Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210-219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine "interior" and "exterior" species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Plant interspecies competition for sunlight: a mathematical model of canopy partitioning

We examine the influence of canopy partitioning on the outcome of competition between two plant species that interact only by mutually shading each other. This analysis is based on a Kolmogorov-type canopy partitioning model for plant species with clonal growth form and fixed vertical leaf profiles (Vance and Nevai in J. Theor. Biol., 2007, to appear). We show that canopy partitioning is necessary for the stable coexistence of the two competing plant species. We also use implicit methods to show that, under certain conditions, the species’ nullclines can intersect at most once. We use nullcline endpoint analysis to show that when the nullclines do intersect, and in such a way that they cross, then the resulting equilibrium point is always stable. We also construct surfaces that divide parameter space into regions within which the various outcomes of competition occur, and then study parameter dependence in the locations of these surfaces. The analysis presented here and in a companion paper (Nevai and Vance, The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model, in review) together shows that canopy partitioning is both necessary and, under appropriate parameter values, sufficient for the stable coexistence of two hypothetical plant species whose structure and growth are described by our model. A. L. Nevai was supported in part by the National Institutes of Health, National Research Service Award (T32-GM008185) from the National Institute of General Medical Sciences (NIGMS).     

Interspecific competition in perennial sedentary organisms: An individual-based model

We investigated a mathematical model of the dynamics of the ecological system consisting of two competing perennial species, each of which leads a sedentary life. It is an individual-based model, in which the growth of each individual is described. The rate of this growth is weakened by competition from neighboring individuals. The strength of the competitors' influence depends on their size and distance to them. The conditions, in which the competitive exclusion of one of the competitors and the coexistence of both competitors take place are provided. The influence of the parameters responsible for the strength of competition, the degree of competitive asymmetry, and consideration of the importance of specific elements of the spatial structure of this ecological system on the results of the competition were analyzed. Both species co-exist when they are equal competitors. Permanent coexistence is possible only when interspecific competition is weaker than intraspecific. When interspecific competition is stronger, the coexistence of equal interspecific competitors is random. Both species have equal probability of extinction. If species are not equal competitors, the stronger one wins. This result can be modified by different strengths of intraspecific competition. The weaker interspecific competitor can permanently coexist with stronger one, when its individuals suffer stronger intraspecific competition.     

Modelling canopy CO2 fluxes: are ‘big‐leaf’ simplifications justified?

• 1 The ‘big‐leaf’ approach to calculating the carbon balance of plant canopies is evaluated for inclusion in the ETEMA model framework. This approach assumes that canopy carbon fluxes have the same relative responses to the environment as any single leaf, and that the scaling from leaf to canopy is therefore linear.
• 2 A series of model simulations was performed with two models of leaf photosynthesis, three distributions of canopy nitrogen, and two levels of canopy radiation detail. Leaf‐ and canopy‐level responses to light and nitrogen, both as instantaneous rates and daily integrals, are presented.
• 3 Observed leaf nitrogen contents of unshaded leaves are over 40% lower than the big‐leaf approach requires. Scaling from these leaves to the canopy using the big‐leaf approach may underestimate canopy photosynthesis by ~20%. A leaf photosynthesis model that treats within‐leaf light extinction displays characteristics that contradict the big‐leaf theory. Observed distributions of canopy nitrogen are closer to those required to optimize this model than the homogeneous model used in the big‐leaf approach.
• 4 It is theoretically consistent to use the big‐leaf approach with the homogeneous photosynthesis model to estimate canopy carbon fluxes if canopy nitrogen and leaf area are known and if the distribution of nitrogen is assumed optimal. However, real nitrogen profiles are not optimal for this photosynthesis model, and caution is necessary in using the big‐leaf approach to scale satellite estimates of leaf physiology to canopies. Accurate prediction of canopy carbon fluxes requires canopy nitrogen, leaf area, declining nitrogen with canopy depth, the heterogeneous model of leaf photosynthesis and the separation of sunlit and shaded leaves. The exact nitrogen profile is not critical, but realistic distributions can be predicted using a simple model of canopy nitrogen allocation.

     

Habitat loss alters effects of intransitive higher-order competition on biodiversity: a new metapopulation framework

Recent studies have suggested that intransitive competition, as opposed to hierarchical competition, allows more species to coexist. Furthermore, it is recognized that the prevalent paradigm, which assumes that species interactions are exclusively pairwise, may be insufficient. More importantly, whether and how habitat loss, a key driver of biodiversity loss, can alter these complex competition structures (and therefore species coexistence) remain unclear. We thus present a new, simple yet comprehensive metapopulation framework that can account for any competition pattern and more complex higher-order interactions (HOIs) among species. We find that competitive intransitivity increases community diversity and that HOIs generally enhance this effect. Essentially, intransitivity promotes species richness by preventing the dominance of a few species, unlike the hierarchical competition, while HOIs facilitate species coexistence through stabilizing community fluctuations. However, variation in species’ vital rates and habitat loss can weaken or even reverse such higher-order effects, as their interaction can lead to a more rapid decline in competitive intransitivity under HOIs. Thus, it is essential to correctly identify the most appropriate interaction model for a given system before models are used to inform conservation efforts. Overall, our simple model framework provides a more parsimonious explanation for biodiversity maintenance than the existing theory.     

Species coexistence under resource competition with intraspecific and interspecific direct competition in a chemostat

Competition theory has developed separately for direct competition and for exploitative competition. However, the combined effects of the two types of competition on species coexistence remain unclear. To examine how intraspecific and interspecific direct competition contributes to the coexistence of species competing for a single resource, we constructed a chemostat-type resource competition model. With general functions for intraspecific and interspecific direct competition, we derived necessary and sufficient conditions (except for a critical case that rarely occurs in a biological sense) that determine the number of stably coexisting species. From these conditions, we found that the number of coexisting species is determined just by the invasibility of each species into subcommunities with a smaller number of species. In addition, using a combination of rigorous mathematical theory and a simple graphical method, we can demonstrate how the stronger intraspecific direct competition facilitates species invasion, leading to a larger number of coexisting species.     

Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning

Can a difference in the heights at which plants place their leaves, a pattern we call canopy partitioning, make it possible for two competing plant species to coexist? To find out, we examine a model of clonal plants living in a nonseasonal environment that relates the dynamical behavior and competitive abilities of plant populations to the structural and functional features of the plants that form them. This examination emphasizes whole plant performance in the vertical light gradient caused by self-shading. This first of three related papers formulates a prototype single species Canopy Structure Model from biological first principles and shows how all plant properties work together to determine population persistence and equilibrium abundance. Population persistence is favored, and equilibrium abundance is increased, by high irradiance, high maximum photosynthesis rate, rapid saturation of the photosynthetic response to increased irradiance, low tissue respiration rate, small amounts of stem and root tissue necessary to support the needs of leaves, and low density of leaf, stem, and root tissues. In particular, equilibrium abundance decreases as mean leaf height increases because of the increased cost of manufacturing and maintaining stem tissue. All conclusions arise from this formulation by straightforward analysis. The argument concludes by stating this formulation's straightforward extension, called a Canopy Partitioning Model, to two competing species.     

A general model of local competition for space

Local competition for space across a wide array of taxa typically involves three mechanisms that we denote here as expansion (spreading into unoccupied habitat), lottery (replacing dead competitors), and overgrowth (encroaching on competitors along zones of contact). By formulating and analysing a simple, general model incorporating these features, we identify ecological conditions and life‐history features that lead to stable coexistence or competitive exclusion (with or without initial‐condition dependence) and gain insight by linking these to case studies in the literature. We demonstrate the importance of contact inhibition, a little‐studied feature of overgrowth, and we show how life‐history tradeoffs may influence and be influenced by local competition for space. The general model we present can help indicate whether local interactions are sufficient to explain patterns of coexistence or exclusion and can serve as the foundation for more specific, realistic models of spatial competition.     

Species packing in nonsmooth competition models

Despite the potential for competition to generate equilibrium coexistence of infinitely tightly packed species along a trait axis, prior work has shown that the classical expectation of system-specific limits to the similarity of stably coexisting species is sound. A key reason is that known instances of continuous coexistence are fragile, requiring fine-tuning of parameters: A small alteration of the parameters leads back to the classical limiting similarity predictions. Here we present, but then cast aside, a new theoretical challenge to the expectation of limiting similarity. Robust continuous coexistence can arise if competition between species is modeled as a nonsmooth function of their differences—specifically, if the competition kernel (differential response of species’ growth rates to changes in the density of other species along the trait axis) has a nondifferentiable sharp peak at zero trait difference. We will say that these kernels possess a “kink.” The difference in predicted behavior stems from the fact that smooth kernels do not change to a first-order approximation around their maxima, creating strong competitive interactions between similar species. “Kinked” kernels, on the other hand, decrease linearly even for small species differences, reducing interspecific competition compared with intraspecific competition for arbitrarily small species differences. We investigate what mechanisms would lead to kinked kernels in the first place. It turns out that discontinuities in resource utilization generate them. We argue that such sudden jumps in the utilization of resources are unrealistic, and therefore, one should expect kernels to be smooth in reality.     

不同栖息地状态下物种竞争模式及模拟研究与应用

物种竞争是影响生态系统演化的重要生态过程之一.而物种在受人类影响出现不同程度毁坏的栖息地上的演化又是非常复杂的,因此研究物种演化对栖息地毁坏的响应是非常必要的.在Tilman研究工作的基础上,将竞争系数引入集合种群动力模式,建立了多物种集合种群竞争共存的数学模型,并对5-物种集合种群在不同栖息地状态下的竞争动态进行了计算机模拟研究.结果表明：（1）不同结构的群落（q值不同）,物种之间的竞争排斥作用强度不同,优势物种明显的群落,物种之间的排斥强度大;（2）随着栖息地毁坏程度的增加,对优势物种的负面影响逐渐减小,而对弱势物种的负面影响逐渐增加;（3）随着栖息地恢复幅度的增加,优势物种和弱势物种之间的竞争越强烈,优势物种受到的竞争排斥加大,而弱势物种逐渐变强,出现了强者变弱、弱者变强的格局;（4）物种竞争排斥与共存受迁移扩散能力和竞争能力影响很大,竞争共存的条件是其竞争能力与扩散能力呈非线性负相关关系;（5）竞争共存的物种的强弱序列发生了变化.     

Species coexistence in resource-limited patterned ecosystems is facilitated by the interplay of spatial self-organisation and intraspecific competition

The exploration of mechanisms that enable species coexistence under competition for a sole limiting resource is widespread across ecology. Two examples of such facilitative processes are intraspecific competition and spatial self-organisation. These processes determine the outcome of competitive dynamics in many resource-limited patterned ecosystems, classical examples of which include dryland vegetation patterns, intertidal mussel beds and subalpine ribbon forests. Previous theoretical investigations have explained coexistence within patterned ecosystems by making strong assumptions on the differences between species (e.g. contrasting dispersal behaviours or different functional responses to resource availability). In this paper, I show that the interplay between the detrimental effects of intraspecific competition and the facilitative nature of self-organisation forms a coexistence mechanism that does not rely on species-specific assumptions and captures coexistence across a wide range of the environmental stress gradient. I use a theoretical model that captures the interactions of two generic consumer species with an explicitly modelled resource to show that coexistence relies on a balance between species' colonisation abilities and their local competitiveness, provided intraspecific competition is sufficiently strong. Crucially, the requirements on species' self-limitation for coexistence to occur differ on opposite ends of the resource input spectrum. For low resource levels, coexistence is facilitated by strong intraspecific dynamics of the species superior in its colonisation abilities, but for larger volumes of resource input, strong intraspecific competition of the locally superior species enables coexistence. Results presented in this paper also highlight the importance of hysteresis in understanding tipping points, in particular extinction events. Finally, the theoretical framework provides insights into spatial species distributions within single patches, supporting verbal hypotheses on coexistence of herbaceous and woody species in dryland vegetation patterns and suggesting potential empirical tests in the context of other patterned ecosystems.     

Apparent competition and recovery from infection

We use mathematical models to analyse how the recovery rate from infection influences the fitness of a host in a setting of interspecific competition. We show that sub-optimal immunity against pathogens can be advantageous for the host in the presence of cross-species infection. Weaker immunity allows the parasite to be used as a biological weapon, and this increases the fitness of the host relative to a competitor. A parameter region is observed in which the outcome of competition depends on the initial conditions. We extend this model and consider the dynamics in a spatial setting and find that the outcome depends on the migration rate of the host species. At low migration rates, coexistence of the host species is possible across space. For higher migration rates, the host species characterized by a lower recovery rate can invade the territory of its competitor. Finally, we study these dynamics in an evolutionary setting. Although a lower recovery rate from infection can increase the competitive ability of a species, we find that evolution maximizes the recovery rate and minimizes parasite burden. The models presented are related to the concept of apparent competition, and our results are discussed in relation to both theoretical and empirical studies.     

Leaf investment and light partitioning among leaves of different genotypes of the clonal plant Potentilla reptans in a dense stand after 5 years of competition

Background and Aims

While within-species competition for light is generally found to be asymmetric – larger plants absorbing more than proportional amounts of light – between-species competition tends to be more symmetric. Here, the light capture was analysed in a 5-year-old competition experiment that started with ten genotypes of the clonal plant Potentilla reptans. The following hypotheses were tested: (a) if different genotypes would do better in different layers of the canopy, thereby promoting coexistence, and (b) if leaves and genotypes with higher total mass captured more than proportional amounts of light, possibly explaining the observed dominance of the abundant genotypes.

Methods

In eight plots, 100 leaves were harvested at various depths in the canopy and their genotype determined to test for differences in leaf biomass allocation, leaf characteristics and the resulting light capture, calculated through a canopy model using the actual vertical light and leaf area profiles. Light capture was related to biomass to determine whether light competition between genotypes was asymmetric.

Key Results

All genotypes could reach the top of the canopy. The genotypes differed in morphology, but did not differ significantly in light capture per unit mass (Φ_mass) for leaves with the laminae placed at the same light levels. Light capture did increase disproportionately with leaf mass for all genotypes. However, the more abundant genotypes did not capture disproportionately more light relative to their mass than less-abundant genotypes.

Conclusions

Vertical niche differentiation in light acquisition does not appear to be a factor that could promote coexistence between these genotypes. Contrary to what is generally assumed, light competition among genetic individuals of the same species was size-symmetric, even if taller individual leaves did capture disproportionately more light. The observed shifts in genotype frequency cannot therefore be explained by asymmetric competition for light.Key words: Potentilla reptans, light, competition, symmetric, clonal, genotype, investment, petiole, canopy, allocation     

生物间非传递性竞争研究进展

生物间的竞争关系是决定群落中物种共存和生物多样性的关键因素。传统研究主要关注物种两两之间的竞争作用, 而对多物种相互竞争形成的网络研究相对较少。近年来, 类似于“石头-剪刀-布”游戏的非传递性竞争被认为是一种重要的物种共存和生物多样性的维持机制, 越来越受到生态学家们的关注。本文首先回顾了非传递性竞争定义的发展过程, 并介绍了非传递环的不同结构。其次介绍了基于竞争结局矩阵以及入侵增长率的非传递性竞争度量指标, 并对比不同指标的特点与适用情形。随后通过多个研究实例介绍了非传递性竞争在自然群落中的普遍性, 并指明物种之间的权衡是非传递性竞争产生的生物学机制。最后介绍了非传递性竞争对生物多样性与生态系统功能的影响。非传递性竞争本质上是物种两两之间相互作用的组合, 是只包含单一作用类型的特殊网络结构。因此, 非传递性竞争如何影响生物多样性-生态系统功能关系和群落稳定性, 如何受到环境与高阶相互作用的影响, 以及如何将竞争网络拓展到包含不同相互作用类型的生态网络, 将是未来非传递性竞争研究的重要方向。对非传递性竞争的研究有助于整合生物间的各种相互作用, 构建更加现实合理的生态网络, 并加深对物种共存和生物多样性维持机制的认识, 进而有助于指导生物多样性的保护和恢复工作。     

基于竞争和扩散能力的非捕食-被捕食集合种群系统的竞争机制

对于非捕食 被捕食(食饵)生态系统,强弱物种之间存在一定的竞争影响.在不考虑栖息地毁坏的情况下,引进双向竞争机制,将Tilman的单向竞争模式推广为n集合种群双向竞争模型,并对6-集合种群的竞争动态进行了计算机模拟研究.结果表明,在平衡态,种群竞争共存的条件是其竞争能力与扩散能力呈现指数型负相关关系,竞争的结果使物种的强弱序列发生变化;物种竞争排除与共存受迁移扩散能力和竞争能力影响很大,在局域斑块上竞争排斥的集合种群在广域尺度上可以竞争共存,即逃亡共存.     

Early-season effects of supplemented solar UV-B radiation on seedling emergence, canopy structure, simulated stand photosynthesis and competition for light

Mixtures and monocultures of wheat (Triticum aestivum) and wild oat (Avena fatua), a common weedy competitor of wheat, were exposed to enhanced solar UV-B radiation simulating a 20% reduction in stratospheric ozone to assess the timing and seasonal development of the UV-B effects on light competition in these species. Results from two years of field study revealed that UV-B enhancement had no detectable effect on the magnitude or timing of seedling emergence in either species. End-of-season measurements showed significant UV-B inhibition of leaf insertion height in wild oat in mixture and monoculture in the second year (irrigated year) but not in the first year (drought year). Leaf insertion height of wheat was not affected by UV-B in either year. The UV-B treatment had no detectable effect on monoculture or total (combined species) mixture LAI but did significantly increase (5–7%) the fractional contribution of wheat to the mixture LAI after four weeks of growth in both years. In addition, the UV-B treatment had subtle effects on LAI height profiles with early season mixtures showing significant reductions in wild oat LAI in lower canopy layers in both years while midseason Year 2 mixtures showed significant reductions in wild oat LAI in upper canopy layers. The changes in canopy structure were found to significantly increase (6–7%) the proportional simulated clear sky canopy photosynthesis and light interception of wheat in mixture. These findings, and others, indicate that the effects of UV-B enhancement on competition are realized very early in canopy development and provide additional support for the hypothesis that UV-B enhancement may shift the balance of competition between these species indirectly by altering competitive interactions for light.     

Carbon allocation and competition maintain variation in plant root mutualisms

Plants engage in multiple root symbioses that offer varying degrees of benefit. We asked how variation in partner quality persists using a resource‐ratio model of population growth. We considered the plant's ability to preferentially allocate carbon to mutualists and competition for plant carbon between mutualist and nonmutualist symbionts. We treated carbon as two nutritionally interchangeable, but temporally separated, resources—carbon allocated indiscriminately for the construction of the symbiosis, and carbon preferentially allocated to the mutualist after symbiosis establishment and assessment. This approach demonstrated that coexistence of mutualists and nonmutualists is possible when fidelity of the plant to the mutualist and the cost of mutualism mediate resource competition. Furthermore, it allowed us to trace symbiont population dynamics given varying degrees of carbon allocation. Specifically, coexistence occurs at intermediate levels of preferential allocation. Our findings are consistent with previous empirical studies as well the application of biological market theory to plantroot symbioses.     

Reproductive efficiency and shade avoidance plasticity under simulated competition

Plant strategy and life‐history theories make different predictions about reproductive efficiency under competition. While strategy theory suggests under intense competition iteroparous perennial plants delay reproduction and semelparous annuals reproduce quickly, life‐history theory predicts both annual and perennial plants increase resource allocation to reproduction under intense competition. We tested (1) how simulated competition influences reproductive efficiency and competitive ability (CA) of different plant life histories and growth forms; (2) whether life history or growth form is associated with CA; (3) whether shade avoidance plasticity is connected to reproductive efficiency under simulated competition. We examined plastic responses of 11 herbaceous species representing different life histories and growth forms to simulated competition (spectral shade). We found that both annual and perennial plants invested more to reproduction under simulated competition in accordance with life‐history theory predictions. There was no significant difference between competitive abilities of different life histories, but across growth forms, erect species expressed greater CA (in terms of leaf number) than other growth forms. We also found that shade avoidance plasticity can increase the reproductive efficiency by capitalizing on the early life resource acquisition and conversion of these resources into reproduction. Therefore, we suggest that a reassessment of the interpretation of shade avoidance plasticity is necessary by revealing its role in reproduction, not only in competition of plants.     

Tolerance to apical and leaf damage of Raphanus raphanistrum in different competitive regimes

Tolerance to herbivory is an adaptation that promotes regrowth and maintains fitness in plants after herbivore damage. Here, we hypothesized that the effect of competition on tolerance can be different for different genotypes within a species and we tested how tolerance is affected by competitive regime and damage type. We inflicted apical or leaf damage in siblings of 29 families of an annual plant Raphanus raphanistrum (Brassicaceae) grown at high or low competition. There was a negative correlation of family tolerance levels between competition treatments: plant families with high tolerance to apical damage in the low competition treatment had low tolerance to apical damage in the high competition treatment and vice versa. We found no costs of tolerance, in terms of a trade‐off between tolerance to apical and leaf damage or between tolerance and competitive ability, or an allocation cost in terms of reduced fitness of highly tolerant families in the undamaged state. High tolerance bound to a specific competitive regime may entail a cost in terms of low tolerance if competitive regime changes. This could act as a factor maintaining genetic variation for tolerance.     

The Enemy of My Enemy Hypothesis: Why Coexisting with Grasses May Be an Adaptive Strategy for Savanna Trees

Savannas are characterized by the coexistence of trees and flammable grasses. Yet, tree–grass coexistence has been labeled as paradoxical—how do these two functional groups coexist over such an extensive area, despite being generally predisposed to excluding each other? For instance, many trees develop dense canopies that limit grass growth, and many grasses facilitate frequent/intense fires, increasing tree mortality. This study revisits tree–grass coexistence with a model of hierarchical competition between pyrogenic grasses, “forest trees” adapted to closed-canopy competition, and “savanna trees” that are inferior competitors in closed-canopy communities, but more resistant to fire. The assumptions of this model are supported by empirical observations, including a systematic review of savanna and forest tree community composition reported here. In general, the model simulations show that when savanna trees exert weaker competitive effects on grasses, a self-reinforcing grass community is maintained, which limits forest tree expansion while still allowing savanna trees to persist (albeit as a subdominant to grasses). When savanna trees exert strong competitive effects on grasses, savanna trees cover increases initially, but as grasses decline their inhibitory effect on forest trees weakens, allowing forest trees to expand and exclude grasses and savanna trees. Rather than paradoxical, these results suggest that having weaker competitive effects on grasses may be advantageous for savanna trees, leading to greater long-term abundance and stability. We label this the “enemy of my enemy hypothesis,” which might apply to species coexistence in communities defined by hierarchical competition or with species capable of generating strong ecological feedbacks.