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.     

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.     

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.     

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.     

A discrete model of avian influenza with seasonal reproduction and transmission

In this paper, we formulate a discrete-time model with the reproductive and overwintering periods to assess the impact of avian influenza transmission in poultry. It is shown that the disease is extinct if the basic reproduction number is less than one and is persistent if the basic reproductive number is greater than one. Furthermore, the model admits a closed invariant cycle, which means that avian influenza fluctuates in poultry.     

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

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.     

Size-dependent allocation to sexual and vegetative reproduction in four clonal composites

Summary Populations of Silphium speciosum, Vernonia baldwinii, Solidago canadensis and Pityopsis graminifolia were studied to determine whether biomass allocation to sexual and vegetative reproduction and the balance between them were size-dependent and whether interpopulation differences in allocation patterns could be predicted from differences in population size distributions. All four species showed strong linear relationships between inflorescence mass and vegetative mass with negative y-intercepts. As a result, sexual reproductive effort (SRE) was a monotonically increasing function of ramet size. Genet size was a poor predictor of SRE. In each species, the regression parameters of these relationships differed significantly between burned and unburned habitats indicating size-independent interpopulation differences in patterns of reproductive effort as well as sizedependent effects. Interpopulation variation in vegetative reproductive effort (VRE) was greater than variation in SRE, but neither VRE nor the pattern of partitioning of VRE among daughter rhizomes showed significant relationships to plant size.     

Extremely reduced sexual reproduction in the clonal cactus Echinopsis thelegona

Sexual and asexual reproduction may occur simultaneously in some plant species. Under certain environmental conditions asexual reproduction is predominant, which generates ecological consequences on sexual fecundity. In this context, we studied the reproductive ecology of the creeping clonal cactus Echinopsis thelegona in La Bodega (Salta, Argentina), where low fruit and seed production was preliminarily observed. Specifically, we studied the flower availability, fruit and seed production, reproductive system, floral visitors and effects of neighbor ramets on reproductive success. The number of available flowers per day was low, and fruit production was low or null as well. Echinopsis thelegona is self-incompatible. Although its flowers have sphingophilous traits, we did not find evidence of moths visiting them. Only native and exotic bees visited the flowers of this species, though with a low number of visits. Therefore, nocturnal visitors were not registered as pollinators of E. thelegona, perhaps because of their low local abundances. On the other hand, bees behaved as inefficient pollinators of E. thelegona due to their foraging behavior, which moves pollen within individuals. A pollen-addition experiment revealed that there is no fruit production at short distances among ramets. Therefore, the combination of self-incompatibility, low flower production and low local abundance of pollinators could account for the low fruit production observed in this species. This finding suggests that populations of E. thelegona persist by clonality and that they tend toward the loss of sexuality.     

Simulating the evolution of a clonal trait in plants with sexual and vegetative reproduction

Aims Phenotypic optimality models neglect genetics. However, especially when heterozygous genotypes are fittest, evolving allele, genotype and phenotype frequencies may not correspond to predicted optima. This was not previously addressed for organisms with complex life histories.Methods Therefore, we modelled the evolution of a fitness-relevant trait of clonal plants, stolon internode length. We explored the likely case of an asymmetric unimodal fitness profile with three model types. In constant selection models (CSMs), which are gametic, but not spatially explicit, evolving allele frequencies in the one-locus and five-loci cases did not correspond to optimum stolon internode length predicted by the spatially explicit, but not gametic, phenotypic model. This deviation was due to the asymmetry of the fitness profile. Gametic, spatially explicit individual-based (SEIB) modeling allowed us relaxing the CSM assumptions of constant selection with exclusively sexual reproduction.Important findings For entirely vegetative or sexual reproduction, predictions of the gametic SEIB model were close to the ones of spatially explicit non-gametic phenotypic models, but for mixed modes of reproduction they approximated those of gametic, not spatially explicit CSMs. Thus, in contrast to gametic SEIB models, phenotypic models and, especially for few loci, also CSMs can be very misleading. We conclude that the evolution of traits governed by few quantitative trait loci appears hardly predictable by simple models, that genetic algorithms aiming at technical optimization may actually miss the optimum and that selection may lead to loci with smaller effects in derived compared with ancestral lines.     

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.     

Shifting from clonal to sexual reproduction in aphids: physiological and developmental aspects

Developmental biology is one of the fastest growing and fascinating research fields in life sciences. Among the wide range of embryonic development, a fundamental difference exists between organisms with sexual or asexual development. Aphids are unusual organisms which display alternative pathways of sexual and asexual development, the orientation of the pathway being determined by environmental conditions. These insects offer an adapted system in which to study developmental plasticity, because a side-by-side comparison of sexual and asexual development can be made in individuals with the same genotype. In this review, we describe the developmental mechanisms that have evolved in aphids for alternative sexual and asexual reproduction. In particular, we discuss how environmental cues orientate the reproductive mode of aphids from signal perception to endocrine regulation, and propose a comparative analysis of sexual and asexual gametogenesis and embryogenesis, which has been possible due to the development of molecular methods. As a result of the recent development of genomic resources in aphids, we expect these species will permit major advances in the study of the genomic basis underlying the choice of developmental fate and multiple reproduction strategies.     

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.     

Multiple attractors in a discrete competition model

The existence of multiple attractors in a competition model implies that the question of coexistence vs. extinction can depend on initial conditions. A discrete stage-structured model of two competing species is derived from a well-tested single-species model of insect populations, and is shown to exhibit multiple attractors for parameter values similar to those used in laboratory experiments which demonstrated chaos in population dynamics. The corresponding basins of attraction are investigated and shown to have very complex structures, and the initial stage structure of the populations is shown to have a significant impact on final outcomes.     

Resource competition and patterns of sexual reproduction in sympatric sibling rotifer species

We present experimental data on the reproduction patterns of three sympatric sibling rotifer species, belonging to the Brachionus plicatilis complex, as they compete for food. Variation existed in the amount and pattern of sexual reproduction among the three cyclical parthenogenetic species. Competitive exclusion between the three Brachionus species was related to a higher investment in sexual reproduction by the inferior competitor. In general, no correlation was found between sexual reproduction and population density in a given species. However, when pairs of competing species were compared, a negative relationship between sexual investment differences (between species) and average population density differences was frequently found. From the results we conclude that: (1) the characteristic sexual pattern of each species has some implications for the competitive outcome; (2) allocation of resources into sexual reproduction does not mediate coexistence; and (3) the response of sexual reproduction to competitive interaction is unlikely to determine competition outcome. Nevertheless, our results suggest that the competitively inferior species invests more in sexual reproduction, as a response to resource limitation, which would accelerate its exclusion.     

Rare sexual reproduction events in the clonal reproduction system of introduced populations of the little fire ant

A unique reproductive system has previously been described in Wasmannia auropunctata, a widespread invasive ant species, where males are produced clonally, female queens are parthenogens, and female workers are produced sexually. However, these findings were mostly based on samples originating from only a limited part of the native range of the species in South America. We used microsatellite markers to uncover the reproductive modes displayed by a large number of nests collected in various invasive W. auropunctata populations introduced 40 years ago into New Caledonia, where the species now forms a single 450-km-long supercolony. Although the main reproduction system in New Caledonia remained clonality for both male and female reproductives, we found evidence of rare sexual reproduction events that led to the production of both new queen and male clonal lineages. All clonal lineages observed in New Caledonia potentially derived from sexual reproduction, recombination, and mutation events from a single female and a single male genotype. Hence, the male and female gene pools are not strictly separated in New Caledonia and the two sexes do not follow independent evolutionary trajectories. Our results also suggest genetic determination for both parthenogenesis and caste. We discuss the evolutionary implications of the emergence of sex in the clonal reproduction system of introduced populations of W. auropunctata.     

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     

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.