A theorem for the invasion triangle and its applicability for invasion biology

The triangle conceptual model is a construct that is foundational across several fields of the natural sciences including the study of diseases, invasive species, and fire. The invasion triangle incorporates the complex ecological and evolutionary interactions between the qualities of the abiotic environment, the invader, and the biotic interactions that describes or predicts the impacts of the invasive species. Although the triangle concept is widely used among fields, to date there has not been an analytical implementation of the model. Current modelling in invasion biology often only considers the effects of one or two factors on the outcomes of species introductions. A mathematical implementation of the triangle model will allow a more comprehensive consideration of the various ecological factors. Here, we provide the first mathematical theorem for an interpretation of the invasion triangle that allows for the consideration of time. This new analytical development of the triangle is flexible, and can be used to model the spatial and temporal population dynamics observed in invasions. We also describe the conditions under which invasion is maintained when factors change with opposing effects. In this interpretation, the lower limits for invasion are explicitly defined and each component can move independently. The complexity of the interactions between factors contributing to invasions is integrated into the single model, such as those suggested by major invasion hypotheses. We briefly describe how the theorem can be applied to account for various phenomena in range dynamics using rapid range expansion and the time lag in invasions as examples. Future work can explicitly define the interdependence among components to suit more specific questions.     

Using range filling rather than prevalence of invasive plant species for management prioritisation: the case of Spathodea campanulata in the Society Islands (South Pacific)

Biological invasion science lacks standardised measures of invasion success that would provide effective prioritisation of invasive species and invaded areas management. Prevalence (area of occupancy) of invasive species is often used as proxy of their success but this metric ignores the extent to which a species fills its potential distribution. This study aims to estimate the performance of invasive tree species by computing the ratio between the compressed canopy area (CCA), assessed through remote sensing, and their potential distribution, estimated using invasive species distribution modelling. This index of ‘range filling’ (RF) has applicability to a broad set of invasive plant species in any biome. A case study is provided using the invasive African tulip tree Spathodea campanulata (Bignoniaceae) on three small tropical oceanic islands (South Pacific) exhibiting different invasion levels to test for differences between CCA and RF. The results show that the RF of Spathodea campanulata varied within islands depending on elevation but not proportionally to the CCA of the species. Another key result was that the RF of the species and its CCA provided different between-island perspectives on the invasions and lead to distinct ranking among islands to prioritise for management. Therefore, managers should disregard species’ prevalence as a measure of success and rather weight it with potential distribution to quantify how an invader is performing in a given environment.     

Revisiting the use of the invasive species concept: An empirical approach

Invasion science has not been developed without controversies. Two questions that are still unsolved are: what is an invasive species?, and are invasive species an inherent conservation problem? These questions have led to discussions about effects versus origins. In contrast to the definitional problems, a unified framework describing invasion as a step‐by‐step process has been widely accepted. I conducted a bibliographic search with two separate databases searching for (i) evidence of less use of controversial terms over time; (ii) how many articles defined ‘invasive species’; (iii) the criteria used to define a species as invasive; and (iv) in which stage of the invasion continuum were species labelled as invasive located. My results show that controversial terms are widely used, that authors rarely define ‘invasive species’ and, often, it is very complicated to determine which criterion they used. In addition, only a fraction of the species labelled as invasive could be classified as such according to the unified framework of invasion stages. This is not a merely semantic issue, because invasive is a strong and value‐laden term that is used to guide environmental agendas. The uncritical use of a key concept could hamper research, complicate communication among peers and produce mixed results.     

Managing invasive species in cities: a decision support framework applied to Cape Town

It has been suggested that existing frameworks for guiding management of invasive species in rural areas and protected areas are inadequate for dealing with invasions in urban settings. A framework for selecting appropriate goals for managing invasive species in urban areas was developed by Gaertner et al. (Landsc Urban Plan 151:1–9, 2016). This framework groups species into three management approaches (control priority, active engagement, and tolerance) depending on their real or perceived benefits and their potential to generate negative impacts. This study tests the practical application of the framework using the example of Cape Town. We assess the suitability of the framework to support decision-making for managing invasive species in urban ecosystems using a questionnaire-based survey of members of the public, and an e-mail-based survey and a workshop with invasion biology researchers and managers. Specifically, we (1) determine the differences in perceptions regarding the benefits and impacts of invasive species between the public, managers and researchers; (2) investigate how consistently managers and researchers group invasive species into the three management categories; and (3) identify, with the help of managers and researchers, issues linked to the framework and give suggestions to overcome the identified issues. We found no clear pattern in the perceptions of the public, managers and researchers regarding perceived benefits and negative impacts of invasive species. Instead, the answers were widely scattered among all groups for most of the species that were considered. However, using the framework leads to a higher consistency among managers in placing the species into management categories, compared to invasive species grouping without guidance of the framework. We conclude that decision-support frameworks can assist managers in placing invasive species into management categories. However, even more specific guidelines on the use of invasive species management frameworks in urban areas are needed.     

Impact: Toward a Framework for Understanding the Ecological Effects of Invaders

Although ecologists commonly talk about the impacts of nonindigenous species, little formal attention has been given to defining what we mean by impact, or connecting ecological theory with particular measures of impact. The resulting lack of generalizations regarding invasion impacts is more than an academic problem; we need to be able to distinguish invaders with minor effects from those with large effects in order to prioritize management efforts. This paper focuses on defining, evaluating, and comparing a variety of measures of impact drawn from empirical examples and theoretical reasoning. We begin by arguing that the total impact of an invader includes three fundamental dimensions: range, abundance, and the per-capita or per-biomass effect of the invader. Then we summarize previous approaches to measuring impact at different organizational levels, and suggest some new approaches. Reviewing mathematical models of impact, we argue that theoretical studies using community assembly models could act as a basis for better empirical studies and monitoring programs, as well as provide a clearer understanding of the relationship among different types of impact. We then discuss some of the particular challenges that come from the need to prioritize invasive species in a management or policy context. We end with recommendations about how the field of invasion biology might proceed in order to build a general framework for understanding and predicting impacts. In particular, we advocate studies designed to explore the correlations among different measures: Are the results of complex multivariate methods adequately captured by simple composite metrics such as species richness? How well are impacts on native populations correlated with impacts on ecosystem functions? Are there useful bioindicators for invasion impacts? To what extent does the impact of an invasive species depend on the system in which it is measured? Three approaches would provide new insights in this line of inquiry: (1) studies that measure impacts at multiple scales and multiple levels of organization, (2) studies that synthesize currently available data on different response variables, and (3) models designed to guide empirical work and explore generalities.     

A neutral terminology to define 'invasive' species

The use of simple terms to articulate ecological concepts can confuse ideological debates and undermine management efforts. This problem is particularly acute in studies of nonindigenous species, which alternatively have been called ‘exotic’, ‘introduced’, ‘invasive’ and ‘naturalised’, among others. Attempts to redefine commonly used terminology have proven difficult because authors are often partial to particular definitions. In an attempt to form a consensus on invasion terminology, we synthesize an invasional framework based on current models that break the invasion process into a series of consecutive, obligatory stages. Unlike previous efforts, we propose a neutral terminology based on this framework. This ‘stage‐based’ terminology can be used to supplement terms with ambiguous meanings (e.g. invasive, introduced, naturalized, weedy, etc.), and thereby improve clarity of future studies. This approach is based on the concept of ‘propagule pressure’ and has the additional benefit of identifying factors affecting the success of species at each stage. Under this framework, invasions can be more objectively understood as biogeographical, rather than taxonomic, phenomena; and author preferences in the use of existing terminology can be addressed. An example of this recommended protocol might be: ‘We examined distribution data to contrast the characteristics of invasive species (stages IVa and V) and noninvasive species (stages III and IVb)’.     

Applying competition theory to invasion: resource impacts indicate invasion mechanisms in California shrublands

Despite widespread work documenting invasion, it remains a challenge to determine invasion mechanisms and incorporate them into invasive species management. Competition theory presents a strong model for evaluating the role of resource reduction and requirements in invasion. Additionally, alternative models suggest fluctuations in resources, niche differences, or non-resource priority effects are key factors determining invasion success. We propose a comparative framework that incorporates resource impacts of native and invasive species, performance in controlled invasion trials, and long-term natural invasion patterns to elucidate relative importance of these invasion mechanisms. To demonstrate this framework, we established monocultures of two representative native and two invasive plant species in Southern California’s coastal sage scrub (CSS), measured resource impacts (i.e., R*), and conducted invasion trials to test whether resource impacts predicted invasion success. We then related experimental results to field invasion patterns. Compared to exotic herbaceous species, native shrubs were associated with greater resource depletion of key resources: light, soil water (at multiple depths), and soil inorganic nitrogen (particularly at depth). In invasion trials, natives resisted invasion by the exotics, as resource depletion measures would predict. However, these results did not follow long-term natural invasion patterns indicating that these exotic species invade areas once dominated by native shrubland. Applying our results to the invasion framework, we conclude that disturbance, or a similar mechanism causing resources to fluctuate, is needed for exotics to invade CSS habitats. This resource-based comparative analysis of invasion mechanisms can point out important processes and help suggest effective management actions.     

Environment and dispersal paths override life strategies and residence time in determining regional patterns of invasion by alien plants

We describe a novel dissimilarity framework to analyze spatial patterns of species diversity and illustrate it with alien plant invasions in Northern Portugal. We used this framework to test the hypothesis that patterns of alien invasive plant species richness and composition are differently affected by differences in climate, land use and landscape connectivity (i.e. Geographic distance as a proxy and vectorial objects that facilitate dispersal such as roads and rivers) between pairs of localities at the regional scale. We further evaluated possible effects of plant life strategies (Grime's C-S-R) and residence time. Each locality consisted of a 1 km² landscape mosaic in which all alien invasive species were recorded by visiting all habitat types.Multi-model inference revealed that dissimilarity in species richness is more influenced by environmental distance (particularly climate), whereas geographic distance (proxies for dispersal limitations) is more important to explain dissimilarity in species composition, with a prevailing role for ecotones and roads. However, only minor differences were found in the responses of the three C-S-R strategies. Some effect of residence time was found, but only for dissimilarity in species richness. Our results also indicated that environmental conditions (e.g. climate conditions) limit the number of alien species invading a given site, but that the presence of dispersal corridors determines the paths of invasion and therefore the pool of species reaching each site. As geographic distances (e.g. ecotones and roads) tend to explain invasion at our regional scale highlights the need to consider the management of alien invasions in the context of integrated landscape planning. Alien species management should include (but not be limited to) the mitigation of dispersal pathways along linear infrastructures. Our results therefore highlight potentially useful applications of the novel multimodel framework to the anticipation and management of plant invasions.     

Impacts of invasive biota in forest ecosystems in an aboveground–belowground context

Forest ecosystems world-wide are being subjected to invasion by organisms representing all domains of life. Here we use a combined aboveground-belowground approach to provide a conceptual framework for assessing how forests respond to biological invasions. We first address mechanisms by which invasive plants and aboveground and belowground consumers impact on forests, and highlight that although we have a growing understanding of the determinants of the effects of invasive plants, for invasive consumers we have yet to move from a series of iconic case studies to the development of general principles. We also address the effects of invasive biota in the context of the drivers of invasion, co-invasion and invasional meltdown, the issue of simultaneous species gains and losses, and forest restoration and recovery post-invasion. We then highlight areas that would benefit from further work, particularly regarding underlying mechanisms, determinants of context-dependency of invader effects, and linkages between causes and consequences of invasion. In concluding, we emphasize that biological invaders have the potential for large-scale and long-term impacts on forest processes, and consideration of these impacts in an aboveground-belowground context will enable better prediction of future responses of forests to invaders and their management as well as of restoration efforts.     

Ecology and management of invasive Pinaceae around the world: progress and challenges

Many species in the family Pinaceae are invaders. These species are relatively easy to control because of some of their intrinsic characteristics and because they are highly visible and easy to eliminate. Many Pinaceae species have been well studied because of their use in forestry and their invasive behavior in many countries. The impacts of invasive Pinaceae are not only ecological, but also economic and social. We review the ecology and management of Pinaceae invasions and explore how restoration of invaded areas should be addressed. There are many ways to prevent invasions and to deal with them. Planting less invasive species, better site selection, and invasion monitoring are used successfully in different parts of the world to prevent invasion. Mechanical and chemical methods are used effectively to control Pinaceae invasions. Control is more effective at the early stages of invasion. Old invasions are more problematic as their elimination is more expensive, and the restoration of native vegetation is challenging. In some areas, native vegetation cannot thrive after Pinaceae have been removed, and weeds colonize cleared areas. More attention is needed to prevent the initiation and spread of invasions by focusing control interventions at early stages of invasion. Finding new ways of dealing sustainably with conflicts of interest between foresters and conservationists is crucial. Non-native Pinaceae are important parts of the economies and landscapes in several countries and they will continue to play such a role in the future. Despite the numerous challenges facing Pinaceae invasion management, several approaches can be successful at controlling them. Proper application of current techniques and development of more efficient ones is needed if the goal of maximizing benefits and minimizing negative impacts is to be achieved.     

Secondary invasion: When invasion success is contingent on other invaders altering the properties of recipient ecosystems

Positive interactions between exotic species may increase ecosystem‐level impacts and potentially facilitate the entry and spread of other exotic species. Invader‐facilitated invasion success—”secondary invasion”—is a key conceptual aspect of the well‐known invasional meltdown hypothesis, but remains poorly defined and empirically underexplored. Drawing from heuristic models and published empirical studies, we explore this form of “secondary invasion” and discuss the phenomenon within the recognized conceptual framework of the determinants of invasion success. The term “secondary invasion” has been used haphazardly in the literature to refer to multiple invasion phenomena, most of which have other more accepted titles. Our usage of the term secondary invasion is akin to “invader‐facilitated invasion,” which we define as the phenomenon in which the invasion success of one exotic species is contingent on the presence, influence, and impacts of one or more other exotic species. We present case studies of secondary invasion whereby primary invaders facilitate the entry or establishment of exotic species into communities where they were previously excluded from becoming invasive. Our synthesis, discussion, and conceptual framework of this type of secondary invasion provides a useful reference to better explain how invasive species can alter key properties of recipient ecosystems that can ultimately determine the invasion success of other species. This study increases our appreciation for complex interactions following invasion and highlights the impacts of invasive species themselves as possible determinants of invasion success. We anticipate that highlighting “secondary invasion” in this way will enable studies reporting similar phenomena to be identified and linked through consistent terminology.     

French broom (Teline monspessulana) invasion in south-central Chile depends on factors operating at different spatial scales

The interaction between plant attributes, environmental factors and the history of human intervention determines which species show a more invasive behavior and which areas are more likely to be invaded. Identifying which factors favor the invasion and which constraint it, remains an essential goal for understanding the process of invasion and to provide a stronger scientific basis for designing management actions to reduce susceptibility to invasion. In this paper we analyze the relative role of different environmental factors in plant invasion, in particular the comparison between human versus climatic and biotic variables at different spatial scales. Furthermore, we show how these factors interact to influence the distribution and abundance of an invasive plant. We have built mixed models to explain the presence and abundance of a renowned invasive species in the south-central area of Chile, Teline monspessulana (French broom, Fabaceae). We recorded the presence and cover of this invasive shrub in a series of transects along the main roads of the study area. For each transect we recorded variables at different spatial scales related to climate and land use. We found that the presence and cover of T. monspessulana are strongly related to proximity to roads, urban areas and locations with higher rainfall. Although this shrub is present in most of the landscape, it is most abundant in shrublands and forest plantations. In a complex human-dominated landscape matrix, shrublands and forest plantations act as a source of propagules from which new areas can be invaded. Interestingly, the presence of this species can be better predicted by combining predictors taken at two hierarchical levels, the so-called transect and plot scales, whereas the species abundance only depends on factors at the plot level. The presence and abundance of this species is highly dependent on human disturbance, and the occurrence of certain landscape elements closely linked to land management is critical to the performance of this species. Maintaining a dense tree canopy cover and reducing the use of fire could help reduce the rate of expansion of this species across the landscape.     

Niche based distribution modelling of an invasive alien plant: effects of population status, propagule pressure and invasion history

Forecasting the spatial spread of invasive species is important to inform management planning. Niche-based species distribution models offer a well-developed framework for assessing the potential range of species. However, these models assume equilibrium between the species’ distribution and its ecological requirements. During range expansion, invasive species are not in such equilibrium due to both dispersal limitation and frequent casual occurrence in sites unsuitable to persistent populations. In this article we use the example of the invasive annual plant Ambrosia artemisiifolia in Austria to evaluate if model accuracy can be enhanced in such non-equilibrium situations by taking account of propagule pressure and by restricting model calibration to naturalized populations. Moreover, we test if model accuracy increases during invasion history using distribution data from 1984 to 2005. The results suggest that models calibrated with naturalized populations are much more accurate than those based on the total set of records. Proxies of propagule pressure slightly but significantly improve goodness of fit, accuracy, and Type I and II error rates of models calibrated with all available records but have less consistent effects on models of naturalized populations. Model accuracy did not increase during the recent invasion history, probably because the species is still far from an equilibrium distribution. We conclude that even a coarse assessment of population status with records of invasive species delivers important information for predictive modelling and that proxies of propagule pressure should be included into such models at least during early to intermediate stages of the invasion history.     

Gridlock and beltways: the genetic context of urban invasions

The rapid expansion of urban land across the globe presents new and numerous opportunities for invasive species to spread and flourish. Ecologists historically rejected urban ecosystems as important environments for ecology and evolution research but are beginning to recognize the importance of these systems in shaping the biology of invasion. Urbanization can aid the introduction, establishment, and spread of invaders, and these processes have substantial consequences on native species and ecosystems. Therefore, it is valuable to understand how urban areas influence populations at all stages in the invasion process. Population genetic tools are essential to explore the driving forces of invasive species dispersal, connectivity, and adaptation within cities. In this review, we synthesize current research about the influence of urban landscapes on invasion genetics dynamics. We conclude that urban areas are not only points of entry for many invasive species, they also facilitate population establishment, are pools for genetic diversity, and provide corridors for further spread both within and out of cities. We recommend the continued use of genetic studies to inform invasive species management and to understand the underlying ecological and evolutionary processes governing successful invasion.     

Advancing impact prediction and hypothesis testing in invasion ecology using a comparative functional response approach

Invasion ecology urgently requires predictive methodologies that can forecast the ecological impacts of existing, emerging and potential invasive species. We argue that many ecologically damaging invaders are characterised by their more efficient use of resources. Consequently, comparison of the classical ‘functional response’ (relationship between resource use and availability) between invasive and trophically analogous native species may allow prediction of invader ecological impact. We review the utility of species trait comparisons and the history and context of the use of functional responses in invasion ecology, then present our framework for the use of comparative functional responses. We show that functional response analyses, by describing the resource use of species over a range of resource availabilities, avoids many pitfalls of ‘snapshot’ assessments of resource use. Our framework demonstrates how comparisons of invader and native functional responses, within and between Type II and III functional responses, allow testing of the likely population-level outcomes of invasions for affected species. Furthermore, we describe how recent studies support the predictive capacity of this method; for example, the invasive ‘bloody red shrimp’ Hemimysis anomala shows higher Type II functional responses than native mysids and this corroborates, and could have predicted, actual invader impacts in the field. The comparative functional response method can also be used to examine differences in the impact of two or more invaders, two or more populations of the same invader, and the abiotic (e.g. temperature) and biotic (e.g. parasitism) context-dependencies of invader impacts. Our framework may also address the previous lack of rigour in testing major hypotheses in invasion ecology, such as the ‘enemy release’ and ‘biotic resistance’ hypotheses, as our approach explicitly considers demographic consequences for impacted resources, such as native and invasive prey species. We also identify potential challenges in the application of comparative functional responses in invasion ecology. These include incorporation of numerical responses, multiple predator effects and trait-mediated indirect interactions, replacement versus non-replacement study designs and the inclusion of functional responses in risk assessment frameworks. In future, the generation of sufficient case studies for a meta-analysis could test the overall hypothesis that comparative functional responses can indeed predict invasive species impacts.     

A conceptual framework for understanding arthropod predator and parasitoid invasions

Arthropods make up the largest group of invasive alien species (IAS) worldwide. Although invasion research has been biased towards alien plants and vertebrates, it has suggested potential mechanisms for the success of IAS and provided a theoretical framework for further investigation. Here we address key concepts from invasion biology that are essential to our understanding of the success of invasive alien arthropod predators and parasitoids including human intervention, environmental characteristics, propagule pressure, biological traits, and biological interactions. To gain a greater understanding of the factors most likely to influence the different stages of invasion (arrival, establishment, and spread) for alien arthropod predators and parasitoids, we use a comparative approach to compare and contrast the differential success of invasions by alien phytophagous and carnivorous arthropods. Insights gained from this comparison suggest that future research will require a multitrophic approach in order to enhance our understanding of invasions at higher trophic levels.     

基于功能性状多维性理解外来物种入侵的达尔文归化难题

达尔文归化难题是进行外来物种入侵预测的重要理论依据,然而,达尔文归化假说和预适应假说却预测了2种截然不同的结果。事实上,达尔文归化难题争论的焦点是物种间的差异性还是相似性促进了外来物种的成功入侵,究其原因可能是忽略了功能性状的多维性。所谓功能性状的多维性,就是不同的功能性状可能代表着不同的生态功能轴,外来物种的入侵是多个维度上不同生态过程的综合结果。本研究以现代物种共存理论为基础,构建了一个基于环境过滤和适合度差异2个维度的入侵预测模型框架,不同维度对应着不同的功能性状以及不同的种间相似性关系。该预测模型表明,在环境过滤维度上与本地物种性状趋同,同时,在适合度维度上与本地物种性状趋异的外来物种是潜在的入侵物种,而其危害程度主要取决于本地群落的构建过程。该模型框架可为外来物种入侵预警提供理论依据,也可为生物多样性保护、外来物种的防治与管理等提供实践指导。     

A predictive framework and review of the ecological impacts of exotic plant invasions on reptiles and amphibians

The invasive spread of exotic plants in native vegetation can pose serious threats to native faunal assemblages. This is of particular concern for reptiles and amphibians because they form a significant component of the world's vertebrate fauna, play a pivotal role in ecosystem functioning and are often neglected in biodiversity research. A framework to predict how exotic plant invasion will affect reptile and amphibian assemblages is imperative for conservation, management and the identification of research priorities. Here, we present a new predictive framework that integrates three mechanistic models. These models are based on exotic plant invasion altering: (1) habitat structure; (2) herbivory and predator‐prey interactions; (3) the reproductive success of reptile and amphibian species and assemblages. We present a series of testable predictions from these models that arise from the interplay over time among three exotic plant traits (growth form, area of coverage, taxonomic distinctiveness) and six traits of reptiles and amphibians (body size, lifespan, home range size, habitat specialisation, diet, reproductive strategy). A literature review provided robust empirical evidence of exotic plant impacts on reptiles and amphibians from each of the three model mechanisms. Evidence relating to the role of body size and diet was less clear‐cut, indicating the need for further research. The literature provided limited empirical support for many of the other model predictions. This was not, however, because findings contradicted our model predictions but because research in this area is sparse. In particular, the small number of studies specifically examining the effects of exotic plants on amphibians highlights the pressing need for quantitative research in this area. There is enormous scope for detailed empirical investigation of interactions between exotic plants and reptile and amphibian species and assemblages. The framework presented here and further testing of predictions will provide a basis for informing and prioritising environmental management and exotic plant control efforts.     

Species distribution models support the need of international cooperation towards successful management of plant invasions

To protect native biodiversity and habitats from the negative impacts of biological invasions, comprehensive studies and measures to anticipate invasions are required, especially across countries in a transfrontier context. Species distribution models (SDMs) can be particularly useful to integrate different types of data and predict the distribution of invasive species across borders, both for current conditions and under scenarios of future environmental changes. We used SDMs to test whether predicting invasions and potential spatial conflicts with protected areas in a transfrontier context, under current and future climatic conditions, would provide additional insights on the patterns and drivers of invasion when compared to models obtained from predictions for individual regions/countries (different modelling strategies). The framework was tested with the invasive alien plant Acacia dealbata in North of Portugal/NW Spain Euro-region, where the species is predicted to increase its distribution under future climatic conditions. While SDMs fitted in a transfrontier context and using “the national strategy (with Portugal calibration data) presented similar patterns, the distribution of the invasive species was higher in the former. The transfrontier strategy expectedly allowed to capture a more complete and accurate representation of the species’ niche. Predictions obtained in a transfrontier context are therefore more suitable to support resource prioritisation for anticipation and monitoring impacts of biological invasions, while also providing additional support for international cooperation when tackling issues of global change. Our proposed framework provided useful information on the potential patterns of invasion by A. dealbata in a transfrontier context, with an emphasis on protected areas. This information is crucial for decision-makers focusing on the prevention of invasions by alien species inside protected areas in a transfrontier context, opening a new way for collaborative management of invasions.