Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.     

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

Modeling the Circulation of a Disease Between Two Host Populations on non Coincident Spatial Domains

We derive a reaction–diffusion system modeling the spatial propagation of a disease with kinetics occurring on distinct spatial domains. This corresponds to the actual invasion of a disease from a species living in a given spatial domain toward a second species living in a different spatial domain. We study the global existence of solutions and discuss the long time behavior of solutions. Then we consider a special case, based on a model of brain worm infection from white-tailed deer to moose populations, for which we discuss the invasion success/failure process and disprove a conjecture stated in an earlier work.     

自然保护区空间特征和地块最优化选择方法

自然保护区是保护物种和生态系统的有效方式,用于建立自然保护区的资源却是稀缺的,这就提出这样的问题：1)自然保护区在空间上应具有什么特征才是有效的;2)如何从许多备选地块中选择一部分组成自然保护区对稀缺资源进行最优化分配。为促进物种生存繁衍,在地块选择时应考虑保护区空间特征。这篇文章综述了该领域最近的研究进展,重点介绍了结合各种空间特征的保护区地块最优化选择模型。介绍了4个主要的空间特征：1)连续,2)间隔和距离,3)边界和集约,4)面积以及核心区和缓冲区。以前用启发式算法求解这些问题,但研究已显示该法不能保证资源的最优化分配。空间特征也可用线性整数规划模拟,用最优化软件求得最优解。目前的线性整数规划模型和软件还不能有效解决大型的保护区地块选择问题,计算效率易成为实际应用的瓶颈。文章概括了我国目前自然保护区设计领域的研究状况和面临的问题,最后讨论了该领域新的研究方向。     

Spreading speeds and traveling waves in competitive recursion systems

This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.     

Effective Sizes and Dynamics of Uniparentally and Diparentally Inherited Genes

Models to determine the temporal dynamics and spatial heterogeneity for maternally and paternally inherited genes were derived for populations that may or may not exhibit spatial subdivision. Results were compared to those for diparentally inherited genes. The models permit definition of parameters for mean and variance of litter sizes, breeding group (subpopulation) sizes, and numbers of female mates per male, dispersal rates, and multiple paternity. Exact solutions for asymptotic effective size and spatial divergence (F(LS)) for maternal and paternal genes are derived. It is shown that solutions for effective size and F(LS) are transformations of the same quadratic equation. When compared to values for diparentally inherited genes, it is shown that effective sizes for maternal genes may be considerably higher when female dispersal is low as in many mammalian taxa. Likewise, effective sizes for paternal genes may be higher than for diparentally inherited traits when male dispersal is relatively low, as in many species of birds. The traditional assumption that the effective size for maternal genes is approximately equal to the number of females is seldom realized. Spatial heterogeneity and temporal dynamics of genes are inextricably linked as is shown by the interdependency of effective size and spatial heterogeneity.     

A method for analysing spatial scales of variation in composition of assemblages

In several areas of research on ecological assemblages, it is useful to be able to analyse patterns of spatial variation at various scales. Multivariate analyses of dissimilarity or similarity in assemblages of species are limited by problems of non-independence caused by repeated use of the sample-units. Where rank-order procedures are used, no comparative quantitative measurements of dissimilarity at different scales are produced. An alternative method is described that uses the sample's average assemblage (or centroid). These estimates are themselves averaged to give centroids for larger spatial scales. Dissimilarities from the centroids at each scale are then calculated using independent replicates for each scale from those in each sample. The dissimilarity measures can then be examined by analysis of variance to detect spatial scales of differences for each sample at every level of a hierarchy of scales. The method is illustrated using data from mangrove forests and rocky shores, involving up to 97 taxonomic groups (species, other taxa). Differences among assemblages at the scales of sites (tens of meters apart) or locations at shores (hundreds of meters apart) were identified. Consequences of different numbers of replicates are discussed, with some potential problems (and their solutions) in application. Received: 14 November 1997 / Accepted: 14 September 1998     

Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model

The present work employs a nonlocal delay reaction-diffusion model to study the impacts of the density dependent birth function, maturation time delay and population dispersal on single species dynamics (i.e., extinction, survival, extinction-survival). It is shown that the maturation time and the birth function are two major factors determining the fate of single species. Whereas the dispersal acts as a subsidiary factor that only affects the spatial patterns of population densities. When the birth function has a compensating density dependence, maturation time delay cannot destabilize the population survival at the positive equilibrium. Nevertheless, when the birth function has an over-compensating density dependence, the population densities of single species fluctuate in the spatial domain due to the increased maturation time delay. With the Allee effect and over-compensating density dependence, the increases in the maturation time may cause extinction of the single species in the entire spatial domain. The numerical simulations suggest that the solutions of the general model may temporarily remain nearby a stationary wave pulse or a stationary wavefront of the reduced model. The former indicates the survival of single species in a narrow region of the spatial domain. Whereas the latter represents the survival in the entire left-half or right-half of the spatial domain.     

Wave solutions for the deterministic non-reducible n-type epidemic

A model is formulated to describe the spatial spread of an epidemic involving n types of individual. This encompasses the measles, host-vector and carrier-borne epidemics, and in addition rabies involving several species of animal. The existence, uniqueness and non-existence of wave solutions for different speeds are established for this model.     

Dispersal, Environmental Correlation, and Spatial Synchrony in Population Dynamics

Many species exhibit widespread spatial synchrony in population fluctuations. This pattern is of great ecological interest and can be a source of concern when the species is rare or endangered. Both dispersal and spatial correlations in the environment have been implicated as possible causes of this pattern, but these two factors have rarely been studied in combination. We develop a spatially structured population model, simple enough to obtain analytic solutions for the population correlation, that incorporates both dispersal and environmental correlation. We ask whether these two synchronizing factors contribute additively to the total spatial population covariance. We find that there is always an interaction between these two factors and that this interaction is small only when one or both of the environmental correlation and the dispersal rate are small. The interaction is opposite in sign to the environmental correlation; so, in the normal case of positive environmental correlation across sites, the population synchrony will be lower than predicted by simply adding the effects of dispersal and environmental correlation. We also find that population synchrony declines as the strength of population regulation increases. These results indicate that dispersal and environmental correlation need to be considered in combination as explanations for observed patterns of population synchrony.     

Prioritizing global marine mammal habitats using density maps in place of range maps

Despite lessons from terrestrial systems, conservation efforts in marine systems continue to focus on identifying priority sites for protection based on high species richness inferred from range maps. Range maps oversimplify spatial variability in animal distributions by assuming uniform distribution within range and de facto giving equal weight to critical and marginal habitats. We used Marxan ver. 2.43 to compare species richness‐based systematic reserve network solutions using information about marine mammal range and relative abundance. At a global scale, reserve network solutions were strongly sensitive to model inputs and assumptions. Solutions based on different input data overlapped by a third at most, with agreement as low as 10% in some cases. At a regional scale, species richness was inversely related to density, such that species richness hotspots excluded highest‐density areas for all species. Based on these findings, we caution that species‐richness estimates derived from range maps and used as input in conservation planning exercises may inadvertently lead to protection of largely marginal habitat.     

The performance of range maps and species distribution models representing the geographic variation of species richness at different resolutions

Aim The method used to generate hypotheses about species distributions, in addition to spatial scale, may affect the biodiversity patterns that are then observed. We compared the performance of range maps and MaxEnt species distribution models at different spatial resolutions by examining the degree of similarity between predicted species richness and composition against observed values from well‐surveyed cells (WSCs). Location Mexico. Methods We estimated amphibian richness distributions at five spatial resolutions (from 0.083° to 2°) by overlaying 370 individual range maps or MaxEnt predictions, comparing the similarity of the spatial patterns and correlating predicted values with the observed values for WSCs. Additionally, we looked at species composition and assessed commission and omission errors associated with each method. Results MaxEnt predictions reveal greater geographic differences in richness between species rich and species poor regions than the range maps did at the five resolutions assessed. Correlations between species richness values estimated by either of the two procedures and the observed values from the WSCs increased with decreasing resolution. The slopes of the regressions between the predicted and observed values indicate that MaxEnt overpredicts observed species richness at all of the resolutions used, while range maps underpredict them, except at the finest resolution. Prediction errors did not vary significantly between methods at any resolution and tended to decrease with decreasing resolution. The accuracy of both procedures was clearly different when commission and omission errors were examined separately. Main conclusions Despite the congruent increase in the geographic richness patterns obtained from both procedures as resolution decreases, the maps created with these methods cannot be used interchangeably because of notable differences in the species compositions they report.     

Gap pattern and colonization opportunities in plant communities: effects of species richness, mortality, and spatial aggregation

We investigate how perturbations that induce mortality transform original spatial patterns in plant communities into binary spatial patterns of survivors and perished individuals. By means of computer simulation, we analyse effects of average mortality, interspecific variation of mortality around the mean, spatial distribution of the species (clumping degree), and species richness. Gap spatial pattern is quantified by four spatial indices or landscape metrics (gap area, density, shape and coherence). In single‐species communities, the emerging gap patterns are subject to critical phenomena: opportunities for colonizers to establish increase with mortality, but more rapidly at specific mortality thresholds. In multi‐species communities, neither species richness nor interspecific variation of mortality influences gap spatial pattern when community assembly is random. Colonization opportunities would therefore not be affected by local species extinction in such a system, nor by the presence of species with divergent sensitivities to perturbation. In a community that is highly spatially aggregated, increases in interspecific mortality variation shift the pattern towards fewer gaps that are larger and more isodiametric, which suggests increased establishment chances for colonizers. Similar changes are induced in communities characterized by large interspecific mortality differences if clumping degree is increased. Loss of species richness only modifies gap spatial pattern to a substantial extent if mortality variation is high: in this case, depauperate communities exhibit a wider variety of colonization opportunities (more gaps which are on average smaller, but the largest gap is larger) than species‐richer ones. These findings may explain the contrast between the negative diversity‐invasibility relationship often found in small‐scale experimental studies and the positive diversity‐invasibility relationship found in observational studies at larger scale. They also demonstrate that the pre‐disturbance spatial structure of a community significantly affects colonization opportunities for alien species, and is therefore a likely determinant of the trajectory of secondary succession following perturbation.     

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.     

The shape of a species' spatial abundance distribution

Aim The shape of a species' spatial abundance distribution may change with spatial scale. We predict that the shape will typically change from strictly downward‐sloping (falling) to humped (rising then falling) as the spatial scale increases. The prediction, motivated in part by central limit reasoning, is intended for common or abundant species over reasonably homogeneous habitats. We test the prediction using data on 800+ tree species, one ant species, and 14 bird species. Location Tree data are for Malaysia and Central America; ant and bird data are for North America. Methods For each species, histograms and relative mode shape statistics are compared across multiple spatial scales. Results The predicted pattern is broadly confirmed. Main conclusions The emerging hump pattern is a candidate for listing among macroecological regularities. In appropriate contexts, spatial theories might be asked to predict the pattern.     

宜昌百里荒草山草坡群落物种分布的空间趋势分析

通过基于CCA的趋势面分析和空间插值方法,研究了宜昌百里荒山地草场的群落结构空间变化,以及群落结构空间趋势与主要环境因子的相关性。结果表明,该群落物种空间中的群落结构面和物理空间中的空间趋势面可以很好地吻合,说明该群落的结构由一种具有强烈空间结构化特征的机制控制。对群落结构和空间趋势影响最显著的环境因素是土壤有效磷。     

Predation may defeat spatial spread of infection

A model of a phytoplankton-zooplankton prey-predator system with viral infection of phytoplankton is investigated. Virus particles (V) are taken into account by an explicit equation. Phytoplankton is split into a susceptible (S) and an infected (I) class. A lytic infection is considered, thus, infected phytoplankton cells stop reproducing as soon as the infection starts and die at an increased mortality rate. Zooplankton (Z) is grazing on both susceptible and infected phytoplankton following a Holling-type II functional response. After the local dynamics of the V-S-I-Z system is analysed, numerical solutions of a stochastic reaction-diffusion model of the four species are presented. These show a spatial competition between zooplankton and viruses, although these two species are not explicitly coupled by the model equations.     

Allee效应对具有生境恢复的集合种群的影响

Allee效应与种群的灭绝密切相关,其研究对生态保护和管理至关重要。Allee效应对物种续存是潜在的干扰因素,濒危物种更容易受其影响,可能会增加生存于生境破碎化斑块的濒危物种的死亡风险,因此研究Allee效应对种群的动态和续存的影响是必要的。从包含由生物有机体对环境的修复产生的Allee效应的集合种群模型出发,引入由其他机制形成的Allee效应,建立了常微分动力系统模型和基于网格模型的元胞自动机模型。通过理论分析和计算机模拟表明:(1)强Allee效应不利于具有生境恢复的集合种群的续存;(2)生境恢复有利于种群续存;(3)局部扩散影响了集合种群的空间结构、动态行为和稳定性,生境斑块之间的局部作用将会减缓或消除集合种群的Allee效应,有利于集合种群的续存。