Effects of convective and dispersive interactions on the stability of two species

The evolution and local stability of a system of two interacting species in a finite two-dimensional habitat is investigated by taking into account the effects of self- and cross-dispersion and convection of the species. In absence of cross-dispersion, an equilibrium state which is stable without dispersion is always stable with dispersion provided that the dispersion coefficients of the two species are equal. However, when the dispersion coefficients of the two species are different, the possibility of self-dispersive instability arises. It is also pointed out that the cross-dispersion of species may lead to stability or instability depending upon the nature and the magnitude of the cross-dispersive interactions in comparison to the self-dispersive interactions. The self-convective movement of species increases the stability of the equilibrium state and can stabilize an otherwise unstable equilibrium state. The effect of cross-convection (in absence of self-dispersion and self-convection) is to stabilize the equilibrium state in a prey-predator model with positive cross-dispersion coefficients for the prey species. Finally, it is shown that if the system is stable under homogeneous boundary conditions it remains so under non-homogeneous boundary conditions.     

Two fish species competition model with nonlinear interactions and equilibrium catches

Two species competition model is built up by assuming the hypothetical second order interactions in order to consider effects of exploitation on two competing fish species with non-linear interactions. Most important characteristic of this model, compared withLotka-Volterra type linear competition model, is that this model can possess multiple stable equilibrium points. Therefore there is a possibility that two species keeping the equilibrium state at one stable equilibrium point will be attracted to the other stable equilibrium point after a heavy perturbation. In this model reversible change of the fishing pressure does not always results in that of the equilibrium catch. In this sence MSY concept for single species can not be extended to this model. If there are multiple stable equilibrium points, the change of the dominant fish species, catastrophic and irreversible change of each equilibrium catch may be observed when the perturbation by the exploitation is added. This phenomenon immediately reminds us of the change of the dominant fish species between Japanese common mackerel and Pacific saury in the northwest Pacific Ocean. In case of the management of two competing fish species with nonlinear interactions, the consideration on the balance between the fishing pressure for each species may be as important as the decision on the catch limit for each species. MSY level for each species based on the single-species theory could be quite erroneous.     

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.     

Regime shift and robustness of organism-created environments: A model for microbial ecosystems

Organism-environment interactions are different from organism-resource interactions in two respects: (1) resources can only be consumed by organisms whereas environmental conditions can be increased or decreased depending on the species; (2) high resource conditions generally stimulate the growth of organisms, whereas extreme environmental conditions are not necessarily favored because each species usually has an optimum range for growth. To investigate the properties of an organism-environment feedback system, we analyze a model for microbial ecosystems in which a single microorganism species can modify the environmental pH. We demonstrate that the equilibrium level of the environmental pH can be partially regulated at a relatively constant value even if the pH in the influx to the ecosystem changes over a wide range. For species that acidify the medium, the equilibrium pH is somewhat lower than the pH optimal for the species. The pH-stabilizing effect of microorganisms is stronger if their growth is self-limited by the environmental pH. When the influx becomes sufficiently alkaline, the population of the organism suddenly disappears and the environmental pH changes abruptly. The system shows bi-stability and hysteresis and therefore differs from a standard resource competition model composed of a single species that consumes resources.     

Using mean first passage times to quantify equilibrium resilience in perturbed intraguild predation systems

Regime shift inducibility depends on equilibrium resilience, which depends on species interactions. When species interactions include intraguild predation (IGP), integrated pest management may induce regime shifts because enhancing the abundance of intraguild predators simultaneously increases competition with, and predation on, invasive prey. To explore the dynamical consequences of such manipulations, we use a bistable, deterministic IGP model with stochastic removals that perturb invader density from the high-density equilibrium. We quantify the combined effects of IGP and such perturbations in terms of mean first passage times (MFPTs) to target invader densities such as thresholds between regimes. Analytical MFPTs compare favorably with those generated by Monte Carlo numerical solutions of the stochastically perturbed IGP model. MFPTs can therefore usefully quantify equilibrium resilience in terms of perturbation schedules. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator–prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator–prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.     

Nonlinearity in interspecific interactions in response to climate change: Cod and haddock as an example

Climate change has profound ecological effects, yet our understanding of how trophic interactions among species are affected by climate change is still patchy. The sympatric Atlantic haddock and cod are co‐occurring across the North Atlantic. They compete for food at younger stages and thereafter the former is preyed by the latter. Climate change might affect the interaction and coexistence of these two species. Particularly, the increase in sea temperature (ST) has been shown to affect distribution, population growth and trophic interactions in marine systems. We used 33‐year long time series of haddock and cod abundances estimates from two data sources (acoustic and trawl survey) to analyse the dynamic effect of climate on the coexistence of these two sympatric species in the Arcto‐Boreal Barents Sea. Using a Bayesian state‐space threshold model, we demonstrated that long‐term climate variation, as expressed by changes of ST, affected species demography through different influences on density‐independent processes. The interaction between cod and haddock has shifted in the last two decades due to an increase in ST, altering the equilibrium abundances and the dynamics of the system. During warm years (ST over ca. 4°C), the increase in the cod abundance negatively affected haddock abundance while it did not during cold years. This change in interactions therefore changed the equilibrium population size with a higher population size during warm years. Our analyses show that long‐term climate change in the Arcto‐Boreal system can generate differences in the equilibrium conditions of species assemblages.     

Coexistence in MacArthur-style consumer-resource models

The nature of and conditions for permanent coexistence of consumers and resources are characterized in a family of models that generalize MacArthur's consumer-resource model. The generalization is of the resource dynamics, which need not be of Lotka-Volterra form but are subject only to certain restrictions loose enough to admit many resource dynamics of biological interest. For any such model, (1) if there is an interior equilibrium, then it is globally attracting, else some boundary equilibrium is globally attracting-thus permanent coexistence is coexistence at a globally attracting equilibrium; (2) there is an interior equilibrium if and only if for any species, the equilibrium approached in the absence of that species and the presence of the others is invasible by that species--thus permanent coexistence is equivalent to mutual invasibility; (3) for resources without direct interactions, the conditions for permanent coexistence of the consumers admit an instructive formulation in terms of regression statistics. The significance and limitations of the models and results are discussed.     

Habitat selection,interspecific interactions and landscape composition

Summary I argue here that, from the perspective of any individual, most landscapes are composed of only three basic types of habitats. These are: (1) source habitat in which reproduction exceeds mortality and the expected per capita growth rate is greater than one; (2) sink habitat, in which limited, reproduction is possible but will not on average, compensate for mortality and the per capita rate of growth is between zero and one; and (3) unusable habitat, which comprises the matrix of all habitats that are never exploited by the species in question, and in which patches of source and sink habitats are embedded. Unlike earlier source-sink models, this model explicitly considers the effects that substituting one type of habitat for another has on the equilibrium size of a population and the interactions between species which can use both source and sink habitats. The model demonstrates that the equilibrium size of a species' population can sometimes be increased by substituting unusable habitat for sink habitat. Thus, even though the average patch quality in the landscape may be decreased, the overall quality of the landscape can increase. For two species with distinct habitat preferences, interactions between species can vary qualitatively as well as quantitatively as a function of the relative abundances of each of the habitat types. The model also shows that the interactions between species are particularly sensitive to the relative costs of moving between patches and sampling patches to determine their quality. Recent fragmentation of natural landscapes may increase the cost of searching for usable (source or sink) patches. Under some conditions, the interspecific interactions may be substantially more negative (competitive) than the interactions that evolved in the original natural landscape, further reducing population sizes and increasing the likelihood of competitive exclusion in fragmented modern landscapes.     

Microbial colonization dynamics in temporary aquatic systems

Most studies of community development in insular systems have investigated the colonization dynamics of only a portion of the biotic community using islands that have the benefit of prior biotic modification. Two experiments assessed the predictive ability of equilibrium island theory with regard to the development of the protistan community in temporary aquatic islands (100-1 plastic swimming pools) void of any prior biotic modification. The first experiment ran for 170 d (March–September 1985) and manipulated the access of islands to animals that represent potentially important dispersal pathways for microbes. A second experiment (May–June 1986) investigated in greater detail the early stages of species accrual in the absence of dispersal pathways other than via the atmosphere. Polyurethane foam substrates were used as sampling devices to increase the accuracy and replicability of sampling and provide habitat for colonists. Sampling was determined to be asymptotic. Species accrual was asymptotic in both experiments, although it initially lagged behind that predicted by theory. Autotrophs approached equilibrium faster than heterotrophs, but at a lower species richness. The predicted number of autotroph species at equilibrium was lower in islands that were excluded from contact with birds compared to less exclosed islands. The colonization dynamics of the entire community was not significantly different among islands having different degrees of exclosure. In both experiments, rates of species immigration were nonmonotonic with respect to time and species richness. This relationship appeared to reflect the importance of species interactions during the initial accrual phase before equilibrium. Rates of extinction were positively correlated with both these parameters, although they tended to decrease with time during the equilibrium period in less exclosed islands. Turnover at equilibrium was significant and resulted in directional changes in species composition over time. Assortative processes appeared to be important since later colonists exhibited greater persistence. Colonizing species generally fall into three autecological categories: 1) those that were ubiquitous and had temporally predictable patterns of immigration (successional species); 2) those possessing temporally predictable distributions but not spatially ubiquitous distributions (dispersal limited species); 3) those that showed little temporal or spatial predictability in immigration (transient or allochthonous species). Individual islands exhibited various degrees of fluctuation in species number during the predicted equilibrium which were poorly correlated with exogenous environmental variables and physicochemical habitat parameters. The presence of predacious mosquito larvae(Culex spp.) invariably resulted in a sharp decreases in microbial species richness, while documented contact with rodents was followed by an increase in species number in an island so contaminated. Several aspects of microbial colonization of temporary waters that contradict equilibrium predictions appear to be strongly influenced by microbe-microbe as well as macrobe-microbe interactions.     

Global Stability of a System of Two Interacting Species with Convective and Dispersive Migration

Using Liapunov's direct method, effects of convective and dispersive migration on the global stability of the equilibrium state of a system of two interacting species are investigated. It is shown that the stable equilibrium state without dispersal remains so with dispersal. Further, it is pointed out that stability or instability of the equilibrium state of the system is not affected by convective migration. These results are justified in cases of a system of mutualistic interactions of species and a prey-predator system with functional response.     

Non-ideal tracer sedimentation equilibrium: a powerful tool for the characterization of macromolecular interactions in crowded solutions

Non-ideal tracer sedimentation equilibrium is a technique devised to quantify the effect of high concentrations of unrelated macromolecules on the self- or hetero-associations of dilute macromolecules. Principles and experimental techniques are reviewed, and previous experimental work summarized. A new analysis of experimental data is presented that requires no a priori assumptions regarding the nature of weak repulsive interactions between solute species and the concentrated (crowding) species.     

Resource conversion: a generalizable mechanism for resource-mediated positive species interactions

Positive species interactions are ubiquitous in natural communities, but the mechanisms through which they operate are poorly understood. One proposed mechanism is resource conversion – the conversion by a benefactor species of a resource from a resource state that is inaccessible to a potential beneficiary species into a resource state that is accessible. Such conversion often occurs as a byproduct of resource consumption, and sometimes in exchange for non-resource benefits to the benefactor species. At least five known classes of interactions, including both facilitative and mutualistic ones, may be classified as resource conversion interactions. We formulated a generalizable mathematical model for resource conversion interactions and examined two model variants that represent processing chain and nurse plant interactions. We examined the conditions under which these conformed to the stress-gradient hypothesis (SGH), which predicts increased interaction benefits in more stressful environments. These yielded four key insights: 1) resource conversion interactions can be positive (towards the resource recipient) only when facilitator-mediated resource conversion is more efficient than the baseline, spontaneous, facilitator-independent resource conversion; 2) the sign of resource conversion interaction outcomes never switches (e.g. from net positive to net negative) with changing levels of resource availability, when all other parameters are kept constant; 3) processing chain interactions at equilibrium can never be positive in a manner that conforms to the SGH; 4) nurse plant interactions can be positive and conform to the SGH, although the manner in which they do depends largely on how resource stress is defined, and the environmental supply rate of surface soil moisture. The first two insights are likely to be generalizable across all resource conversion interactions. The general agreement of the model with empirical studies suggest that resource conversion is the mechanism underlying the aforementioned interactions, and an ecologically meaningful way of classifying these previously unassociated positive species interactions.     

Salt-dependent interconversion of inner histone oligomers.

The inner histone complex, extracted from chicken erythrocyte chromatin in 2 M NaCL AT pH 7.4, has been characterized by sedimentation equilibrium and sedimentation velocity. High speed sedimentation equilibrium studies indicate that in 2 M NaCl the inner histones are a weakly associating system with contributions from species ranging in molecular weight from dimer to octamer. The appearance of a single boundary (3.8S at 2 M NaCl) in sedimentation velocity studies conducted over a wide range of protein concentrations and ionic conditions indicates that the various histone oligomers present are in rapid equilibrium with one another. At higher salts the equilibrium is shifted to favor higher molecular weight species; in 4 M NaCl essentially all of the histone is octameric at protein concentrations above 0.2 mg/ml. The facile interconversion of histone oligomers suggests that small alterations in histone-histone interactions may be responsible for changes in nucleosome conformations during various biological processes.     

Environmental variability uncovers disruptive effects of species' interactions on population dynamics

How species respond to changes in environmental variability has been shown for single species, but the question remains whether these results are transferable to species when incorporated in ecological communities. Here, we address this issue by analysing the same species exposed to a range of environmental variabilities when (i) isolated or (ii) embedded in a food web. We find that all species in food webs exposed to temporally uncorrelated environments (white noise) show the same type of dynamics as isolated species, whereas species in food webs exposed to positively autocorrelated environments (red noise) can respond completely differently compared with isolated species. This is owing to species following their equilibrium densities in a positively autocorrelated environment that in turn enables species–species interactions to come into play. Our results give new insights into species'' response to environmental variation. They especially highlight the importance of considering both species'' interactions and environmental autocorrelation when studying population dynamics in a fluctuating environment.     

Sedimentation equilibrium in a solution containing an arbitrary number of solute species at arbitrary concentrations: theory and application to concentrated solutions of ribonuclease

Simple expressions are derived describing the equilibrium concentration gradient of each species in a solution containing an arbitrary number of solute species at arbitrary concentration, as a function of the concentration of all species. Quantitative relationships between the species gradients and experimentally observable signal gradients are presented. The expressions are model-free and take into account both attractive and repulsive interactions between all species. In order to analyze data obtained from strongly nonideal solutions, a statistical thermodynamic model for repulsive solute-solute interactions is required. The relations obtained are utilized to analyze the dependence of the equilibrium gradient of ribonuclease A in phosphate-buffered saline, pH 7.4, upon total protein concentration. Experimental results are interpreted in the context of a model for weak self-association leading to the formation of significant amounts of oligomers at total protein concentrations exceeding 25 g/l.     

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

Using sensitivity analysis to identify keystone species and keystone links in size‐based food webs

Human‐induced alterations in the birth and mortality rates of species and in the strength of interactions within and between species can lead to changes in the structure and resilience of ecological communities. Recent research points to the importance of considering the distribution of body sizes of species when exploring the response of communities to such perturbations. Here, we present a new size‐based approach for assessing the sensitivity and elasticity of community structure (species equilibrium abundances) and resilience (rate of return to equilibrium) to changes in the intrinsic growth rate of species and in the strengths of species interactions. We apply this approach on two natural systems, the pelagic communities of the Baltic Sea and Lake Vättern, to illustrate how it can be used to identify potential keystone species and keystone links. We find that the keystone status of a species is closely linked to its body size. The analysis also suggests that communities are structurally and dynamically more sensitive to changes in the effects of prey on their consumers than in the effects of consumers on their prey. Moreover, we discuss how community sensitivity analysis can be used to study and compare the fragility of communities with different body size distributions by measuring the mean sensitivity or elasticity over all species or all interaction links in a community. We believe that the community sensitivity analysis developed here holds some promise for identifying species and links that are critical for the structural and dynamic robustness of ecological communities.     

Multilocus genetics and the coevolution of quantitative traits

We develop and analyze an explicit multilocus genetic model of coevolution. We assume that interactions between two species (mutualists, competitors, or victim and exploiter) are mediated by a pair of additive quantitative traits that are also subject to direct stabilizing selection toward intermediate optima. Using a weak-selection approximation, we derive analytical results for a symmetric case with equal locus effects and no mutation, and we complement these results by numerical simulations of more general cases. We show that mutualistic and competitive interactions always result in coevolution toward a stable equilibrium with no more than one polymorphic locus per species. Victim-exploiter interactions can lead to different dynamic regimes including evolution toward stable equilibria, cycles, and chaos. At equilibrium, the victim is often characterized by a very large genetic variance, whereas the exploiter is polymorphic in no more than one locus. Compared to related one-locus or quantitative genetic models, the multilocus model exhibits two major new properties. First, the equilibrium structure is considerably more complex. We derive detailed conditions for the existence and stability of various classes of equilibria and demonstrate the possibility of multiple simultaneously stable states. Second, the genetic variances change dynamically, which in turn significantly affects the dynamics of the mean trait values. In particular, the dynamics tend to be destabilized by an increase in the number of loci.     

Effects of convective and dispersive migration on the linear stability of a two species system with mutualistic interactions and functional response

In this paper, effects of convective and dispersive migration on the linear stability of the equilibrium state of a two species system with mutualistic interactions and functional response have been investigated. In both finite and semi-infinite habitats, it has been shown that the otherwise stable equilibrium state without dispersal remains so with dispersal also, both under flux and reservior conditions. In the case of finite habitat, the degree of stability increases as dispersal coefficients of the two species increase. The effect of convective migration also is to stabilize the equilibrium state in this case.