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 plant–pollinator–robber systems

Plant–pollinator–robber systems are considered, where the plants and pollinators are mutualists, the plants and nectar robbers are in a parasitic relation, and the pollinators and nectar robbers consume a common limiting resource without interfering competition. My aim is to show a mechanism by which pollination–mutualism could persist when there exist nectar robbers. Through the dynamics of a plant–pollinator–robber model, it is shown that (i) when the plants alone (i.e., without pollination–mutualism) cannot provide sufficient resources for the robbers’ survival but pollination–mutualism can persist in the plant–pollinator system, the pollination–mutualism may lead to invasion of the robbers, while the pollinators will not be driven into extinction by the robbers’ invasion. (ii) When the plants alone cannot support the robbers’ survival but persistence of pollination–mutualism in the plant–pollinator system is density-dependent, the pollinators and robbers could coexist if the robbers’ efficiency in translating the plant–robber interactions into fitness is intermediate and the initial densities of the three species are in an appropriate region. (iii) When the plants alone can support the robbers’ survival, the pollinators will not be driven into extinction by the robbers if their efficiency in translating the plant–pollinator interactions into fitness is relatively larger than that of the robbers. The analysis leads to an explanation for the persistence of pollination–mutualism in the presence of nectar robbers in real situations.     

The exploitation of mutualisms

Mutualisms (interspecific cooperative interactions) are ubiquitously exploited by organisms that obtain the benefits mutualists offer, while delivering no benefits in return. The natural history of these exploiters is well-described, but relatively little effort has yet been devoted to analysing their ecological or evolutionary significance for mutualism. Exploitation is not a unitary phenomenon, but a set of loosely related phenomena: exploiters may follow mixed strategies or pure strategies at either the species or individual level, may or may not be derived from mutualists, and may or may not inflict significant costs on mutualisms. The evolutionary implications of these different forms of exploitation, especially the threats they pose to the stability of mutualism, have as yet been minimally explored. Studies of this issue are usually framed in terms of a "temptation to defect" that generates a destabilizing conflict of interest between partners. I argue that this idea is in fact rather inappropriate for interpreting most observed forms of exploitation in mutualisms. I suggest several alternative and testable ideas for how mutualism can persist in the face of exploitation.     

The Stability and Persistence of Mutualisms Embedded in Community Interactions

In this paper we argue that two-species models of mutualism may be oversimplifications of the real world that lead to erroneous predictions. We present a four-species model of a pollination mutualism embedded in other types of community interactions. Conclusions derived from two-species models about the destabilizing effect of mutualisms are misleading when applied to the present scenario; although the mutualisms are locally destabilizing, the effect is more than canceled by an increased chance of feasibility. The crucial difference is the interaction of the mutualists with other species in a larger web. Furthermore, community persistence (without unrealistic population explosion), arguably a superior ecological criterion, is greatly enhanced by the presence of mutualisms. Therefore, we predict that mutualisms should be common in the real world, a prediction matching empirial findings and in contrast to the predictions from local stability analysis of basic two-species models. This method of stabilizing a mutualism appears superior in some ways to the often-used method of introducing density dependence in the strength of the mutualism, because it permits obligate mutualisms to exist even at low densities, again matching empirical findings. Lastly, this study is an example of how complex model assemblages can behave qualitatively differently from analogous simpler ones.     

A mathematical model of facultative mutualism with populations interacting in a food chain

Facultative mutualism with populations interacting in a food chain is modeled by a system of four autonomous ordinary differential equations. Two cases are considered: mutualism with the prey and mutualism with the first predator. In both cases persistence and extinction criteria are developed in terms of the invariant flows on the boundaries.     

Understanding the coevolutionary dynamics of mutualism with population genomics

Decades of research on the evolution of mutualism has generated a wealth of possible ways whereby mutually beneficial interactions between species persist in spite of the apparent advantages to individuals that accept the benefits of mutualism without reciprocating — but identifying how any particular empirical system is stabilized against cheating remains challenging. Different hypothesized models of mutualism stability predict different forms of coevolutionary selection, and emerging high‐throughput sequencing methods allow examination of the selective histories of mutualism genes and, thereby, the form of selection acting on those genes. Here, I review the evolutionary theory of mutualism stability and identify how differing models make contrasting predictions for the population genomic diversity and geographic differentiation of mutualism‐related genes. As an example of the possibilities offered by genomic data, I analyze genes with roles in the symbiosis of Medicago truncatula and nitrogen‐fixing rhizobial bacteria, the first classic mutualism in which extensive genomic resources have been developed for both partners. Medicago truncatula symbiosis genes, as a group, differ from the rest of the genome, but they vary in the form of selection indicated by their diversity and differentiation — some show signs of selection expected from roles in sanctioning noncooperative symbionts, while others show evidence of balancing selection expected from coevolution with symbiont signaling factors. I then assess the current state of development for similar resources in other mutualistic interactions and look ahead to identify ways in which modern sequencing technology can best inform our understanding of mutualists and mutualism.     

The Benefits of Mutualism: A Conceptual Framework

There are three general mechanisms by which phenotypic benefits are transferred between unrelated organisms. First, one organism may purloin benefits from another by preying on or parasitizing the other organism. Second, one organism may enjoy benefits that are incidental to or a by-product of the self-serving traits of another organism. Third, an organism may invest in another organism if that investment produces return benefits which outweigh the cost of the investment. Interactions in which both parties gain a net benefit are mutualistic. The three mechanisms by which benefits are transferred between organisms can be combined in pairs to produce six possible kinds of original or 'basal' mutualisms that can arise from an amutualistic state. A review of the literature suggests that most or all interspecific mutualism have origins in three of the six possible kinds of basal mutualism. Each of these three basal mutualisms have byproduct benefits flowing in at least one direction. The transfer of by-product benefits and investment are common to both intra- and interspecific mutualisms, so that some interspecific mutualisms have intraspecific analogs. A basal mutualism may evolve to the point where each party invests in the other, sometimes obscuring the nature of the original interaction along the way. Two prominent models for the evolution of mutualism do not include by-product benefits: Roughgarden's model for the evolution of the damsel-fish anemone mutualism and the 'Tit-for-Tat' model of reciprocity. Using the conceptual framework presented here, including in particular by-product benefits, I have shown how it is possible to construct more parsimonious alternatives to both models.     

Pathways to mutualism breakdown

Mutualisms are ubiquitous in nature despite the widely held view that they are unstable interactions. Models predict that mutualists might often evolve into parasites, can abandon their partners to live autonomously and are also vulnerable to extinction. Yet a basic empirical question, whether mutualisms commonly break down, has been mostly overlooked. As we discuss here, recent progress in molecular systematics helps address this question. Newly constructed phylogenies reveal that parasites as well as autonomous (non-mutualist) taxa are nested within ancestrally mutualistic clades. Although models have focused on the propensity of mutualism to become parasitic, such shifts appear relatively rarely. By contrast, diverse systems exhibit reversions to autonomy, and this might be a common and unexplored endpoint to mutualism.     

Analysis of three species models of mutualism in predator-prey and competitive systems

Mutualism arises in many qualitatively different ways, but previous 2-species models of mutualism correspond to only a few of these. Following Addicott and Freedman (1983), we present two models of mutualism in which interactions among three species lead to mutualism between two of them. One model involves interactions among a predator, a mutualist-prey, and a mutualist, while the other involves interactions among a competitor, a mutualist-competitor, and a mutualist. Given biologically reasonable constraints upon the functions in the model, we present (1) the conditions for boundedness of solutions, (2) the equilibria and their local stability, and (3) the conditions for the existence of small amplitude periodic solutions, a behavior not predicted by 2-species models of mutualism. The models include costs to the mutualist-prey or mutualist-competitor of associating with the mutualist. Analysis of special cases shows that mutualism can cause the extinction of the predator, or the reversal of competitive outcomes.     

Patterns of patchy spread in multi-species reaction–diffusion models

Spread of populations in space often takes place via formation, interaction and propagation of separated patches of high species density, without formation of continuous fronts. This type of spread is called a ‘patchy spread’. In earlier models, this phenomenon was considered to be a result of a pronounced environmental or/and demographic stochasticity. Recently, it was found that a patchy spread can arise in a fully deterministic predator–prey system and in models of infectious diseases; in each case the process takes place in a homogeneous environment. It is well recognized that the observed patterns of patchy spread in nature are a result of interplay between stochastic and deterministic factors. However, the models considering deterministic mechanism of patchy spread are developed and studied much less compared to those based on stochastic mechanisms. A further progress in the understanding of the role of deterministic factors in the patchy spread would be extremely helpful. Here we apply multi-species reaction–diffusion models of two spatial dimensions in a homogeneous environment. We demonstrate that patterns of patchy spread are rather common for the considered approach, in particular, they arise both in mutualism and competition models influenced by predation. We show that this phenomenon can occur in a system without a strong Allee effect, contrary to what was assumed to be crucial in earlier models. We show, as well, a pattern of patchy spread having significantly different speeds in different spatial directions. We analyze basic features of spatiotemporal dynamics of patchy spread common for the reaction–diffusion approach. We discuss in which ecosystems we would observe patterns of deterministic patchy spread due to the considered mechanism.     

EXPLAINING MUTUALISM VARIATION: A NEW EVOLUTIONARY PARADOX?

The paradox of mutualism is typically framed as the persistence of interspecific cooperation, despite the potential advantages of cheating. Thus, mutualism research has tended to focus on stabilizing mechanisms that prevent the invasion of low‐quality partners. These mechanisms alone cannot explain the persistence of variation for partner quality observed in nature, leaving a large gap in our understanding of how mutualisms evolve. Studying partner quality variation is necessary for applying genetically explicit models to predict evolution in natural populations, a necessary step for understanding the origins of mutualisms as well as their ongoing dynamics. An evolutionary genetic approach, which is focused on naturally occurring mutualist variation, can potentially synthesize the currently disconnected fields of mutualism evolution and coevolutionary genetics. We outline explanations for the maintenance of genetic variation for mutualism and suggest approaches necessary to address them.     

Modelling nutritional mutualisms: challenges and opportunities for data integration

Nutritional mutualisms are ancient, widespread, and profoundly influential in biological communities and ecosystems. Although much is known about these interactions, comprehensive answers to fundamental questions, such as how resource availability and structured interactions influence mutualism persistence, are still lacking. Mathematical modelling of nutritional mutualisms has great potential to facilitate the search for comprehensive answers to these and other fundamental questions by connecting the physiological and genomic underpinnings of mutualisms with ecological and evolutionary processes. In particular, when integrated with empirical data, models enable understanding of underlying mechanisms and generalisation of principles beyond the particulars of a given system. Here, we demonstrate how mathematical models can be integrated with data to address questions of mutualism persistence at four biological scales: cell, individual, population, and community. We highlight select studies where data has been or could be integrated with models to either inform model structure or test model predictions. We also point out opportunities to increase model rigour through tighter integration with data, and describe areas in which data is urgently needed. We focus on plant‐microbe systems, for which a wealth of empirical data is available, but the principles and approaches can be generally applied to any nutritional mutualism.     

基于集合种群理论框架的榕树-榕小蜂互惠系统动力学模型

互惠共生指双方物种都能通过对方增加适合度的种间关系。榕树与榕小蜂的传粉关系是自然界中典型的互惠共生系统,这种互惠关系发生在榕果中,也就是榕果是种间作用的场所,榕小蜂在其中传粉和产卵。由此,可以把榕果看作榕小蜂的生境斑块,利用集合种群理论框架构建榕树与榕小蜂互惠关系的动力学模型,研究这个互惠系统的稳定性和续存条件。由于这里的生境斑块(榕果)的动态变化性(榕果产生和掉落),模型不同于传统的生境斑块固定不变的集合种群模型,增加了描述生境动态的维度。模型表明:(1)榕果产生率足够大(大于一个阈值)是榕树与榕小蜂互惠系统能够续存的必要条件;(2)榕树和榕小蜂互惠系统存在双稳态现象(Allee效应),这个互惠系统续存需要种群大小超过一个阈值,换言之,种群大小低于这个阈值时,系统必然灭绝;(3)榕果产生率增加使榕小蜂种群增加,但不会影响未被占据的榕果数量。我们的模型不但可用来研究榕树与榕小蜂互惠系统的动力学性质,而且也是集合种群理论斑块动态化的发展。     

Global stability of obligate mutualism in community modules with facultative mutualists

Mutualism is a fundamental building block of ecological communities and an important driver of biotic evolution. Classic theory suggests that a pairwise two‐species obligate mutualism is fragile, with a large perturbation potentially driving both mutualist populations into extinction. In nature, however, there are many cases of pairwise obligate mutualism. Such pairwise obligate mutualisms are occasionally associated with additional interactions with facultative mutualists. Here, we use a mathematical model to show that when a two‐species obligate mutualism has a single additional link to a third facultative mutualist, the obligate mutualism can become permanently persistent. In the model, a facultative mutualist interacts with one of two inter‐dependent obligate mutualists, and the facultative mutualist enhances the persistence not only of its directly interacting obligate mutualist, but also that of the other obligate mutualist indirectly, enabling the permanent coexistence of the three mutualist species. The effect of the facultative mutualist is strong; it can allow a three‐species permanent coexistence even when two obligate mutualists by themselves are not sustainable (i.e. not locally stable). These results suggest that facultative mutualists can play a pivotal role for the persistence of obligate mutualisms, and contribute to a better understanding on the mechanisms maintaining more complex mutualistic networks of multiple species.     

The qualitative approach in investigating the role of species interactions on stability of natural communities

Community models with competition and mutualism are qualitatively analyzed using the methodology of loop analysis combined with computer stochastic simulation. The concept of "moving equilibrium" in the growth rate of the species is discussed in 14 "tables of predictions", presented as analytical tools that can help to shed light on controversial ecological issues such as direct versus indirect interaction and positive feedback effects on stability. While the stochastic simulation shows that only little or no difference exists in probability of stability between models with competition and models with mutualism, the related tables of predictions show that the networks among links are able to activate indirect interactions, with both negative and positive effects, between any pair of species. This phenomenon makes it difficult to determine how much stability is related to the direct interactions.     

The contribution of cancellous bone to long bone strength and rigidity

A recent article (Burr and Piotrowski, 1982) suggested that structural analyses of long bone cross-sectional geometry will be inaccurate and should be considered inappropriate when cancellous bone accounts for 10-15% or more of the cross-sectional area. Consideration of material property differences between compact and cancellous bone, however, indicates that even significant proportions of cancellous bone (10-40% of total cross-sectional area) will very likely have negligible effects on bone strength and rigidity, and can be effectively ignored in geometrical analyses of diaphyseal sections. In metaphyseal and epiphyseal regions, however, geometric analyses of section properties such as area moments of inertia are inappropriate, both because of significant trabecular bone effects, and because of the inherent constraints of mechanical beam models.     

Short-term depression and transient memory in sensory cortex

Persistent neuronal activity is usually studied in the context of short-term memory localized in central cortical areas. Recent studies show that early sensory areas also can have persistent representations of stimuli which emerge quickly (over tens of milliseconds) and decay slowly (over seconds). Traditional positive feedback models cannot explain sensory persistence for at least two reasons: (i) They show attractor dynamics, with transient perturbations resulting in a quasi-permanent change of system state, whereas sensory systems return to the original state after a transient. (ii) As we show, those positive feedback models which decay to baseline lose their persistence when their recurrent connections are subject to short-term depression, a common property of excitatory connections in early sensory areas. Dual time constant network behavior has also been implemented by nonlinear afferents producing a large transient input followed by much smaller steady state input. We show that such networks require unphysiologically large onset transients to produce the rise and decay observed in sensory areas. Our study explores how memory and persistence can be implemented in another model class, derivative feedback networks. We show that these networks can operate with two vastly different time courses, changing their state quickly when new information is coming in but retaining it for a long time, and that these capabilities are robust to short-term depression. Specifically, derivative feedback networks with short-term depression that acts differentially on positive and negative feedback projections are capable of dynamically changing their time constant, thus allowing fast onset and slow decay of responses without requiring unrealistically large input transients.     

What is the role of predation on stability of natural communities? A theoretical investigation.

A basic question in ecology concerns the role of species interaction on dynamics of natural communities. In this framework, ecologists have considered predation, competition, mutualism, the three most important interactions, highlighting their specific effects on distribution and abundance of species, providing knowledge about phenomena like coexistence and extinction. This paper seeks to identify the effects of predation on stability of natural communities by mathematical models. Simple multispecies community models, organized in trophic levels, are analyzed by means of a qualitative technique, loop analysis, combined with a computer calculation procedure. Results do not support the hypothesis of predation as a stabilizing factor. Rather, the outcomes of the analysis suggest that predation may or may not stabilize a community. This depends on the predator's behaviour and on the network of the community.     

Selective feedback between dispersal distance and the stability of mutualism

The evolution and maintenance of mutually beneficial interactions has been one of the oldest problems for evolutionary theory. For cooperation to be stable, mechanisms such as spatial population structure must exist that prevent non‐cooperative individuals from invading cooperative groups. Selection for certain traits like increased dispersal can erode that structure. Here, I used a spatially explicit individual based dual lattice computer simulation to investigate how the evolution of dispersal interacts with the evolution of mutualism and how this interaction affects the stability of mutualism in the face of non‐mutualists. I ran simulations manipulating the self‐structuring phenotype, dispersal distance, over a range of environmental conditions, as well as letting both dispersal and mutualism evolve independently, with and without a cost of dispersal. I found that environmental productivity is negatively correlated with the stability of mutualism, and that the stability of mutualism relied on the ability of mutualists to evolve shorter dispersal distances than non‐mutualists. The inclusion of a dispersal cost essentially fixed the upper limit of dispersal, and therefore limits the ability of non‐mutualists to evolve higher average dispersal than mutualists, but as costs are relaxed, the differences are recovered. These results show how selection on seemingly unrelated traits can align suites of traits into holistic life history strategies.