A simple stochastic model for complex coextinctions in mutualistic networks: robustness decreases with connectance

Understanding and predicting species extinctions and coextinctions is a major goal of ecological research in the face of a biodiversity crisis. Typically, models based on network topology are used to simulate coextinctions in mutualistic networks. However, such topological models neglect two key biological features of species interactions: variation in the intrinsic dependence of species on the mutualism, and variation in the relative importance of each interacting partner. By incorporating both types of variation, we developed a stochastic coextinction model capable of simulating extinction cascades far more complex than those observed in previous topological models. Using a set of empirical mutualistic networks, we show that the traditional topological model may either underestimate or overestimate the number and likelihood of coextinctions, depending on the intrinsic dependence of species on the mutualism. More importantly, contrary to topological models, our stochastic model predicts extinction cascades to be more likely in highly connected mutualistic communities.     

On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist

Two basic models of mutualism are presented in which interactions among three species lead to mutualism between two of them. The models represent 2-species predator-prey or competition systems in which a third species acts as a mutualist with either the predator, the prey, or one of the competitors. The models include the assumptions that there is a cost of associating with the mutualist and that the mutualist population grows much more slowly than the other two populations. Special cases of these two models correspond to six qualitatively different types of mutualistic benefit, all of which are known to occur in nature: deterring predation, increasing prey availability, feeding on (or competing with) a predator, increasing competitive interactions, decreasing competitive interactions, and feeding on (or competing with) a competitor. These models and their special cases are subjected to a local stability analysis. The results show that mutualism based upon deterring predation, competing with a predator, or decreasing competitive interactions enhances local stability, while mutualism based upon increasing prey availability or increasing competitive interactions reduces local stability. These results clearly reject the idea that mutualism is an inherently unstable process, and reinforces the idea that each different kind of mutualism will have to be considered separately. Compared to 2-species models of mutualism, the 3-species models provide a more realistic representation of the structure of many mutualistic systems, the mechanisms by which one species benefits another, and the regulation of the interaction.     

Dynamics of mutualist populations that are demographically open

1. Few theoretical studies have examined the impact of immigration and emigration on mutualist population dynamics, but a recent empirical study (A.R. Thompson Oecologia, 143, 61-69) on mutualistic fish and shrimp showed that immigration can prevent population collapse, and that intraspecific competition for a mutualistic partner can curb population expansion. To understand in a theoretical context the implications of these results, and to assess their generality, we present a two-species model that accounts explicitly for immigration and emigration, as well as distinguishing the impacts of mutualism on birth rates, death rates and habitat acquisition. 2. The model confirms that immigration can stabilize mutualistic populations, and predicts that high immigration, along with enhanced reproduction and/or reduced mortality through mutualism, can cause population sizes to increase until habitat availability curbs further expansion. 3. We explore in detail the effects of different forms of habitat limitation on mutualistic populations. Habitat availability commonly limits the density of both populations if mutualists acquire shelter independently. If a mutualist depends on a partner for habitat, densities of that mutualist are capped by the amount of space provided by that partner. The density of the shelter-provider is limited by the environment. 4. If a mutualism solely augments reproduction, and most locally produced individuals leave the focal patch, then the mutualism will have a minimal effect on local dynamics. If the mutualism operates by reducing rates of death or enhancing habitat availability, and there is at least some immigration, then mutualism will affect local dynamics. This finding may be particularly relevant in marine systems, where there is high variability (among species and locations) in the extent to which progeny disperse from natal locations. 5. Overall, our results demonstrate that the consequences of immigration and emigration for the dynamics of mutualists depend strongly on which demographic rate is influenced by mutualism. 6. By relating our model to a variety of terrestrial and aquatic systems, we provide a general framework to guide future empirical studies of the dynamics of mutualistic populations.     

Strategy Diversity Stabilizes Mutualism through Investment Cycles,Phase Polymorphism,and Spatial Bubbles

There is continuing interest in understanding factors that facilitate the evolution and stability of cooperation within and between species. Such interactions will often involve plasticity in investment behavior, in response to the interacting partner''s investments. Our aim here is to investigate the evolution and stability of reciprocal investment behavior in interspecific interactions, a key phenomenon strongly supported by experimental observations. In particular, we present a comprehensive analysis of a continuous reciprocal investment game between mutualists, both in well-mixed and spatially structured populations, and we demonstrate a series of novel mechanisms for maintaining interspecific mutualism. We demonstrate that mutualistic partners invariably follow investment cycles, during which mutualism first increases, before both partners eventually reduce their investments to zero, so that these cycles always conclude with full defection. We show that the key mechanism for stabilizing mutualism is phase polymorphism along the investment cycle. Although mutualistic partners perpetually change their strategies, the community-level distribution of investment levels becomes stationary. In spatially structured populations, the maintenance of polymorphism is further facilitated by dynamic mosaic structures, in which mutualistic partners form expanding and collapsing spatial bubbles or clusters. Additionally, we reveal strategy-diversity thresholds, both for well-mixed and spatially structured mutualistic communities, and discuss factors for meeting these thresholds, and thus maintaining mutualism. Our results demonstrate that interspecific mutualism, when considered as plastic investment behavior, can be unstable, and, in agreement with empirical observations, may involve a polymorphism of investment levels, varying both in space and in time. Identifying the mechanisms maintaining such polymorphism, and hence mutualism in natural communities, provides a significant step towards understanding the coevolution and population dynamics of mutualistic interactions.     

Predicting abundances of plants and pollinators using a simple compartmental mutualistic model

Key gaps to be filled in population and community ecology are predicting the strength of species interactions and linking pattern with process to understand species coexistence and their relative abundances. In the case of mutualistic webs, like plant–pollinator networks, advances in understanding species abundances are currently limited, mainly owing to the lack of methodological tools to deal with the intrinsic complexity of mutualisms. Here, we propose an aggregation method leading to a simple compartmental mutualistic population model that captures both qualitatively and quantitatively the size-segregated populations observed in a Mediterranean community of nectar-producing plant species and nectar-searching animal species. We analyse the issue of optimal aggregation level and its connection with the trade-off between realism and overparametrization. We show that aggregation of both plants and pollinators into five size classes or compartments leads to a robust model with only two tunable parameters. Moreover, if, in each compartment, (i) the interaction coefficients fulfil the condition of weak mutualism and (ii) the mutualism is facultative for at least one party of the compartment, then the interactions between different compartments are sufficient to guarantee global stability of the equilibrium population.     

Arbuscular Mycorrhizal Fungal Networks Vary throughout the Growing Season and between Successional Stages

To date, few analyses of mutualistic networks have investigated successional or seasonal dynamics. Combining interaction data from multiple time points likely creates an inaccurate picture of the structure of networks (because these networks are aggregated across time), which may negatively influence their application in ecosystem assessments and conservation. Using a replicated bipartite mutualistic network of arbuscular mycorrhizal (AM) fungal-plant associations, detected using large sample numbers of plants and AM fungi identified through molecular techniques, we test whether the properties of the network are temporally dynamic either between different successional stages or within the growing season. These questions have never been directly tested in the AM fungal-plant mutualism or the vast majority of other mutualisms. We demonstrate the following results: First, our examination of two different successional stages (young and old forest) demonstrated that succession increases the proportion of specialists within the community and decreases the number of interactions. Second, AM fungal-plant mutualism structure changed throughout the growing season as the number of links between partners increased. Third, we observed shifts in associations between AM fungal and plant species throughout the growing season, potentially reflecting changes in biotic and abiotic conditions. Thus, this analysis opens up two entirely new areas of research: 1) identifying what influences changes in plant-AM fungal associations in these networks, and 2) what aspects of temporal variation and succession are of general importance in structuring bipartite networks and plant-AM fungal communities.     

On the evolution of non-specific mutualism

It has been argued that mutualisms are non-specific when mutualistic interactions are weak and transient, and become more specific as interactions increase in strength. However, this runs counter to the observation that there exist tightly linked mutualisms of great antiquity that are highly nonspecific. Here we argue that mutualism generates positive, interspecific, frequency-dependent selection, which acts as a cohesive evolutionary force, discouraging evolution of specificity. A simple mathematical model is constructed to analyse the evolution of a community consisting of two guilds of species with mutualistic between-guild interactions, two competing species in each guild and two genetically distinct phenotypes within each species. With some simplifying assumptions, the trajectories in the neighbourhood of the only interior equilibrium point are determined analytically in terms of interactions between individuals. These show that the equilibrium is locally stable (no evolution) when there is little differentiation between phenotypes in mutualistic and interspecific, competitive interactions. On the other hand, when there is strong differentiation between phenotypes in their mutualistic interactions, the equilibrium is unstable and the community starts to evolve towards non-specificity. There are, however, two forces counteracting this tendency which, if sufficiently potent, cause evolution towards specificity. The first is generated by strong differentiation between phenotypes in interspecific competition; the second is caused by specificity which already exists between species in their mutualistic interactions. Thus, the tendency for non-specificity or specificity to evolve depends on the interplay between antagonistic and mutualistic interactions in the community. We illustrate these results with some numerical examples and, finally, survey some data on specificity of mutualisms in the light of the analysis.     

Photorhabdus: a model for the analysis of pathogenicity and mutualism

Photorhabdus are entomopathogenic members of the family Enterobacteriaceae. In addition to killing insects Photorhabdus also have a mutualistic association with nematodes from the family Heterorhabditidiae. Therefore, the bacteria have a complex life cycle that involves temporally separated pathogenic and mutualistic associations with two different invertebrate hosts. This tripartite Photorhabdus-insect-nematode association provides researchers with a unique opportunity to characterize the prokaryotic contribution to two different symbioses, i.e. pathogenicity and mutualism while also studying the role of the host in determining the outcome of association with the bacteria. In this review I will outline the life cycle of Photorhabdus and describe recent important advances in our understanding of the symbiology of Photorhabdus. Finally, the contribution made by this model to our understanding of the nature of symbiotic associations will be discussed.     

Persistence of mutualisms with bidirectional interactions in a two-species system

In Lotka–Volterra equations (LVEs) of mutualisms, population densities of mutualists will increase infinitely if the mutualisms between them are strong, which is called the divergence problem. In order to avoid the problem, a mutualism system of two species is analyzed in this work. The model is derived from reactions on lattice and has a form similar to that of LVEs. Population densities of species will not increase infinitely because of spatial limitation on the lattice. Stability analysis of the model demonstrates basic mechanisms by which the mutualisms lead to coexistence/extinction of the species. When in coexistence, intermediate mutualistic effect is shown to lead to the maximal density in certain parameter ranges, while a strong or weak mutualistic effect is not so good. Furthermore, the stability analysis exhibits that extremely strong/weak mutualisms will result in extinction of one/both species.     

Stability properties of 2-species models of mutualism: Simulation studies

Summary Most previous analyses of the stability properties of models of mutualism have emphasized the destabilizing effects of mutualism. However, these analyses can be shown to be based upon inappropriate assumptions, or to be applicable only for special cases of mutualism. In this paper three basic 2-species models of mutualism are presented and their six combinations are analyzed by computer simulation for their return time stability and persistence stability. Four out of six models show greater return time stability than an appropriate model without mutualism, and all models show higher persistence stability than the model without mutualism. It is argued that real biological systems can be related to the qualitative structure of each of the basic models of mutualism, and that therefore none of the basic models or their stability properties can be eliminated a priori as being inappropriate. The conclusion follows that while some kinds of mutualistic interactions may be relatively unstable, other mutualisms, probably representing the majority of cases, can be considered to be relatively stable. The limitations of these models and analyses are considered.     

Effective competition determines the global stability of model ecosystems

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.     

Interaction intimacy affects structure and coevolutionary dynamics in mutualistic networks

The structure of mutualistic networks provides clues to processes shaping biodiversity [1-10]. Among them, interaction intimacy, the degree of biological association between partners, leads to differences in specialization patterns [4, 11] and might affect network organization [12]. Here, we investigated potential consequences of interaction intimacy for the structure and coevolution of mutualistic networks. From observed processes of selection on mutualistic interactions, it is expected that symbiotic interactions (high-interaction intimacy) will form species-poor networks characterized by compartmentalization [12, 13], whereas nonsymbiotic interactions (low intimacy) will lead to species-rich, nested networks in which there is a core of generalists and specialists often interact with generalists [3, 5, 7, 12, 14]. We demonstrated an association between interaction intimacy and structure in 19 ant-plant mutualistic networks. Through numerical simulations, we found that network structure of different forms of mutualism affects evolutionary change in distinct ways. Change in one species affects primarily one mutualistic partner in symbiotic interactions but might affect multiple partners in nonsymbiotic interactions. We hypothesize that coevolution in symbiotic interactions is characterized by frequent reciprocal changes between few partners, but coevolution in nonsymbiotic networks might show rare bursts of changes in which many species respond to evolutionary changes in a single species.     

Pollination and seed predation by moths on Silene and allied Caryophyllaceae: evaluating a model system to study the evolution of mutualisms

Nursery pollinators, and the plants they use as hosts for offspring development, function as exemplary models of coevolutionary mutualism. The two pre-eminent examples--fig wasps and yucca moths--show little variation in the interaction: the primary pollinator is an obligate mutualist. By contrast, nursery pollination of certain Caryophyllaceae, including Silene spp., by two nocturnal moth genera, Hadena and Perizoma, ranges from antagonistic to potentially mutualistic, offering an opportunity to test hypotheses about the factors that promote or discourage the evolution of mutualism. Here, we review nursery pollination and host-plant interactions in over 30 caryophyllaceous plants, based on published studies and a survey of researchers investigating pollination, seed predation, and moth morphology and behavior. We detected little direct evidence of mutualism in these moth-plant interactions, but found traits and patterns in both that are nonetheless consistent with the evolution of mutualism and merit further attention.     

Interactions between seed predators/pollinators and their host plants: a first step towards mutualism?

The mutualisms between fig trees and their pollinator fig wasps and between yucca plants and yucca moths are spectacular examples of coevolution. The characteristics of these independently evolved mutualisms have resulted from long‐term processes, the first stages of which are unknown. A fundamental question in the study of mutualism is how these interactions evolve. Seed predator/pollinator and host plant interactions, which may initially be considered as mainly antagonistic, have the potential to provide good model systems for the study of the first stages of evolution towards mutualism. We present here theoretical models assessing the consequences of interactions between specialized seed predator insects and their host plants. These models describe the parameters that affect the fitness of an individual female seed predator and her influence on the fitness of the host plant. In an optimal strategy for the seed predator, the number of eggs laid in each flower depends on the interaction between the adult and larva survival. Along with a growing predation pressure on adults and larvae several eggs must be laid in each flower by the female seed predator to enhance her fitness. However, in a situation where the host plant selectively aborts flowers with a high number of eggs the fitness of the seed predator will seriously decrease. If the cost of selective abortion is less than the cost of seed predation the host plant will maintain fitness. In a mutualistic relationship a balance between the cost and the benefit of the parameters in the fitness models of the seed predator and the host plant has to occur so that the net seed output is larger than zero (0). Any unselfish behaviour or quality of the seed predator that would benefit the host plant in such a way that the net seed output increases might be a first stage in an interaction becoming mutualistic. The models presented here will not only provide a platform for empirical studies on interactions that may swing from parasitism to mutualism, but also for seed predator/pollinator and host plant interactions in general.     

Molecular phylogenies of fig wasps: partial cocladogenesis of pollinators and parasites

Figs (Ficus spp., Moraceae) and their pollinating wasps form an obligate mutualism, which has long been considered a classic case of coevolution and cospeciation. Figs are also exploited by several clades of nonpollinating wasps, which are parasites of the mutualism and whose patterns of speciation have received little attention. We used data from nuclear and mitochondrial DNA regions to estimate the phylogenies of 20 species of Pleistodontes pollinating wasps and 16 species of Sycoscapter nonpollinating wasps associated with Ficus species in the section Malvanthera. We compare the phylogenies of 15 matched Pleistodontes/Sycoscapter species pairs and show that the level of cospeciation is significantly greater than that expected by chance. Our estimates of the maximum level of cospeciation (50 to 64% of nodes) are very similar to those obtained in other recent studies of coevolved parasitic and mutualistic associations. However, we also show that there is not perfect congruence of pollinator and parasite phylogenies (for any substantial clade) and argue that host plant switching is likely to be less constrained for Sycoscapter parasites than for Pleistodontes pollinators. There is perfect correspondence between two terminal clades of two sister species in the respective phylogenies, and rates of molecular evolution in these pairs are similar.     

The origin of a mutualism: a morphological trait promoting the evolution of ant-aphid mutualisms

Mutualisms are mutually beneficial interactions between species and are fundamentally important at all levels of biological organization. It is not clear, however, why one species participates in a particular mutualism whereas another does not. Here we show that pre-existing traits can dispose particular species to evolve a mutualistic interaction. Combining morphological, ecological, and behavioral data in a comparative analysis, we show that resource use in Chaitophorus aphids (Hemiptera: Aphididae) modulates the origin of their mutualism with ants. We demonstrate that aphid species that feed on deeper phloem elements have longer mouthparts, that this inhibits their ability to withdraw their mouthparts and escape predators and that, consequently, this increases their need for protection by mutualist ants.     

Dynamical Transitions in a Pollination–Herbivory Interaction: A Conflict between Mutualism and Antagonism

Plant-pollinator associations are often seen as purely mutualistic, while in reality they can be more complex. Indeed they may also display a diverse array of antagonistic interactions, such as competition and victim–exploiter interactions. In some cases mutualistic and antagonistic interactions are carried-out by the same species but at different life-stages. As a consequence, population structure affects the balance of inter-specific associations, a topic that is receiving increased attention. In this paper, we developed a model that captures the basic features of the interaction between a flowering plant and an insect with a larval stage that feeds on the plant’s vegetative tissues (e.g. leaves) and an adult pollinator stage. Our model is able to display a rich set of dynamics, the most remarkable of which involves victim–exploiter oscillations that allow plants to attain abundances above their carrying capacities and the periodic alternation between states dominated by mutualism or antagonism. Our study indicates that changes in the insect’s life cycle can modify the balance between mutualism and antagonism, causing important qualitative changes in the interaction dynamics. These changes in the life cycle could be caused by a variety of external drivers, such as temperature, plant nutrients, pesticides and changes in the diet of adult pollinators.     

A BIOLOGICAL MARKET ANALYSIS OF THE PLANT‐MYCORRHIZAL SYMBIOSIS

It has been argued that cooperative behavior in the plant‐mycorrhizal mutualism resembles trade in a market economy and can be understood using economic tools. Here, we assess the validity of this “biological market” analogy by investigating whether a market mechanism—that is, competition between partners over the price at which they provide goods—could be the outcome of natural selection. Then, we consider the conditions under which this market mechanism is sufficient to maintain mutualistic trade. We find that: (i) as in a market, individuals are favored to divide resources among trading partners in direct relation to the relative amount of resources received, termed linear proportional discrimination; (ii) mutualistic trade is more likely to be favored when individuals are able to interact with more partners of both species, and when there is a greater relative difference between the species in their ability to directly acquire different resources; (iii) if trade is favored, then either one or both species is favored to give up acquiring one resource directly, and vice versa. We then formulate testable predictions as to how environmental changes and coevolved responses of plants and mycorrhizal fungi will influence plant fitness (crop yields) in agricultural ecosystems.     

Coevolution in temporally variable environments

Many potentially mutualistic interactions are conditional, with selection that varies between mutualism and antagonism over space and time. We develop a genetic model of temporally variable coevolution that incorporates stochastic fluctuations between mutualism and antagonism. We use this model to determine conditions necessary for the coevolution of matching traits between a host and a conditional mutualist. Using an analytical approximation, we show that matching traits will coevolve when the geometric mean interaction is mutualistic. When this condition does not hold, polymorphism and trait mismatching are maintained, and coevolutionary cycles may result. Numerical simulations verify this prediction and suggest that it remains robust in the presence of temporal autocorrelation. These results are compared with those from spatial models with unrestricted movement. The comparisons demonstrate that gene flow is unnecessary for generating empirical patterns predicted by the geographic mosaic theory of coevolution.     

Direct and indirect effects of invasions of predators on a multiple-species community

A Lotka-Volterra system for a multiple species community with two trophic levels is analyzed to illustrate how community structure is reorganized upon invasions of predators. The lower trophic level is assumed to consist of interfering competing species, some of which are preferentially consumed by invading predators. Effects of invading predators on the lower trophic level are investigated in terms of predator-mediated coexistence and predator-induced instability. Competitive interactions between species in the lower trophic level result in indirect mutualism or indirect competition between predators depending on which competitors are preyed upon.