Empirical observations in marine ecosystems have suggested a balance of biological and advection time scales as a possible explanation of species coexistence. To characterise this scenario, we measure the time to fixation in neutrally evolving populations in chaotic flows. Contrary to intuition the variation of time scales does not interpolate straightforwardly between the no-flow and well-mixed limits; instead we find that fixation is the slowest at intermediate Damköhler numbers, indicating long-lasting coexistence of species. Our analysis shows that this slowdown is due to spatial organisation on an increasingly modularised network. We also find that diffusion can either slow down or speed up fixation, depending on the relative time scales of flow and evolution.
Cyclic dominance between species may yield spiral waves that are known to provide a mechanism enabling persistent species coexistence. This observation holds true even in presence of spatial heterogeneity in the form of quenched disorder. In this work we study the effects on spatio-temporal patterns and species coexistence of structured spatial heterogeneity in the form of habitats that locally provide one of the species with an advantage. Performing extensive numerical simulations of systems with three and six species we show that these structured habitats destabilize spiral waves. Analyzing …
Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species allowing other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the parameter setting the rate at which one species invades another changes, from a coexistence t…
Local coexistence of species in large ecosystems is traditionally explained within the broad framework of niche theory. However, its rationale hardly justifies rich biodiversity observed in nearly homogeneous environments. Here we consider a consumer-resource model in which a coarse-graining procedure accounts for a variety of ecological mechanisms and leads to effective spatial effects which favour species coexistence. Herein, we provide conditions for several species to live in an environment with very few resources. In fact, the model displays two different phases depending on whether the n…
This letter introduced a new R package 'coexist' which can perform species coexistence simulation and analysis. The package was initially developed for understanding the role of different combinations of varying species growth rates, dispersal ability, habitat heterogeneity, and competition interaction on influencing species coexistence under fluctuating environment. Further versions could introduce more coexistence models under different ecological conditions. The source package is available at the Google Code (https://code.google.com/p/coexist/) and R CRAN repository (http://cran.r-project.o…
We introduce an asymmetric noisy voter model to study the joint effect of immigration and a competition-dispersal tradeoff in the dynamics of two species competing for space in regular lattices. Individuals of one species can invade a nearest-neighbor site in the lattice, while individuals of the other species are able to invade sites at any distance but are less competitive locally, i.e., they establish with a probability $g \le 1$. The model also accounts for immigration, modeled as an external noise that may spontaneously replace an individual at a lattice site by another individual of the …
Despite the general acknowledgment of the role of niche and fitness differences in community dynamics, species abundance has been coined as a relevant feature not just regarding niche perspectives, but also according to neutral perspectives. Here we explore a minimum probabilistic stochastic model to evaluate the role of populations relative initial and total abundances on species chances to outcompete each other and their persistence in time (i.e., unstable coexistence). The present results show that taking into account the stochasticity in demographic properties and conservation of individua…
The outcome of competition among species is influenced by the spatial distribution of species and effects such as demographic stochasticity, immigration fluxes, and the existence of preferred habitats. We introduce an individual-based model describing the competition of two species and incorporating all the above ingredients. We find that the presence of habitat preference --- generating spatial niches --- strongly stabilizes the coexistence of the two species. Eliminating habitat preference --- neutral dynamics --- the model generates patterns, such as distribution of population sizes, practi…
The savanna biome is characterised by a continuous vegetation cover, comprised of herbaceous and woody plants. The coexistence of species in arid savannas, where water availability is the main limiting resource for plant growth, provides an apparent contradiction to the classical principle of competitive exclusion. Previous theoretical work using nonspatial models has focussed on the development of an understanding of coexistence mechanisms through the consideration of resource niche separation and ecosystem disturbances. In this paper, we propose that a spatial self-organisation principle, ca…
Species sharing a prey or a predator species may go extinct due to exploitative or apparent competition. We examine whether evolution of the shared species acts as a coexistence mechanism and to what extent the answer depends on the genetic architecture underlying trait evolution. In our models of exploitative and apparent competition, the shared species evolves its defense or prey use. Evolving species are either haploid or diploid. A single locus pleiotropically determines prey nutritional quality and predator attack rates. When pleiotropy is sufficiently antagonistic (e.g. nutritional prey …
Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality r…
The dynamics of many microbial ecosystems are driven by cross-feeding interactions, in which metabolites excreted by some species are metabolised further by others. The population dynamics of such ecosystems are governed by frequency-dependent selection, which allows for stable coexistence of two or more species. We have analysed a model of cross-feeding based on the replicator equation, with the aim of establishing criteria for coexistence in ecosystems containing three species, given the information of the three species' ability to coexist in their three separate pairs, i.e. the long term dy…
Statistical Physics has proved essential to analyze multi-agent environments. Motivated by the empirical observation of various non-equilibrium features in Barro Colorado and other ecological systems, we analyze a plant-species abundance model, presenting analytical evidence of scale-invariant plant clusters and non-trivial emergent modular correlations. Such first theoretical confirmation of a scale-invariant region, based on percolation processes, reproduces the key features in actual ecological ecosystems and can confer the most stable equilibrium for ecosystems with vast biodiversity.
In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d=1, 2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor …
We present properties of Lotka-Volterra equations describing ecological competition among a large number of competing species. First we extend to the case of a non-homogeneous niche space stability conditions for solutions representing species coexistence. Second, we discuss mechanisms leading to species clustering and obtain an analytical solution for a lumped state in a specific instance of the system. We also discuss how realistic ecological interactions may result in different types of competition coefficients.
Resource competition theory predicts coexistence and exclusion patterns based on species R*s, the minimum resource values required for a species to persist. A central assumption of the theory is that all species have equal access to resources. However, many systems are characterized by preemption exploitation, where some species deplete resources before their competitors can access them (e.g., asymmetric light competition, contest competition among animals). We hypothesize that coexistence under preemption requires an R*-preemption tradeoff, i.e., the species with the priority access should ha…
Neutral theories have played a crucial and revolutionary role in fields such as population genetics and biogeography. These theories are critical by definition, in the sense that the overall growth rate of each single allele/species/type vanishes. Thus each species in a neutral model sits at the edge between invasion and extinction, allowing for the coexistence of symmetric/neutral types. However, in finite systems, mono-dominated states are ineludibly reached in relatively short times owing to demographic fluctuations, thus leaving us with an unsatisfactory framework to rationalize empiricall…
Food webs with intraguild predation (IGP) are widespread in natural habitats. Their adaptation and resilience behaviour is principal for understanding restructuring of ecological communities. In spite of the importance of IGP food webs their behaviour even for the simplest 3-species systems has not been fully explored. One fundamental question is how an increase of diversity of the lowest trophic level impacts the persistence of higher trophic levels in IGP relationships. We analyze a 3-species food web model with a heterogeneous resources and IGP. The model consists of two predators directly …
We study a class of the stochastic May-Leonard models, with three species dominating each other in a cyclic nonhierarchical way, according to the rock-paper-scissors game. We introduce an unevenness in the system, by considering that one of the species is weaker because of a lower selection probability. The simulation results show that the pattern formation is drastically affected by the presence of the weaker species, with no spiral waves arising immediately from random initial conditions. Instead, single-species spatial domains cyclically dominate the entire territory until a region occupied…
How do species coexist? A framework known as Modern Coexistence Theory measures mechanisms of coexistence by comparing a species perturbed to low density (the invader) to other species that remain at their typical densities (the residents); this invader-resident comparison measures a rare-species advantage that results from specialization. However, there are several reasonable ways (i.e., methods) to compare invaders and residents, each differing in practicality and biological interpretation. Here, using theoretical arguments and case studies, we compare four such methods for calculating coexi…
Environmental fluctuations have important consequences in the organization of ecological communities, and understanding how such a variability influences the biodiversity of an ecosystem is a major question in ecology. In this paper, we analyze the case of two species competing for the resources within the framework of the neutral theory in the presence of environmental noise, devoting special attention on how such a variability modulates species fitness. The environment is dichotomous and stochastically alternates between periods favoring one of the species while disfavoring the other one, pr…
The paradox of the plankton highlights the apparent contradiction between Gause's law of competitive exclusion and the observed diversity of phytoplankton. It is well known that phytoplankton dynamics depend heavily on two main resources: light and nutrients. Here we treat light as a continuum of resources rather than a single resource by considering the visible light spectrum. We propose a spatially explicit reaction-diffusion-advection model to explore under what circumstance coexistence is possible from mathematical and biological perspectives. Furthermore, we provide biological context as …
Ecological trade-offs between species are often invoked to explain species coexistence in ecological communities. However, few mathematical models have been proposed for which coexistence conditions can be characterized explicitly in terms of a trade-off. Here we present a model of a plant community which allows such a characterization. In the model plant species compete for sites where each site has a fixed stress condition. Species differ both in stress tolerance and competitive ability. Stress tolerance is quantified as the fraction of sites with stress conditions low enough to allow establ…
When three species compete cyclically in a well-mixed, stochastic system of $N$ individuals, extinction is known to typically occur at times scaling as the system size $N$. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a "social temperament" to be either introverted or extroverted, leading them to cut or add links respectively, then long-living state in which all species coexist can occur when both introverts and extroverts a…
To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When…
Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among $m$ ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for $m=2$ no coexistence of the two species is usually possible, if $m \ge 3$ we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coex…
Selection Rules for Species Coexistence in a Hierarchical May-Leonard Model
Raja Rammohun Roy Mahavidyalaya（拉贾·拉姆莫汉·罗伊大学）；Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology Hyderabad（海德拉巴信息技术学院计算自然科学和生物信息中心）；Sister Nibedita Govt. General Degree College for Girls（西比尔·尼贝迪塔女子政府综合学位学院）；School of Science, Constructor University（Constructor大学理学院）；Department of Mathematics, Nalanda University（那烂陀大学数学系）
Explaining coexistence in species-rich communities of primary producers remains a challenge for ecologists because of their likely competition for shared resources. Following Hutchinson's seminal suggestion, many theoreticians have tried to create diversity through a fluctuating environment, which impairs or slows down competitive exclusion. However, fluctuating-environment models often only produce a dozen of coexisting species at best. Here, we investigate how to create richer communities in fluctuating environments, using an empirically parameterized model. Building on the forced Lotka-Volt…
Current theories about mechanisms promoting species co-existence in diverse communities assume that species only interact ecologically. Species are treated as discrete evolutionary entities, even though abundant empirical evidence indicates that patterns of gene flow such as selfing and hybridization frequently occur in plant and animal groups. Here, we allow mating behavior to respond to local species composition and abundance in a data-driven meta-community model of species co-existence. While individuals primarily out-cross, they also maintain some small capacity for selfing and hybridizati…
Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
物种的循环优势已被确定为维持生物多样性的一种潜在机制，见例如B。Kerr，M. a.莱利，M。W. Feldman和B. J. M. Bohannan [Nature {\bf 418}，171（2002）]和B. Kirkup和M. a. Riley [Nature {\bf 428}，412（2004）].通过数值模拟支持的分析方法，我们解决了这个问题，通过研究一个典型的非空间三种群随机系统，即"石头剪刀布"或循环洛特卡-沃尔泰拉模型的属性。虽然确定性的方法（速率方程）预测的物种的共存导致定期（但中性稳定）振荡的人口密度，我们表明，在系统中产生的波动与\emph{finite number of agents}大大改变了这一局面，并负责灭绝：经过足够长的时间后，三个物种中的两个灭绝。作为主要发现，我们提供了在给定时间的灭绝概率的解析估计和数值计算。我们还讨论了我们的研究结果的影响，一类广泛的竞争人口系统。
Patterned vegetation is a characteristic feature of many dryland ecosystems. While plant densities on the ecosystem-wide scale are typically low, a spatial self-organisation principle leads to the occurrence of alternating patches of high biomass and patches of bare soil. Nevertheless, intraspecific competition dynamics other than competition for water over long spatial scales are commonly ignored in mathematical models for vegetation patterns. In this chapter, I address the impact of local intraspecific competition on a modelling framework for banded vegetation patterns. Firstly, I show that …
Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics which fluctuate due to random drift. Dependence of species selection advantages on the environment introduces additional possibilities for the evolutionary dynamics. We analyse a simple model in which a random environment allows competing species to coexist for a long time before a fixation of a single species happens. In our analysis we use stability in a lin…
Eco-evolutionary frameworks can explain certain features of communities in which ecological and evolutionary processes occur over comparable timescales. Here, we investigate whether an evolutionary dynamics may interact with the spatial structure of a prey-predator community in which both species show limited mobility and predator perceptual ranges are subject to natural selection. In these conditions, our results unveil an eco-evolutionary feedback between species spatial mixing and predators perceptual range: different levels of mixing select for different perceptual ranges, which in turn re…
We analyze the long term behavior of interacting populations which can be controlled through harvesting. The dynamics is assumed to be discrete in time and stochastic due to the effect of environmental fluctuations. We present extinction and coexistence criteria when there are one or two interacting species. We then use these tools in order to see when harvesting leads to extinction or persistence of species, as well as what the optimal harvesting strategies, which maximize the expected long term yield, look like. For single species systems, we show under certain conditions that the optimal ha…
Decoding species coexistence: A reinforcement learning perspective
School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理与信息技术学院）；School of Physics and Electronic Science, East China Normal University（华东师范大学物理与电子科学学院）；School of Computer Science, Soochow University（苏州大学计算机科学与技术学院）；School of Physics, Ningxia University（宁夏大学物理学院）
We introduce and analyze a spatial Lotka-Volterra competition model with local and nonlocal interactions. We study two alternative classes of nonlocal competition that differ in how each species' characteristics determine the range of the nonlocal interactions. In both cases, nonlocal interactions can create spatial patterns of population densities in which highly populated clumps alternate with unpopulated regions. This non-populated regions provide spatial niches for a weaker competitor to establish in the community and persist in conditions in which local models predict competitive exclusio…
Neutral species facilitate coexistence among cyclically competing species under birth and death processes
School of Statistics and Mathematics, Yunnan University of Finance and Economics（云南财经大学统计与数学学院）；School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Department of Applied Mathematics, Colleage of Applied Sciences, Kyung Hee University（韩国庆熙大学应用科学学院应用数学系）；G-LAMP NEXUS Institute, Kyung Hee University（韩国庆熙大学G-LAMP NEXUS研究所）
I examine the effect of exogenous spatial heterogeneity on the coexistence of competing species using a simple model of non-hierarchical competition for site occupancy on a lattice. The sites on the lattice are divided into two types representing two different habitats or spatial resources. The model features no temporal variability, hierarchical competition, type-dependent interactions or other features traditionally known to support more competing species than there are resources. Nonetheless, stable coexistence of two habitat specialists and a generalist is observed in this model for a rang…
If two species exhibit different nonlinear responses to a single shared resource, and if each species modifies the resource dynamics such that this favors its competitor, they may stably coexist. This coexistence mechanism, known as relative nonlinearity of competition, is well understood theoretically, but less is known about its evolutionary properties and its prevalence in real communities. We address this challenge by using adaptive dynamics theory and individual-based simulations to compare community stabilization and evolutionary stability of species that coexist by relative nonlinearity…
We examine the two-dimensional extension of the model of Kessler and Sander of competition between two species identical except for dispersion rates. In this class of models, the spatial inhomogeneity of reproduction rates gives rise to an implicit cost of dispersal, due to the tendency to leave favorable locations. Then, as in the Hamilton-May model with its explicit dispersal cost, the tradeoff between dispersal case and the beneficial role of dispersal in limiting fluctuations, leads to an advantage of one dispersal rate over another, and the eventual extinction of the disadvantaged species…
How large ecosystems can create and maintain the remarkable biodiversity we see in nature is probably one of the biggest open questions in science, attracting attention from different fields, from Theoretical Ecology to Mathematics and Physics. In this context, modeling the stable coexistence of species competing for limited resources is a particularly challenging task. From a mathematical point of view, coexistence in competitive dynamics can be achieved when dominance among species forms intransitive loops. However, these relationships usually lead to species' relative abundances neutrally c…
Social interaction between microbes can be described at many levels of details, ranging from the biochemistry of cell-cell interactions to the ecological dynamics of populations. Choosing the best level to model microbial communities without losing generality remains a challenge. Here we propose to model cross-feeding interactions at an intermediate level between genome-scale metabolic models of individual species and consumer-resource models of ecosystems, which is suitable to empirical data. We applied our method to three published examples of multi-strain Escherichia coli communities with i…
Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov…
