taxis, whereas CheY1 alone mediates no taxis, but is needed for the full tactic response; (ii) CheY2 is the main response regulator directing chemokinesis and smooth swimming in response to attractant, whereas CheY1 contributes little to chemokinesis, but interferes with smooth swimming; (iii) in a CheY2-overproducing strain, flagellar rotary speed increases upon addition and decreases upon removal of attractant; (iv) both CheY2 and CheY1 require phosphorylation by CheA for activity. We conclude that addition of attractant causes inhibition of CheA kinase and removal causes activation, and that consequent production of CheY1-P and CheY2-P acts to slow the flagellar motor. The action of the chief regulator, CheY2-P, on flagellar rotation is modulated by CheY1, probably by competition for phosphate from CheA.