Parasite-mediated and direct competition in a two-host shared macroparasite system

This paper investigates the local dynamical behaviour of a deterministic model describing two host species experiencing three forms of competition: direct competition, apparent competition mediated by macroparasites, and intra-specific (density-dependent) competition. The problem of algebraic intractability is sidestepped by adopting a geometric approach, in which an array of maps is constructed in parameter space, each structured by bifurcation surfaces which mark qualitative changes in system behaviour. The maps provide both a succinct and a comprehensive overview of the stability and feasibility structure of the system equilibria, from which can be deduced the possible modes of local dynamical behaviour. A detailed examination of these maps shows that (i) the system is highly sensitive to the effect of infection on fecundity with synchronous sustained cycles readily generated by Hopf bifurcations; (ii) for a broad range of parameter values, pertinent to actual biological systems, apparent competition mediated by macroparasites is sufficient, on its own, to explain host exclusion; (iii) direct competition reinforces parasite-mediated competition to expand the host exclusion region; and (iv) the condition for host exclusion can be expressed simply in a form which holds for both micro- and macroparasite models and which involves just two key indices, measuring tolerance to the infection and the strength of direct competition. The techniques used in this paper are not restricted to the analysis of host-parasite systems but can be applied to a wide range of nonlinear population models. They are therefore as relevant to the analysis of such general issues as exploitative competition and trophic interactions as they are to specific epidemiological problems.     

Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

On the preservation of vessel bifurcations during flow-mediated angiogenic remodelling

During developmental angiogenesis, endothelial cells respond to shear stress by migrating and remodelling the initially hyperbranched plexus, removing certain vessels whilst maintaining others. In this study, we argue that the key regulator of vessel preservation is cell decision behaviour at bifurcations. At flow-convergent bifurcations where migration paths diverge, cells must finely tune migration along both possible paths if the bifurcation is to persist. Experiments have demonstrated that disrupting the cells’ ability to sense shear or the junction forces transmitted between cells impacts the preservation of bifurcations during the remodelling process. However, how these migratory cues integrate during cell decision making remains poorly understood. Therefore, we present the first agent-based model of endothelial cell flow-mediated migration suitable for interrogating the mechanisms behind bifurcation stability. The model simulates flow in a bifurcated vessel network composed of agents representing endothelial cells arranged into a lumen which migrate against flow. Upon approaching a bifurcation where more than one migration path exists, agents refer to a stochastic bifurcation rule which models the decision cells make as a combination of flow-based and collective-based migratory cues. With this rule, cells favour branches with relatively larger shear stress or cell number. We found that cells must integrate both cues nearly equally to maximise bifurcation stability. In simulations with stable bifurcations, we found competitive oscillations between flow and collective cues, and simulations that lost the bifurcation were unable to maintain these oscillations. The competition between these two cues is haemodynamic in origin, and demonstrates that a natural defence against bifurcation loss during remodelling exists: as vessel lumens narrow due to cell efflux, resistance to flow and shear stress increases, attracting new cells to enter and rescue the vessel from regression. Our work provides theoretical insight into the role of junction force transmission has in stabilising vasculature during remodelling and as an emergent mechanism to avoid functional shunting.     

Spatial interplay of plant competition and consumer foraging mediate plant coexistence and drive the invasion ratchet

Indirect effects may play an important role in structuring plant communities. Using a spatially explicit model of consumer foraging and plant competition, we demonstrate how the relationship between the spatial area over which plants compete and the spatial scale of consumer behaviour can determine the outcome of competition when one plant species provides a refuge for mobile consumers (i.e. refuge-mediated apparent competition). Once an initial population of the invader is established, complete invasion may be inevitable because of an ever-advancing invasion front ratchets forward driven by a feeding front of mobile consumers. Because the spatial extent of apparent competition determines the area available for colonization, consumers may also dictate the rate at which an invasion occurs. We find that, as long as refuge-mediated apparent competition is sufficiently localized, invasion is possible even in systems characterized by low overall levels of consumer pressure. Moreover, we show that a stable equilibrium can result in which both resident and invading plants coexist, suggesting that spatial heterogeneity created by refuge-mediated apparent competition may be important in mediating coexistence in plant communities. The spatial interplay of consumer behaviour and plant competition may be an underappreciated mechanism affecting the composition, diversity and spatial pattern of plant communities.     

Female competition and aggression: interdisciplinary perspectives

This paper introduces a Theme Issue combining interdisciplinary perspectives in the study of female competition and aggression. Despite a history of being largely overlooked, evidence is now accumulating for the widespread evolutionary significance of female competition. Here, we provide a synthesis of contributions to this Theme Issue on humans and other vertebrates, and highlight directions for future research. Females compete for resources needed to survive and reproduce, and for preferred mates. Although female aggression takes diverse forms, under most circumstances relatively low-risk competitive strategies are favoured, most probably due to constraints of offspring production and care. In social species, dominance relationships and threats of punishment can resolve social conflict without resort to direct aggression, and coalitions or alliances may reduce risk of retaliation. Consistent with these trends, indirect aggression is a low cost but effective form of competition among young women. Costs are also minimized by flexibility in expression of competitive traits, with aggressive behaviour and competitive signalling tailored to social and ecological conditions. Future research on female competition and the proximate mediators of female aggression will be greatly enhanced by opportunities for interdisciplinary exchange, as evidenced by contributions to this Theme Issue.     

Bistability and oscillations in chemical reaction networks

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993; Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks. We have added a dedication of the paper to K. Gatermann.     

Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.     

On competition of predators and prey infection

A class of prey–predator models with infected prey is investigated. Predation terms are either of Holling type II or III, infection is either modelled by mass action or standard incidence. It is shown that the key for understanding the model behaviour is the competition of predators versus infection. In the presented models the predator is not susceptible to the infection and the infection of the prey has no influence on the ability of the predator of catching the prey. However, the prey population can be seen as a resource which both the predators and the infection depend on. The competition for this resource is strong—the principle of competitive exclusion holds for biologically meaningful choices of parameters as long as there is no destabilisation by a Hopf bifurcation. The representation of models in competition diagrams which are introduced in this article can be used for a wide range of competition models which seems to be a promising method with many potential applications.     

Elucidation of direct competition and allosteric modulation of small‐molecular‐weight protein ligands using surface plasmon resonance methods

The present work introduces a surface plasmon resonance‐based method for the discrimination of direct competition and allosteric effects that occur in ternary systems comprising a receptor protein and two small‐molecular‐weight ligands that bind to it. Fatty acid binding protein 4, fructose‐1,6‐bisphosphatase and human serum albumin were used as model receptor molecules to demonstrate the performance of the method. For each of the receptor molecules, pairs of ligand molecules were selected for which either direct competition or an allosteric effect had already been determined by other methods. The method of discrimination introduced here is based on the surface plasmon resonance responses observed at equilibrium when an immobilized receptor protein is brought into contact with binary mixtures of interacting ligands. These experimentally determined responses are compared with the responses calculated using a theoretical model that considers both direct competition and allosteric ligand interaction modes. This study demonstrates that the allosteric ternary complex model, which enables calculation of the fractional occupancy of the protein by each ligand in such ternary systems, is well suited for the theoretical calculation of these types of responses. For all of the ternary systems considered in this work, the experimental and calculated responses in the chosen concentration ratio range were identical within a five‐σ confidence interval when the calculations considered the correct interaction mode of the ligands (direct competition or different types of allosteric regulation), and in case of allosteric modulation, also the correct strength of this effect. This study also demonstrates that the allosteric ternary complex model‐based calculations are well suited to predict the ideal concentration ratio range or even single concentration ratios that can serve as hot spots for discrimination, and such hot spots can drastically reduce the numbers of measurements needed for discrimination between direct competition and distinct modulation modes (neutral, positive or negative allostery). Copyright © 2015 John Wiley & Sons, Ltd.     

Interspecific Competition and Speciation in Endoparasitoids

Ecological speciation occurs when inherent reproductive barriers to gene flow evolve between populations as a result of divergent natural selection. Frequency dependent effects associated with intraspecific resource competition are thought to be one important source of divergent selection facilitating ecological speciation. Interspecific competition may also play an important role in promoting population divergence. Although evidence for interspecific competition in nature is ubiquitous, there is currently little empirical data supporting its role in the speciation process. Here, we discuss two general models in which interspecific competition among species can promote ecological speciation among populations within a species. In both models, interspecific competition is the source of divergent selection driving adaption to different portions of the resource distribution, generating ecological reproductive isolation from other conspecific populations. We propose that the biology of endoparasitoids that attack phytophagous insects make model systems for studying the role of interspecific competition in ecological speciation. We describe details for one such system, the community of endoparasitic braconid wasps attacking Rhagoletis fruit flies, as a potential model for investigating competitive speciation. We conclude by hypothesizing that a model in which interspecific competition forces an inferior competitor to alternative fly hosts may be a common theme contributing to parasitoid diversification in the Rhagoletis-parasitoid system.     

Modelling and simulation of the behaviour of a biofluid in a microchannel biochip separator

This paper reports an investigation into the flow behaviour of a biofluid in a microchannel systems through conceptual analysis and modelling. The application is the design of a microfluidic chip developed for the separation of plasma from blood. The effect of key design features of the microchannels on the flow behaviour of the biofluid is explored. These include geometric features such as the constriction, bending channel, bifurcation and the channel length ratio between the main and side channels. The performance of each design is discussed in terms of separation efficiency of the red blood cells with respect to the rest of the medium. Particular phenomena such as the Fahraeus and Fahraeus–Lindqvist effects, the Zweifach–Fung bifurcation law and the cell-free layer are discussed. In this paper, the fluid is modelled as a single-phase flow assuming either Newtonian or Non-Newtonian behaviour to investigate the effect of the fluid viscosity on both flow and separation efficiency. For a flow rate-controlled Newtonian flow system, it is found that viscosity and outlet pressure have little effect on the velocity distribution through each of the microchannels. For a diluted fluid where the flow in the whole channel system is modelled with a uniform viscosity, less plasma is separated from blood than observed in the non-Newtonian case. This results in an increase in the flow rate ratio between the main and side channels. A comparison of Newtonian and non-Newtonian flows shows that both flows tend to behave identically with an increase in the shear strain rate.     

Review Behaviour and indirect interactions in food webs of plant-inhabiting arthropods

With the increased use of biological control agents, artificial food webs are created in agricultural crops and the interactions between plants, herbivores and natural enemies change from simple tritrophic interactions to more complex food web interactions. Therefore, herbivore densities will not only be determined by direct predator–prey interactions and direct and indirect defence of plants against herbivores, but also by other direct and indirect interactions such as apparent competition, intraguild predation, resource competition, etc. Although these interactions have received considerable attention in theory and experiments, little is known about their impact on biological control. In this paper, we first present a review of indirect food web interactions in biological control systems. We propose to distinguish between numerical indirect interactions, which are interactions where one species affects densities of another species through an effect on the numbers of an intermediate species and functional indirect interactions, defined as changes in the way that two species interact through the presence of a third species. It is argued that functional indirect interactions are important in food webs and deserve more attention. Subsequently, we discuss experimental results on interactions in an artificial food web consisting of pests and natural enemies on greenhouse cucumber. The two pest species are the two-spotted spider mite Tetranychus urticae and the western flower thrips, Frankliniella occidentalis. Their natural enemies are the predatory mite Phytoseiulus persimilis, which is commonly used for spider mite control and the predatory mites Neoseiulus cucumeris and Iphiseius degenerans and the predatory bug Orius laevigatus, all natural enemies of thrips. First, we analyse the possible interactions between these seven species and we continue by discussing how functional indirect interactions, particularly the behaviour of arthropods, may change the significance and impact of direct interactions and numerical indirect interactions. It was found that a simple food web of only four species already gives rise to some quite complicated combinations of interactions. Spider mites and thrips interact indirectly through resource competition, but thrips larvae are intraguild predators of spider mites. Some of the natural enemies used for control of the two herbivore species are also intraguild predators. Moreover, spider mites produce a web that is subsequently used by thrips to hide from their predators. We discuss these and other results obtained so far and we conclude with a discussion of the potential impact of functional indirect and direct interactions on food webs and their significance for biological control.     

Modelling and simulation of the behaviour of a biofluid in a microchannel biochip separator

This paper reports an investigation into the flow behaviour of a biofluid in a microchannel systems through conceptual analysis and modelling. The application is the design of a microfluidic chip developed for the separation of plasma from blood. The effect of key design features of the microchannels on the flow behaviour of the biofluid is explored. These include geometric features such as the constriction, bending channel, bifurcation and the channel length ratio between the main and side channels. The performance of each design is discussed in terms of separation efficiency of the red blood cells with respect to the rest of the medium. Particular phenomena such as the Fahraeus and Fahraeus-Lindqvist effects, the Zweifach-Fung bifurcation law and the cell-free layer are discussed. In this paper, the fluid is modelled as a single-phase flow assuming either Newtonian or Non-Newtonian behaviour to investigate the effect of the fluid viscosity on both flow and separation efficiency. For a flow rate-controlled Newtonian flow system, it is found that viscosity and outlet pressure have little effect on the velocity distribution through each of the microchannels. For a diluted fluid where the flow in the whole channel system is modelled with a uniform viscosity, less plasma is separated from blood than observed in the non-Newtonian case. This results in an increase in the flow rate ratio between the main and side channels. A comparison of Newtonian and non-Newtonian flows shows that both flows tend to behave identically with an increase in the shear strain rate.     

Direct and indirect effects of n-species competition

Summary A measure of net interspecific competition is proposed which incorporates both direct and indirect effects between each species pair in a community. Indirect effects of one species on another may be mediated through other species in the community such that changes in the first species induce changes in other species which in turn directly affect the second species. Even in a competitive community these indirect effects are not necessarily always of a competitive nature and may sometimes actually act to oppose the effects of direct competition. Species pairs which are strong competitors in isolation may exhibit little net competition or even appear mutualistic depending on the other species present in the community.An expression for the net competitive effects between species pairs is derived in terms of the elements of the -matrix representation of the direct competitive effects. Examples calculated from the literature suggest that indirect competitive effects may often act to counterbalance direct competition between otherwise strong competitors in natural communities.     

Rate theory models for ion transport through rigid pores. II. Time-dependent analysis of the single-file model with special reference to oscillatory behavior

This article deals with the time-dependent evolution of the single-file movement of ions through channels of both biological and artificial membranes. The single-file transport process may exhibit not only the usual relaxation behaviour but also oscillatory behaviour as a steady state is approached after an initial perturbation. A necessary condition for the occurrence of oscillations is that the system acts sufficiently far from equilibrium. The occurrence of oscillations is due to the interactions within the transport system which are taken into account by the single-file model; these are the electrostatic repulsion between the ions being transported, and the competition of the ions for the free binding sites within the pore. Information about the strength of the interactions can be obtained by measuring the damping of the transport observables (e.g. the electric current): The stronger the inter-ionic repulsion, the more apparent the oscillatory behaviour will become. Furthermore, the damping is influenced by the microscopic structure of the transport system (i.e. the energy profile of the pores). With an increasing degree of microscopicity, i.e. with a decreasing number of binding sites and an increasingly irregular pore profile, the oscillations become more damped. However, a considerable oscillatory behaviour can only be predicted for pores with both a sufficiently regular structure and a sufficiently large number of binding sites. For this class of pores, however, the measurement of the damping represents an appropriate method of gaining information which could exceed that obtainable from the usual methods of measuring stationary quantities (e.g. stationary conductance). Moreover, our goal is to explain theoretically how the oscillatory behaviour can be interpreted in terms of the order inherent in the ionic movement, which is determined by both the external and internal forces and the microscopic properties of the system.     

Rate of forcing and the forecastability of critical transitions

Critical transitions are qualitative changes of state that occur when a stochastic dynamical system is forced through a critical point. Many critical transitions are preceded by characteristic fluctuations that may serve as model‐independent early warning signals, implying that these events may be predictable in applications ranging from physics to biology. In nonbiological systems, the strength of such early warning signals has been shown partly to be determined by the speed at which the transition occurs. It is currently unknown whether biological systems, which are inherently high dimensional and typically display low signal‐to‐noise ratios, also exhibit this property, which would have important implications for how ecosystems are managed, particularly where the forces exerted on a system are anthropogenic. We examine whether the rate of forcing can alter the strength of early warning signals in (1) a model exhibiting a fold bifurcation where a state shift is driven by the harvesting of individuals, and (2) a model exhibiting a transcritical bifurcation where a state shift is driven by increased grazing pressure. These models predict that the rate of forcing can alter the detectability of early warning signals regardless of the underlying bifurcation the system exhibits, but that this result may be more pronounced in fold bifurcations. These findings have important implications for the management of biological populations, particularly harvested systems such as fisheries, and suggest that knowing the class of bifurcations a system will manifest may help discriminate between true‐positive and false‐positive signals.     

Coexistence of plant species in a biodiversity hotspot is stabilized by competition but not by seed predation

Understanding the mechanisms of species coexistence is a key task for ecology. Recent theory predicts that both competition and predation (which causes apparent competition among prey) can either promote or limit species coexistence. Both mechanisms cause negative interactions between individuals, and each mechanism promotes stable coexistence if it causes negative interactions to be stronger between conspecifics than between heterospecifics. However, the relative importance of competition and predation for coexistence in natural communities is poorly known. Here, we study how competition and apparent competition via pre‐dispersal seed predators affect the long‐term fecundity of Protea shrubs in the fire‐prone Fynbos biome (South Africa). These shrubs store all viable seeds produced since the last fire in fire‐proof cones. Competitive effects on cone number and pre‐dispersal seed predation reduce their fecundity and can thus limit recruitment after the next fire. In 27 communities comprising 49 990 shrubs of 22 Protea species, we measured cone number and per‐cone seed predation rate of 2154 and 1755 focal individuals, respectively. Neighbourhood analyses related these measures to individual‐based community maps. We found that conspecific neighbours had stronger competitive effects on cone number than heterospecific neighbours. In contrast, apparent competition via seed predators was comparable between conspecifics and heterospecifics. This indicates that competition stabilizes coexistence of Protea species, whereas pre‐dispersal seed predation does not. Larger neighbours had stronger competitive effects and neighbours with large seed crops exerted stronger apparent competition. For 97% of the focal plants, competition reduced fecundity more than apparent competition. Our results show that even in communities of closely related and ecologically similar species, intraspecific competition can be stronger than interspecific competition. On the other hand, apparent competition through seed predators need not have such a stabilizing effect. These findings illustrate the potential of ‘community demography’, the demographic study of multiple interacting species, for understanding plant coexistence.     

Continuum limits of pattern formation in hexagonal-cell monolayers

Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O’Dea and King (Math Biosci 231(2):172–185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.     

Refuge-mediated apparent competition in plant–consumer interactions

At the intersection of consumer behaviour and plant competition is the concept of refuge-mediated apparent competition: an indirect interaction whereby plants provide a refuge for a shared consumer, subsequently increasing consumer pressure on another plant species. Here, we use a simple model and empirical examples to develop and illustrate the concept of refuge-mediated apparent competition. We find that the likelihood that an inferior competitor will succeed via refuge-mediated apparent competition is greater when competitors have similar resource requirements and when consumers exhibit a strong response to the refuge and high attack rates on the superior competitor. Refuge-mediated apparent competition may create an emergent Allee effect, such that a species invades only if it is sufficiently abundant to alter consumer impact on resident species. This indirect interaction may help explain unresolved patterns observed in biological invasion, such as the different physical structure of invasive exotic plants, the lag phase, and the failure of restoration efforts. Given the ubiquity of refuge-seeking behaviour by consumers and the ability of consumers to alter the outcome of direct competition among plants, refuge-mediated apparent competition may be an underappreciated mechanism affecting the composition and diversity of plant communities.
Ecology Letters (2010) 13: 11–20     

Dilation-producing stresses and strains obtained with the aid of density measurements

Stress systems in a deformed medium are in general of two kinds: (i) dilation-producing and (ii) nondilation-producing. Ordinarily the dilation is computed from a knowledge of the stresses, which in turn are obtained by solving the general elastostatic equations of equilibrium, under specified boundary conditions. If, however, interest is centered only in dilation-producing stresses, or in the event that only such stresses exist in the deformed system, and the density changes in the medium may be experimentally determined or postulateda priori as functions of the coordinates, these stresses may be directly obtained by a simplified procedure of solving for a particular integral of the Poisson equation without appeal to boundary conditions.^** In irrotational systems in which conservative body forces exist, there is a direct relationship between the dilation and the distributed body forces. Reference to both purely physical and biophysical systems is made. This investigation was supported by a PHS research grant, RG-5067, from the Public Health Service.