On phytoplankton aggregation: a view from an IBM approach

In this paper, we build up an individual-based model (IBM) that describes the aggregative behavior in phytoplankton. The processes in play at the individual level (an individual=a phytoplankton cell) are: a random dispersal, a displacement due to the net effect of cells present in a suitable neighborhood (spatial interactions) and a branching (cell division and death). The IBM model provides a virtual world where phytoplankton cells appear to form clusters. Using this model, we explore the spatial structure of phytoplankton and present some numerical simulations that help the understanding of the aggregation phenomenon.     

Mechanisms of Side Branching and Tip Splitting in a Model of Branching Morphogenesis

Recent experimental work in lung morphogenesis has described an elegant pattern of branching phenomena. Two primary forms of branching have been identified: side branching and tip splitting. In our previous study of lung branching morphogenesis, we used a 4 variable partial differential equation (PDE), due to Meinhardt, as our mathematical model to describe the reaction and diffusion of morphogens creating those branched patterns. By altering key parameters in the model, we were able to reproduce all the branching styles and the switch between branching modes. Here, we attempt to explain the branching phenomena described above, as growing out of two fundamental instabilities, one in the longitudinal (growth) direction and the other in the transverse direction. We begin by decoupling the original branching process into two semi-independent sub-processes, 1) a classic activator/inhibitor system along the growing stalk, and 2) the spatial growth of the stalk. We then reduced the full branching model into an activator/inhibitor model that embeds growth of the stalk as a controllable parameter, to explore the mechanisms that determine different branching patterns. We found that, in this model, 1) side branching results from a pattern-formation instability of the activator/inhibitor subsystem in the longitudinal direction. This instability is far from equilibrium, requiring a large inhomogeneity in the initial conditions. It successively creates periodic activator peaks along the growing stalk, each of which later on migrates out and forms a side branch; 2) tip splitting is due to a Turing-style instability along the transversal direction, that creates the spatial splitting of the activator peak into 2 simultaneously-formed peaks at the growing tip, the occurrence of which requires the widening of the growing stalk. Tip splitting is abolished when transversal stalk widening is prevented; 3) when both instabilities are satisfied, tip bifurcation occurs together with side branching.     

A fully continuous individual-based model of tumor cell evolution

The aim of this work is to develop and study a fully continuous individual-based model (IBM) for cancer tumor invasion into a spatial environment of surrounding tissue. The IBM improves previous spatially discrete models, because it is continuous in all variables (including spatial variables), and thus not constrained to lattice frameworks. The IBM includes four types of individual elements: tumor cells, extracellular macromolecules (MM), a matrix degradative enzyme (MDE), and oxygen. The algorithm underlying the IBM is based on the dynamic interaction of these four elements in the spatial environment, with special consideration of mutation phenotypes. A set of stochastic differential equations is formulated to describe the evolution of the IBM in an equivalent way. The IBM is scaled up to a system of partial differential equations (PDE) representing the limiting behavior of the IBM as the number of cells and molecules approaches infinity. Both models (IBM and PDE) are numerically simulated with two kinds of initial conditions: homogeneous MM distribution and heterogeneous MM distribution. With both kinds of initial MM distributions spatial fingering patterns appear in the tumor growth. The output of both simulations is quite similar.     

Towards a resolution of ‘the paradox of the plankton’: A brief overview of the proposed mechanisms

In plankton ecology, it is a fundamental question as to how a large number of competing phytoplankton species coexist in marine ecosystems under a seemingly-limited variety of resources. This ever-green question was first proposed by Hutchinson [Hutchinson, G.E., 1961. The paradox of the plankton. Am. Nat. 95, 137–145] as ‘the paradox of the plankton’. Starting from Hutchinson [Hutchinson, G.E., 1961. The paradox of the plankton. Am. Nat. 95, 137–145], over more than four decades several investigators have put forward varieties of mechanisms for the extreme diversity of phytoplankton species. In this article, within the boundary of our knowledge, we review the literature of the proposed solutions and give a brief overview of the mechanisms proposed so far. The proposed mechanisms that we discuss mainly include spatial and temporal heterogeneity in physical and biological environment, externally imposed or self-generated spatial segregation, horizontal mesoscale turbulence of ocean characterized by coherent vortices, oscillation and chaos generated by several internal and external causes, stable coexistence and compensatory dynamics under fluctuating temperature in resource competition, and finally the role of toxin-producing phytoplankton in maintaining the coexistence and biodiversity of the overall plankton population that we have proposed recently. We find that, although the different mechanisms proposed so far is potentially applicable to specific ecosystems, a universally accepted theory for explaining plankton diversity in natural waters is still an unachieved goal.     

Competitive Asymmetry Reduces Spatial Effects on Size-Structure Dynamics in Plant Populations

The growth of each individual in plant populations was simulatedby a spatial competition model for five density levels and fourdifferent spatial distribution patterns of individuals, varyingfrom highly clumped to regular. The simulation results wereanalysed using the diffusion model for evaluating the effectsof density and distribution pattern on the size-structure dynamicsin relation to the degree of competitive asymmetry. At low densities,changes in statistics of plant weight over time such as mean,coefficient of variation, skewness, and Box-Cox-transformedkurtosis differed greatly among spatial patterns, irrespectiveof the degree of competitive asymmetry. In completely symmetriccompetition, the spatial effect on size-structure dynamics remainedrelatively large irrespective of densities, although mean plantweight became similar among the spatial patterns with increasingdensity. However, the spatial effect diminished with increaseddensity in strongly asymmetric competition, when similar sizedistributions were realized irrespective of the spatial patterns.Therefore, it was concluded that: (1) irrespective of the degreeof competitive asymmetry, spatial pattern is important for size-structuredynamics at low densities; (2) spatial pattern is nearly immaterialunder strongly asymmetric competition at high densities; and(3) under crowded conditions, neighbourhood effects are muchmore apparent at the population level in less asymmetric competition.These processes and outcomes are linked to the forms of thefunctions of mean growth rate of individuals [G(t,x) function]and variance in growth rate [D(t,x) function]. These functionsare variable depending on the spatial pattern under symmetriccompetition, but are rather stable under strongly asymmetriccompetition at high densities irrespective of the spatial patterns.Therefore, size structure under strongly asymmetric competitioncan be regarded as a stable system, whereas that under symmetriccompetition is regarded as a variable system in relation tothe spatial pattern and process. From this, it was inferredthat: (1) the goodness-of-fit of spatial competition modelsfor crowded plant populations is higher in less asymmetric competition;and (2) higher species diversity in plant communities is associatedwith the lower degree of competitive asymmetry.Copyright 1994,1999 Academic Press Asymmetric competition, diffusion model, neighbourhood effect, size-structure stability, spatial competition model, spatial distribution pattern, species diversity, symmetric competition     

Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles

Birth–death–movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth–death–movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.     

Angular smoothing and radial regularization of ODF fields: Application on deterministic crossing fiber tractography

The advent of high angular resolution diffusion imaging (HARDI) has opened up new perspectives for the delineation of crossing and branching fiber pathways. However, image acquisition under clinical conditions with limited measurement time faces the problem of poor spatial and angular resolution and the technique’s high susceptibility to noise. In this paper we present a straightforward spatial filter for ODF fields that uses the data-inherent structural information around a voxel as part of a directionally selective method for angular smoothing and radial regularization (ASRR). Especially in regions where fibers cross (multimodal voxels), the method allows us to reduce noise, improve the accuracy of ODF diffusion peaks, and strengthen signals of non-dominant fibers. Moreover, we propose a dynamic scheme in which regularization is applied only to ODFs classified as multimodal. The approach is quantitatively evaluated on synthetic datasets of various configurations. With an in vivo dataset of a human subject, measured under clinical imaging conditions, we demonstrate the method’s ability to improve tractography of non-dominant transcallosal fiber pathways and the long fibers of the superior longitudinal fasciculus.     

Phytoplankton defence mechanisms: traits and trade‐offs

In aquatic ecosystems, unicellular algae form the basis of the food webs. Theoretical and experimental studies have demonstrated that one of the mechanisms that maintain high diversity of phytoplankton is through predation and the consequent evolution of defence mechanisms. Proposed defence mechanisms in phytoplankton are diverse and include physiological (e.g. toxicity, bioluminescence), morphological (e.g. silica shell, colony formation), and behavioural (e.g. escape response) traits. However, the function of many of the proposed defence mechanisms remains elusive, and the costs and benefits (trade‐offs) are often unquantified or undocumented. Here, we provide an overview of suggested phytoplankton defensive traits and review their experimental support. Wherever possible we quantify the trade‐offs from experimental evidence and theoretical considerations. In many instances, experimental evidence suggests that defences are costless. However, we argue that (i) some costs materialize only under natural conditions, for example, sinking losses, or dependency on the availability of specific nutrients, and (ii) other costs become evident only under resource‐deficient conditions where a rivalry for limiting resources between growth and defence occurs. Based on these findings, we suggest two strategies for quantifying the costs of defence mechanisms in phytoplankton: (i) for the evaluation of defence costs that are realized under natural conditions, a mechanistic understanding of the hypothesized component processes is required; and (ii) the magnitude of the costs (i.e. growth reduction) must be assessed under conditions of resource limitation.     

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.     

Vertical distribution and composition of phytoplankton under the influence of an upper mixed layer

The vertical distribution of phytoplankton is of fundamental importance for the dynamics and structure of aquatic communities. Here, using an advection-reaction-diffusion model, we investigate the distribution and competition of phytoplankton species in a water column, in which inverse resource gradients of light and a nutrient can limit growth of the biomass. This problem poses a challenge for ecologists, as the location of a production layer is not fixed, but rather depends on many internal parameters and environmental factors. In particular, we study the influence of an upper mixed layer (UML) in this system and show that it leads to a variety of dynamic effects: (i) Our model predicts alternative density profiles with a maximum of biomass either within or below the UML, thereby the system may be bistable or the relaxation from an unstable state may require a long-lasting transition. (ii) Reduced mixing in the deep layer can induce oscillations of the biomass; we show that a UML can sustain these oscillations even if the diffusivity is less than the critical mixing for a sinking phytoplankton population. (iii) A UML can strongly modify the outcome of competition between different phytoplankton species, yielding bistability both in the spatial distribution and in the species composition. (iv) A light limited species can obtain a competitive advantage if the diffusivity in the deep layers is reduced below a critical value. This yields a subtle competitive exclusion effect, where the oscillatory states in the deep layers are displaced by steady solutions in the UML. Finally, we present a novel graphical approach for deducing the competition outcome and for the analysis of the role of a UML in aquatic systems.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Harmful algal blooms: combining excitability and competition

Harmful algal blooms (HABs) characterized by a large concentration of toxic species appear rather rarely, but have a severe impact on the whole ecosystem. To study on possible trigger mechanisms for the emergence of HABs, we consider a nutrient-phytoplankton-zooplankton model to find the conditions under which a toxic phytoplankton species is able to form a bloom by winning the competition against its nontoxic competitor. The basic mechanism is related to the excitability of the system, i.e., the ability to develop a large response on certain perturbations. In a large class of models, a HAB results from a combined effect of nutrient enrichment and selective predation on different phytoplankton populations by zooplankton. We show that the severity of HAB is controlled by nutrient enrichment and zooplankton abundance, while the frequency of its occurrence depends on the strength of selectivity of predation. Thereby the intricate interplay between excitability, competition, and selective grazing pressure builds the backbone of the mechanism of the emergence of HABs.     

球形棕囊藻囊体形成中光照、营养盐和共存硅藻的影响

在不同光照和营养盐结构条件下半连续培养球形棕囊藻和3种硅藻,研究光照、营养盐限制和硅藻竞争对球形棕囊藻囊体形成的影响。结果表明:高光照显著促进了藻类的生长,球形棕囊藻在低光环境下几乎不形成囊体。球形棕囊藻和3种硅藻对光限制和P限制更加敏感,而在N限制环境中均具有相对较高的生物量。粒径较小的球形棕囊藻游离单细胞和中肋骨条藻在营养盐和光限制条件下比粒径较大的细胞具有更强的竞争能力。硝酸盐是球形棕囊藻囊体形成的营养基础,但是营养盐结构并未改变棕囊藻囊体形态。具有两种生活史状态有利于球形棕囊藻度过资源限制的环境,从而有利于球形棕囊藻在硅藻藻华之后再次形成藻华。     

How competitive intransitivity and niche overlap affect spatial coexistence

Competitive intransitivity is mostly considered outside the main body of coexistence theories that rely primarily on the role of niche overlap and differentiation. How the interplay of competitive intransitivity and niche overlap jointly affects species coexistence has received little attention. Here, we consider a rock–paper–scissors competition system where interactions between species can represent the full spectra of transitive–intransitive continuum and niche overlap/differentiation under different levels of competition asymmetry. By comparing results from pair approximation that only considers interference competition between neighbouring cells in spatial lattices, with those under the mean-field assumption, we show that 1) species coexistence under transitive competition is only possible at high niche differentiation; 2) in communities with partial or pure intransitive interactions, high levels of niche overlap are not necessary to beget species extinction; and 3) strong spatial clustering can widen the condition for intransitive loops to facilitate species coexistence. The two mechanisms, competitive intransitivity and niche differentiation, can support species persistence and coexistence, either separately or in combination. Finally, the contribution of intransitive loops to species coexistence can be enhanced by strong local spatial correlations, modulated and maximised by moderate competition asymmetry. Our study, therefore, provides a bridge to link intransitive competition to other generic ecological theories of species coexistence.     

Association between temporal and spatial beta diversity in phytoplankton

The rates of temporal and spatial species turnover have been compared in different organisms and scales, revealing that both are not independent but, rather, associated. However, the knowledge is limited for the association between spatial turnover and temporal turnover. Here, we performed two investigations of the phytoplankton composition in the lakes of the Yangtze River catchment in China in the spring and summer of 2012, which covered regional spatial scale and two‐season temporal scale. We analysed the association between temporal and spatial species turnover in phytoplankton. The results showed that 1) the two‐season temporal turnover of phytoplankton varied based on the mean values and the coefficient of variance of environmental variables, and pH was the most important variable negatively affecting the temporal turnover; 2) the spatial beta diversity of phytoplankton in summer was higher than that in spring, and the distance decay pattern was significant in summer, but not in spring; 3) the variation in spatial turnover in spring and summer was attributed to the primary environmental variables (nitrogen, phosphorus and underwater available light) and broader‐scale spatial variables; 4) the proportion of jointly explained variation of spatial Bray–Curtis dissimilarity by the environment and space increased from ~38% (spring) to ~55% (summer), which was mainly due to the variation in spatially structured environmental variables during the two‐season temporal turnover, such as pH and ion concentrations; 5) the community compositions in summer were more similar between the lakes with similar two‐season temporal turnover. These results indicate that the spatial turnover of phytoplankton composition in summer was partially predetermined by the variation in environmental variables and phytoplankton composition during the process of two‐season temporal turnover, and highlight the understanding of temporal variations in spatial beta diversity as well as the underlying assembly mechanisms in phytoplankton.     

Biological control of phytoplankton by the subtropical submerged macrophytes Egeria densa and Potamogeton illinoensis: a mesocosm study

1. In temperate regions, submerged macrophytes can hamper phytoplankton blooms. Such an effect could arise directly, for instance via allelopathy, or indirectly, via competition for nutrients or the positive interaction between submerged macrophytes and zooplankton grazing. However, there is some evidence that the positive interaction between submerged macrophytes and zooplankton grazing is less marked in warmer regions, where the interaction is less well studied, and that negative effects of higher water plants on phytoplankton biomass are weaker. 2. We carried out two consecutive mesocosm experiments in Uruguay (subtropical South America) to study the effects of two common submerged macrophytes from this region (Egeria densa and Potamogeton illinoensis) on phytoplankton biomass, in the absence of zooplankton grazing. We compared phytoplankton development between different macrophyte treatments (no macrophytes, artificial macrophytes, real Egeria and real Potamogeton). We used artificial macrophytes to differentiate between physical effects (i.e. shading, sedimentation and competition with periphyton) and biological effects (i.e. nutrient competition and allelopathy). 3. In Experiment 1, we found no evidence for physical effects of macrophytes on phytoplankton biomass, but both macrophyte species seemed to exert strong biological effects on phytoplankton biomass. Only Egeria affected phytoplankton community structure, particularly tempering the dominance of Scenedesmus. Nutrient addition assays revealed that only Egeria suppressed phytoplankton through nutrient competition. 4. We performed a second mesocosm experiment with the same design, but applying saturating nutrient conditions as a way of excluding the effects of competition for nutrients. This experiment showed that both macrophytes were still able to suppress phytoplankton through biological mechanisms, providing evidence for allelopathic effects. Our results indicate that both common macrophytes are able to keep phytoplankton biomass low, even in the absence of zooplankton grazing.     

Environmental and spatial processes influencing phytoplankton biomass along a reservoirs‐river‐floodplain lakes gradient: A metacommunity approach

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Evolutionary dynamics through multispecies competition

Disruptive selection, emerging from frequency-dependent intraspecific competition can have very exciting evolutionary outcomes. One such outcome is the origin of new species through an evolutionary branching event. Literature on theoretical models investigating the emergence of disruptive selection is vast, with some investigating the sensitivity of the models on assumptions of the competition and carrying capacity functions’ shapes. What is seldom modeled is what happens once the population escapes its effect via increase phenotypic or genotypic variance. The expectation is mixed: disruptive selection could diminish and ultimately disappear or it could still exist leading to further speciation events through multiple evolutionary branching events. Here, we derive the conditions under which disruptive selection drives two subpopulations that originated at a branching point to other points in trait space where each subpopulation again experiences disruptive selection. We show that the general pattern for further branchings require that the competition function to be even narrower than what is required for the first evolutionary branching. However, we also show that the existence of disruptive selection in higher dimensional systems is also sensitive to the shapes of the functions used.     

Myosin II has distinct functions in PNS and CNS myelin sheath formation

The myelin sheath forms by the spiral wrapping of a glial membrane around the axon. The mechanisms responsible for this process are unknown but are likely to involve coordinated changes in the glial cell cytoskeleton. We have found that inhibition of myosin II, a key regulator of actin cytoskeleton dynamics, has remarkably opposite effects on myelin formation by Schwann cells (SC) and oligodendrocytes (OL). Myosin II is necessary for initial interactions between SC and axons, and its inhibition or down-regulation impairs their ability to segregate axons and elongate along them, preventing the formation of a 1:1 relationship, which is critical for peripheral nervous system myelination. In contrast, OL branching, differentiation, and myelin formation are potentiated by inhibition of myosin II. Thus, by controlling the spatial and localized activation of actin polymerization, myosin II regulates SC polarization and OL branching, and by extension their ability to form myelin. Our data indicate that the mechanisms regulating myelination in the peripheral and central nervous systems are distinct.