Convergence to stationary distributions in two-species stochastic competition models

Two sets of sufficient conditions are given for convergence to stationary distributions, for some general models of two species competing in a randomly varying environment. The models are nonlinear stochastic difference equations which define Markov chains. One set of sufficient conditions involves strong continuity and -irreducibility of the transition probability for the chain. The second set has a much weaker irreducibility condition, but is only applicable to monotonic models. The results are applied to a stochastic two-species Ricker model, and to Chesson's lottery model with vacant space, to illustrate how the assumptions can be checked in specific models.     

Diffusion-mediated persistence in two-species competition Lotka-Volterra model

We consider a system composed of two Lotka-Volterra patches connected by diffusion. Each patch has two competitors. Conditions for persistence of the system are given. It is proved that the system can be made persistent under appropriate diffusion coefficients ensuring the instability of boundary equilibria, even if each species is not persistent within each patch. The choice of the coefficients depends closely on the patch dynamics without diffusion.     

Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Coexistence and invasibility in a two-species competition model with habitat-preference

The outcome of competition among species is influenced by the spatial distribution of species and effects such as demographic stochasticity, immigration fluxes, and the existence of preferred habitats. We introduce an individual-based model describing the competition of two species and incorporating all the above ingredients. We find that the presence of habitat preference—generating spatial niches—strongly stabilizes the coexistence of the two species. Eliminating habitat preference—neutral dynamics—the model generates patterns, such as distribution of population sizes, practically identical to those obtained in the presence of habitat preference, provided an higher immigration rate is considered. Notwithstanding the similarity in the population distribution, we show that invasibility properties depend on habitat preference in a non-trivial way. In particular, the neutral model results more invasible or less invasible depending on whether the comparison is made at equal immigration rate or at equal distribution of population size, respectively. We discuss the relevance of these results for the interpretation of invasibility experiments and the species occupancy of preferred habitats.     

A discrete stage-structured two-species competition model with sexual and clonal reproduction

In this paper, we analyse a discrete stage-structured model which is a generalization of the two-species competition model studied in [2]. Motivated by plant populations, each species is assumed to reproduce both sexually and clonally. We show that this model has a dynamical behaviour that is similar to that of the classical continuous two-dimensional Lotka-Volterra model under weak nonlinearities of the Beverton-Holt type. By allowing the species to have different competition efficiencies, we show that it is possible to obtain different dynamics including coexistence, bistability and competitive exclusion, in contrast with the model studied in [2], which exhibits only competitive exclusion behaviour.     

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.     

Analysis of a lottery competition model with limited nutrient availability

How a plant species utilizes a limited nutrient is important for its survival. The purpose of this work is to examine how nutrient utilization mechanisms (for seed production) affect the coexistence of competing plant species. We construct a revised lottery model that uses one of three possible kinds of nutrient utilization functions. In all cases the models suggest that two species can coexist under certain circumstances, but that three species cannot coexist, at least when the nutrient utilization functions are continuous functions of nutrient uptake. However, in the discontinuous case three species can coexist in a state of sustained oscillations. The results suggest that one need pay close attention to the differences in the nutrient utilization mechanisms among competing plant species in order to ascertain the competitive outcome.     

Analysis of a lottery competition model with limited nutrient availability

How a plant species utilizes a limited nutrient is important for its survival. The purpose of this work is to examine how nutrient utilization mechanisms (for seed production) affect the coexistence of competing plant species. We construct a revised lottery model that uses one of three possible kinds of nutrient utilization functions. In all cases the models suggest that two species can coexist under certain circumstances, but that three species cannot coexist, at least when the nutrient utilization functions are continuous functions of nutrient uptake. However, in the discontinuous case three species can coexist in a state of sustained oscillations. The results suggest that one need pay close attention to the differences in the nutrient utilization mechanisms among competing plant species in order to ascertain the competitive outcome.     

Time lags and global stability in two-species competition

Global asymptotic stability and equilibrium coexistence is established in two species Lotka-Volterra-type competition when there are time delays in interspecific interaction terms and the intraspecies competition is stronger than the interspecies competition.     

Conflict between the need to forage and the need to avoid competition: persistence of two-species model

We consider a model in which the need to forage and the need to avoid a competitor are in conflict. The model is composed of two Lotka-Volterra patches. The system has two competitors; one can diffuse between two patches, but the other is confined to one of the patches and cannot diffuse. It is proved that the system can be made persistent under appropriate diffusion conditions that ensure the instability of boundary equilibria, even if the competitive patch is not persistent without diffusion. Further it is shown that the system is globally stable for any diffusion rate if the competition between the two species is weak.     

Metabolic commensalism and competition in a two-species microbial consortium

We analyzed metabolic interactions and the importance of specific structural relationships in a benzyl alcohol-degrading microbial consortium comprising two species, Pseudomonas putida strain R1 and Acinetobacter strain C6, both of which are able to utilize benzyl alcohol as their sole carbon and energy source. The organisms were grown either as surface-attached organisms (biofilms) in flow chambers or as suspended cultures in chemostats. The numbers of CFU of P. putida R1 and Acinetobacter strain C6 were determined in chemostats and from the effluents of the flow chambers. When the two species were grown together in chemostats with limiting concentrations of benzyl alcohol, Acinetobacter strain C6 outnumbered P. putida R1 (500:1), whereas under similar growth conditions in biofilms, P. putida R1 was present in higher numbers than Acinetobacter strain C6 (5:1). In order to explain this difference, investigations of microbial activities and structural relationships were carried out in the biofilms. Insertion into P. putida R1 of a fusion between the growth rate-regulated rRNA promoter rrnBP1 and a gfp gene encoding an unstable variant of the green fluorescent protein made it possible to monitor the physiological activity of P. putida R1 cells at different positions in the biofilms. Combining this with fluorescent in situ hybridization and scanning confocal laser microscopy showed that the two organisms compete or display commensal interactions depending on their relative physical positioning in the biofilm. In the initial phase of biofilm development, the growth activity of P. putida R1 was shown to be higher near microcolonies of Acinetobacter strain C6. High-pressure liquid chromatography analysis showed that in the effluent of the Acinetobacter strain C6 monoculture biofilm the metabolic intermediate benzoate accumulated, whereas in the biculture biofilms this was not the case, suggesting that in these biofilms the excess benzoate produced by Acinetobacter strain C6 leaks into the surrounding environment, from where it is metabolized by P. putida R1. After a few days, Acinetobacter strain C6 colonies were overgrown by P. putida R1 cells and new structures developed, in which microcolonies of Acinetobacter strain C6 cells were established in the upper layer of the biofilm. In this way the two organisms developed structural relationships allowing Acinetobacter strain C6 to be close to the bulk liquid with high concentrations of benzyl alcohol and allowing P. putida R1 to benefit from the benzoate leaking from Acinetobacter strain C6. We conclude that in chemostats, where the organisms cannot establish in fixed positions, the two strains will compete for the primary carbon source, benzyl alcohol, which apparently gives Acinetobacter strain C6 a growth advantage, probably because it converts benzyl alcohol to benzoate with a higher yield per time unit than P. putida R1. In biofilms, however, the organisms establish structured, surface-attached consortia, in which heterogeneous ecological niches develop, and under these conditions competition for the primary carbon source is not the only determinant of biomass and population structure.     

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     

The dynamics of two-species allelopathic competition with optimal harvesting

This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare–Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.     

The dynamics of two-species allelopathic competition with optimal harvesting

This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare-Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.     

The maximum power principle predicts the outcomes of two-species competition experiments

The maximum power principle (MPP) states that biological systems organize to increase power whenever the system constraints allow. The MPP has the potential to explain a variety of ecological patterns because biological power (metabolism) is a component of all ecological interactions. I empirically tested the MPP by reanalyzing three two-species competition experiments by Gause, Vandermeer, and Fox and Morin. These experiments investigated competitive outcomes in microcosms of heterotrophic protists. I introduce metabolic state-space graphs to portray the metabolic trajectories of the communities and show that the steady-state outcomes of these experiments are consistent with the MPP. Winning species were successfully predicted a priori from their status as the species with the highest power when alone. In addition, periods of coexistence, although not predictable a priori, were consistent with the MPP because coexistence states had community-level power that was higher than either species could achieve alone. Thus, the outcomes of all ten trials were the maximum power states, given the options. The results suggest that the maximum power principle may represent a useful energetic organizing principle for communities.     

Extinction times and size of the surviving species in a two-species competition process

We investigate a stochastic model for the competition between two species. Based on percentiles of the maximum number of individuals in the ecosystem, we present an approximating model for which the extinction time can be thought of as a phase-type random variable. We determine formulae for the probabilities of extinction and the moments of the extinction time. We discuss the use of several quasi-stationary assumptions. We include a comparative study between existing asymptotic results, results obtained from a simulation of the process, and our solution.     

Stability analysis of a two-species model with transitions between population interactions

Stability of a simple two-species system is investigated. This model assumes that the kind of inter-specific interactions is not fixed, and that it depends on the system state, i.e., undergoes transitions between different population interactions due to variation in population densities. The main goal is to show the effects of the transitions between different population interactions on the two-species coexistence, and on the stability conditions of multiple equilibria.     

Spatial analysis of two-species interactions

Summary In this paper, I present and discuss some methods for the analysis of univariate and bivariate spatial point pattern data. Examples of such data in ecology include x-y coordinates of organisms in mapped field plots. I illustrate the methods with analyses of data from mapped field plots on Mount St. Helens, Washington state, USA. The statistical methods I emphasize are graphical methods that rely on analysis of distances between organisms. Hypothesis testing for methods like these is easily done using Monte Carlo methods, which I also discuss. For both univariate and bivariate analyses, I find that second-order methods such as K-function plots are often preferable to first-order methods (i.e., QQ-plots). However, for multivariate analyses, these second-order methods are more sensitive to small sample sizes than first-order analyses.     

Assessing determinants of community biomass composition in two-species plant competition studies

A method is proposed for assessing the relative importance of species identity, neighbour species influence and environment as determinants of change in community biomass composition in two-species short-term competition experiments. The method is based on modelling the differences in relative growth rates (RGR) of species (hence called the RGRD method). Using a multiple regression approach it quantifies the effects of initial species abundance, species identity and environment on RGRD and hence on change in community biomass composition. The RGRD approach is relatively simple to use and deals readily with statistical difficulties associated with correlated responses between species from the same stand. It can be easily adapted to analyse sequential harvest data. An example based on data from two-species mixtures of the annual species Stellaria media and Poa annua is used to illustrate the method. The main determinant of change in community biomass composition was species identity, reflected in the difference in growth rates between the species. Change in community composition was not, in general, significantly affected by the influence of neighbours or fertiliser level. The unimportance of the influence of neighbours in affecting the composition of these communities contrasts with the strong role of intra- and interspecific competition in determining the size of individuals of both species (Connolly et al. in Oecologia 82:513–526, 1990).