Discrete time models for two-species competition.

A discrete (difference) single age-class model for two-species competition is presented and its stability properties discussed. It resembles the Lotka-Volterra model in having linear zero growth isoclines, and thus, also in its general requirements for the coexistence of competing species. It differs in allowing the populations to show damped oscillations, stable cycles or even apparent “chaos” if competition is sufficiently severe. A similar two age-class model is discussed where there is both intra- and interspecific competition in one of the developmental stages, but only intraspecific competition in the other. Even this slight increase in complexity leads to markedly different properties. The zero growth curves become nonlinear and up to three equilibria between two competing species are possible.     

Analysis of linear determinacy for spread in cooperative models

 The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models. Received: 7 August 2000 / Revised version: 5 January 2002 / Published online: 17 July 2002     

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.     

Age-specific coexistence in two-species competition

Sufficient conditions are obtained for the existence and linear stability of stationary age distributions in a two-species competition model with age-dependent mortality and fertility functions.     

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.     

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.     

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.     

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.     

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.     

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.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

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.     

Diffusion analysis and stationary distribution of the two-species lottery competition model

The lottery model is a stochastic population model in which juveniles compete for space. Examples include sedentary organisms such as trees in a forest and members of marine benthic communities. The behavior of this model appears to be characteristic of that found in other sorts of stochastic competition models. In a community with two species, it was previously demonstrated that coexistence of the species is possible if adult death rates are small and environmental variation is large. Environmental variation is incorporated by assuming that the birth rates and death rates are random variables. Complicated conditions for coexistence and competitive exclusion have been derived elsewhere. In this paper, simple and easily interpreted conditions are found by using the technique of diffusion approximation. Formulae are given for the stationary distribution and means and variances of population fluctuations. The shape of the stationary distribution allows the stability of the coexistence to be evaluated.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

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.     

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).     

Spreading disease through social groupings in competition

Many animal populations live in social groups which avoid contact with other conspecific groups for at least part of the year. This may give rise to competition between groups for items such as shelter, land and mates. We couple intra-specific group competition with disease dynamics to investigate how infectious diseases may spread through population subgroups, particularly with reference to the contact rates between groups. Our model uses a nonlinear systems of ODEs for which steady-state analysis is carried out in the simplest two-group system. This indicates that coexistence of social groups is possible with the disease or that competitive exclusion occurs with one group dying out whilst the other retains disease. Moreover, we show that in certain circumstances the model can exhibit multistability and we discuss the ecological implications of this result in relation to contact between social groups.