Three-species ecosystems in a solvable model

A detailed discussion of the three-species ecosystems is presented in an exactly solvable model with interactions of the Gompertz form. Three different possibilities, namely, a one-prey-two-predator system, a two-prey-one-predator system and a three-step prey-predator food chain are considered. These systems are studied not only when they include their basic prey-predator interactions, but also when various self-interactions as well as competition between like species, in different possible combinations, are included. It is then inferred, by obtaining and examining the exact solutions, as to when these systems possess stable equilibrium and when not, or when they are purely oscillatory, etc. We also study, within our model, the two-species versus three-species situation. It is seen that there are situations when the three-species system possesses stable equilibrium even under circumstances under which the corresponding two-species system is unstable. We also come across cases when the addition of the third species destroys the possibility of stable equilibrium which the initial two-species system possessed. Some other results also follow. Of particular interest is the one where the initial two-species system is purely oscillatory but the enlarged system, which is a three-step prey-predator chain, has the first and the last populations of the chain rising indefinitely and the middle population remains oscillatory. A comparison of our results with results of other authors, wherever possible, has also been made.     

A Bifurcation Analysis of a Differential Equations Model for Mutualism

We develop from basic principles a two-species differential equations model which exhibits mutualistic population interactions. The model is similar in spirit to a commonly cited model [Dean, A.M., Am. Nat. 121(3), 409–417 (1983)], but corrects problems due to singularities in that model. In addition, we investigate our model in more depth by varying the intrinsic growth rate for each of the species and analyzing the resulting bifurcations in system behavior. We are especially interested in transitions between facultative and obligate mutualism. The model reduces to the familiar Lotka–Volterra model locally, but is more realistic for large populations in the case where mutualist interaction is strong. In particular, our model supports population thresholds necessary for survival in certain cases, but does this without allowing unbounded population growth. Experimental implications are discussed for a lichen population.     

Spatio-Temporal Environmental Correlation and Population Variability in Simple Metacommunities

Natural populations experience environmental conditions that vary across space and over time. This variation is often correlated between localities depending on the geographical separation between them, and different species can respond to local environmental fluctuations similarly or differently, depending on their adaptation. How this emerging structure in environmental correlation (between-patches and between-species) affects spatial community dynamics is an open question. This paper aims at a general understanding of the interactions between the environmental correlation structure and population dynamics in spatial networks of local communities (metacommunities), by studying simple two-patch, two-species systems. Three different pairs of interspecific interactions are considered: competition, consumer–resource interaction, and host–parasitoid interaction. While the results paint a relatively complex picture of the effect of environmental correlation, the interaction between environmental forcing, dispersal, and local interactions can be understood via two mechanisms. While increasing between-patch environmental correlation couples immigration and local densities (destabilising effect), the coupling between local populations under increased between-species environmental correlation can either amplify or dampen population fluctuations, depending on the patterns in density dependence. This work provides a unifying framework for modelling stochastic metacommunities, and forms a foundation for a better understanding of population responses to environmental fluctuations in natural systems.     

Coupled Flip-Flop Model for REM Sleep Regulation in the Rat

Recent experimental studies investigating the neuronal regulation of rapid eye movement (REM) sleep have identified mutually inhibitory synaptic projections among REM sleep-promoting (REM-on) and REM sleep-inhibiting (REM-off) neuronal populations that act to maintain the REM sleep state and control its onset and offset. The control mechanism of mutually inhibitory synaptic interactions mirrors the proposed flip-flop switch for sleep-wake regulation consisting of mutually inhibitory synaptic projections between wake- and sleep-promoting neuronal populations. While a number of synaptic projections have been identified between these REM-on/REM-off populations and wake/sleep-promoting populations, the specific interactions that govern behavioral state transitions have not been completely determined. Using a minimal mathematical model, we investigated behavioral state transition dynamics dictated by a system of coupled flip-flops, one to control transitions between wake and sleep states, and another to control transitions into and out of REM sleep. The model describes the neurotransmitter-mediated inhibitory interactions between a wake- and sleep-promoting population, and between a REM-on and REM-off population. We proposed interactions between the wake/sleep and REM-on/REM-off flip-flops to replicate the behavioral state statistics and probabilities of behavioral state transitions measured from experimental recordings of rat sleep under ad libitum conditions and after 24 h of REM sleep deprivation. Reliable transitions from REM sleep to wake, as dictated by the data, indicated the necessity of an excitatory projection from the REM-on population to the wake-promoting population. To replicate the increase in REM-wake-REM transitions observed after 24 h REM sleep deprivation required that this excitatory projection promote transient activation of the wake-promoting population. Obtaining the reliable wake-nonREM sleep transitions observed in the data required that activity of the wake-promoting population modulated the interaction between the REM-on and REM-off populations. This analysis suggests neuronal processes to be targeted in further experimental studies of the regulatory mechanisms of REM sleep.     

Evidence for emergent behaviour in the community-scale dynamics of a fungal microcosm

A stochastic cellular automaton for modelling the dynamics of a two-species fungal microcosm is presented. The state of each cell in the automaton depends on the state of a predefined neighbourhood via a set of conditional probabilities derived from experiments conducted on pairwise combinations of species. The model is tested by detailed comparison with larger-scale experimental microcosms. By employing different hypotheses which relate the pairwise data to the conditional probabilities in the model, the nature of the local and non-local interactions in the community is explored. The hypothesis that the large-scale dynamics are a consequence of independent interactions between species in a local neighbourhood can be excluded at the 5% significance level. The form of the interdependencies is determined and it is shown that the outcome of the interactions at the local neighbourhood-scale depends on the community-scale patterning of individuals. The dynamics of the microcosm are therefore an emergent property of the system of interacting mycelia that cannot be deduced from a study of the components in isolation.     

Combining Perturbations and Parameter Variation to Influence Mean First Passage Times

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.     

Spreading speed and linear determinacy for two-species competition models

 One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on `linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11]. Received: 7 August 2000 / Revised version: 5 January 2002 / Published online: 17 July 2002     

Positive interactions, discontinuous transitions and species coexistence in plant communities

The population and community level consequences of positive interactions between plants remain poorly explored. In this study we incorporate positive resource-mediated interactions in classic resource competition theory and investigate the main consequences for plant population dynamics and species coexistence. We focus on plant communities for which water infiltration rates exhibit positive dependency on plant biomass and where plant responses can be improved by shading, particularly under water limiting conditions. We show that the effects of these two resource-mediated positive interactions are similar and additive. We predict that positive interactions shift the transition points between different species compositions along environmental gradients and that realized niche widths will expand or shrink. Furthermore, continuous transitions between different community compositions can become discontinuous and bistability or tristability can occur. Moreover, increased infiltration rates may give rise to a new potential coexistence mechanism that we call controlled facilitation.     

Stationary solutions to a system of size-structured populations with nonlinear growth rate

We study stationary solutions to a system of size-structured population models with nonlinear growth rate. Several characterizations of stationary solutions are provided. It is shown that the steady-state problem can be converted into different problems such as two types of eigenvalue problems and a fixed-point problem. In the two-species case, we give an existence result of nonzero stationary solutions by using the fixed-point problem.     

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 general theory of environmental noise in ecological food webs

We examine the effects of environmental noise on populations that are parts of simple two-species food webs. We assume that the species are strongly interacting and that one or the other population is affected by the noise signal. Further assuming that a stable equilibrium with positive population densities exists, we are able to perform a complete frequency analysis of the system. If only one of the populations is subject to noise, the relative noise response by both populations is fully determined by the sign of a single element of the Jacobian matrix. The analysis is readily extended to cases when both species are affected by noise or when the food web has more than two species. The general conclusion about relative responses to noise is then less unambiguous, but the power spectra describing the frequency composition of the population variabilities are nevertheless completely determined. These results are entirely independent on the exact nature of the interaction (i.e., predation, competition, mutualism) between the populations. The results show that the interpretation of the "color" of ecological time series (i.e., the frequency composition of population variability over time) may be complicated by species interactions. The propagation of noise signals through food webs and the importance of web structure for the expected response of all parts of the web to such signals is a challenging field for future studies.     

Order Parameter Dynamics of Body-scaled Hysteresis and Mode Transitions in Grasping Behavior

Several experimental studies have shown that human grasping behavior exhibits a transition from one-handed to two-handed grasping when to-be-grasped objects become larger and larger. The transition point depends on the relative size of objects measured in terms of human body-scales. Most strikingly, the transitions between the two different behavioral ‘modes’ of grasping exhibit hysteresis. That is, one-to-two hand transitions and two-to-one hand transitions occur at different relative object sizes when objects are scaled up or down in size. In our study we approach body-scaled hysteresis and mode transitions in grasping by exploiting the notion that human behavior in general results from self-organization and satisfies appropriately-defined order parameter equations. To this end, grasping transitions and grasping hysteresis are discussed from a theoretical perspective in analogy to cognitive processes defined by Haken’s neural network model for pattern recognition. In doing so, issues such as the exclusivity of grasping modes, biomechanical constraints, mode-mode interactions, single subject behavior and population behavior are explored.     

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Several firing patterns experimentally observed in neural populations have been successfully correlated to animal behavior. Population bursting, hereby regarded as a period of high firing rate followed by a period of quiescence, is typically observed in groups of neurons during behavior. Biophysical membrane-potential models of single cell bursting involve at least three equations. Extending such models to study the collective behavior of neural populations involves thousands of equations and can be very expensive computationally. For this reason, low dimensional population models that capture biophysical aspects of networks are needed. The present paper uses a firing-rate model to study mechanisms that trigger and stop transitions between tonic and phasic population firing. These mechanisms are captured through a two-dimensional system, which can potentially be extended to include interactions between different areas of the nervous system with a small number of equations. The typical behavior of midbrain dopaminergic neurons in the rodent is used as an example to illustrate and interpret our results. The model presented here can be used as a building block to study interactions between networks of neurons. This theoretical approach may help contextualize and understand the factors involved in regulating burst firing in populations and how it may modulate distinct aspects of behavior.     

The impact of interspecific competition on lineage evolution and a rapid peak shift by interdemic genetic mixing in experimental bacterial populations

Epistatic interactions between genes in the genome constrain the accessible evolutionary paths of lineages. Two factors involving epistasis that can affect the evolutionary path and fate of lineages were investigated. The first factor concerns the impact of competition with another species lineage that has different epistatic constraints. Five enteric bacterial populations were evolved by point mutation in medium containing a single limiting resource. Single-species and two-species cultures were used to determine whether different asexual lineages have different capacities for producing variants due to epistatic constraints, and whether their survival is determined by local inter-lineage competition with different species. Local inter-lineage competition quickly resulted in one successful lineage, with another lineage becoming extinct before finding a higher peak. The second factor concerns a peak-shifting process, and whether the sexual recombination between different demes can cause peak shifts was investigated. An Escherichia coli population consisting of a male (Hfr) and female strain (F(-)) was evolved in a single limiting resource and compared to evolving populations containing the male or female strain alone. The E. coli sexual lineage was successful due to its ability to escape lower peaks and reach a higher peak, not because of a rapid approach to the nearest local peak the male or female asexual lineage could reach. The data in this study demonstrate that lineage survivability can be determined by the ability to produce beneficial mutations and checked by local competition between lineages of different species. Interspecific competition may prevent a population from evolving through crossing fitness valleys or adaptive ridges if it requires many generations to achieve peak shifts. The data also show that genomic recombination between different conspecific lineages can rapidly carry the combined lineage to a higher peak.     

Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments

In this article, we develop population game theory, a theory that combines the dynamics of animal behavior with population dynamics. In particular, we study interaction and distribution of two species in a two-patch environment assuming that individuals behave adaptively (i.e., they maximize Darwinian fitness). Either the two species are competing for resources or they are in a predator-prey relationship. Using some recent advances in evolutionary game theory, we extend the classical ideal free distribution (IFD) concept for single species to two interacting species. We study population dynamical consequences of two-species IFD by comparing two systems: one where individuals cannot migrate between habitats and one where migration is possible. For single species, predator-prey interactions, and competing species, we show that these two types of behavior lead to the same population equilibria and corresponding species spatial distributions, provided interspecific competition is patch independent. However, if differences between patches are such that competition is patch dependent, then our predictions strongly depend on whether animals can migrate or not. In particular, we show that when species are settled at their equilibrium population densities in both habitats in the environment where migration between habitats is blocked, then the corresponding species spatial distribution need not be an IFD. Thus, when species are given the opportunity to migrate, they will redistribute to reach an IFD (e.g., under which the two species can completely segregate), and this redistribution will also influence species population equilibrial densities. Alternatively, we also show that when two species are distributed according to the IFD, the corresponding population equilibrium can be unstable.     

Unintended consequences of conservation actions: managing disease in complex ecosystems

Infectious diseases are increasingly recognised to be a major threat to biodiversity. Disease management tools such as control of animal movements and vaccination can be used to mitigate the impact and spread of diseases in targeted species. They can reduce the risk of epidemics and in turn the risks of population decline and extinction. However, all species are embedded in communities and interactions between species can be complex, hence increasing the chance of survival of one species can have repercussions on the whole community structure. In this study, we use an example from the Serengeti ecosystem in Tanzania to explore how a vaccination campaign against Canine Distemper Virus (CDV) targeted at conserving the African lion (Panthera leo), could affect the viability of a coexisting threatened species, the cheetah (Acinonyx jubatus). Assuming that CDV plays a role in lion regulation, our results suggest that a vaccination programme, if successful, risks destabilising the simple two-species system considered, as simulations show that vaccination interventions could almost double the probability of extinction of an isolated cheetah population over the next 60 years. This work uses a simple example to illustrate how predictive modelling can be a useful tool in examining the consequence of vaccination interventions on non-target species. It also highlights the importance of carefully considering linkages between human-intervention, species viability and community structure when planning species-based conservation actions.     

Dynamics in a discrete two-species competition model: coexistence and over-compensation

The dynamic features of an over-compensating discrete two-species competition system with stable coexistence are recaptured, and it is shown how the probabilities of the different possible ecological scenarios, e.g. coexistence, may be calculated when the assumption of no over-compensation is loosened. A Bayesian methodology for calculating the probability that stable oscillations or chaos may occur in plant populations or communities is outlined. The methodology is exemplified using an experimental population of Arabidopsis thaliana. It is concluded that, when making ecological predictions it is preferable and possibly important to test for the possibility of chaotic population dynamics due to over-compensation rather than assuming a priori that over-compensation does not occur.     

Dynamic modeling of cooperative protein secretion in microorganism populations

Interactions between microorganisms can have a crucial effect on their population dynamics. Typically, interactions are mediated through the environment by molecules and proteins that are products of cell metabolism and physiology; they therefore reflect the internal dynamics of the single cell. In this work we aim to integrate single-cell properties of gene expression that affect indirect interactions between microorganisms under challenging conditions, into a quantitative model of population dynamics. Specifically we address the problem of a microbial population secreting a protein that can actively extract a growth-limiting resource, such as a simple sugar or iron, from the environment. The genes coding for the protein can undergo random epigenetic transitions between active and silenced states, and can be repressed by the product of their reaction. We model cooperative and competitive interactions between protein producing and non-producing phenotypes by nonlinear dynamical systems and analyze them both in terms of asymptotic states and of transient dynamics. Our model shows that phenotypic transitions allow a stable coexistence of the two phenotypes, and enables us to make predictions regarding the conditions required for such coexistence and the typical timescales of transient dynamics. It also shows how repression by the reaction product induces a feedback at the population-environment level that can result in limit cycle dynamics. The relation of these results to experiments are discussed.     

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.     

Coexistence via resource partitioning fails to generate an increase in community function

Classic ecological theory suggests that resource partitioning facilitates the coexistence of species by reducing inter-specific competition. A byproduct of this process is an increase in overall community function, because a greater spectrum of resources can be used. In contrast, coexistence facilitated by neutral mechanisms is not expected to increase function. We studied coexistence in laboratory microcosms of the bactivorous ciliates Paramecium aurelia and Colpidium striatum to understand the relationship between function and coexistence mechanism. We quantified population and community-level function (biomass and oxygen consumption), competitive interactions, and resource partitioning. The two ciliates partitioned their bacterial resource along a size axis, with the larger ciliate consuming larger bacteria than the smaller ciliate. Despite this, there was no gain in function at the community level for either biomass or oxygen consumption, and competitive effects were symmetrical within and between species. Because other potential coexistence mechanisms can be ruled out, it is likely that inter-specific interference competition diminished the expected gain in function generated by resource partitioning, leading to a system that appeared competitively neutral even when structured by niche partitioning. We also analyzed several previous studies where two species of protists coexisted and found that the two-species communities showed a broad range of biomass levels relative to the single-species states.