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.     

A competition model for a seasonally fluctuating nutrient

A model of two species consuming a single, limited, periodically added resource is discussed. The model is based on chemostat-type equations, which differ from the classical models of Lotka and Volterra. The model incorporates nonlinear functional response curves of the Holling or Michaelis-Menten type to describe the dependence of the resource-exploitation rate on the amount of resource. Coexistence of two species due to seasonal variation is indicated by numerical studies.     

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.     

Biomass allocation in a clonal vine: Effects of intraspecific competition and nutrient availability

Biomass allocation to roots, rhizomes, runners and climbing stems (i.e. twining axis and attached leaves) was studied inCalystegia sepium L., a clonal vine. In an experiment which took 2 months, nutrient availability (low and high) and intraspecific competition (none, shoot root and both shoot and root) were manipulated. Under low nutrients the highest biomass of climbing stems was found in plants with shoot competition; the lowest was found in plants with both shoot and root competition. Total biomass under high nutrients was also greatest in plants with shoot competition. Thus, plants benefited from climbing up a shared stake rather than separate stakes. Larger plants allocated a higher proportion of biomass to runners in the nutrient-poor environment than in the nutrient-rich environment. This behaviour may increase the chance of finding nutrient-rich patches in the neighbourhood of the mother plant in a heterogeneous environment.     

A population model for limited food competition

In this paper a deterministic differential equation system is proposed to model the population dynamics of a biological community in which two species on the same trophic level compete for a common food, taken to be in limited supply. Food limitation is assumed to be the only inhibition of the growth of the populations and food quantity is assumed to be only affected by consumption. The model is thus designed to mimic a closed experimental situation rather than a natural community.Analytical properties of the solution of the differential equation system are developed and corresponding biological interpretations suggested.Cited laboratory data on the experimental batch community consisting of the marine ciliates Euplotes vannus and Uronema marinum feeding on bacteria motivated the model and supported its analytic properties.     

Sensitivity of the invasive geophyte Oxalis pes-caprae to nutrient availability and competition

BACKGROUND AND AIMS: Invasion by alien plants may be partially related to disturbance-related increases in nutrient availability and decreases of competition with native species, and to superior competitive ability of the invader. Oxalis pes-caprae is an invasive winter geophyte in the Mediterranean Islands that reproduces vegetatively via bulbs. An investigation was made into the relative responses of O. pes-caprae and the native annual grass Lolium rigidum to nutrient availability and to competition with each other in order to understand patterns of invasion in the field. Because Oxalis accumulates oxalic acid in its leaves, which could ameliorate soil phosphorous availability, field observations were made to determine whether the presence of Oxalis alters soil P availability. METHODS: A full-factorial glasshouse experiment was conducted with nutrient availability (high and low) and competition (Lolium alone, Oxalis alone, and Lolium and Oxalis together). Plant performance was assessed by determining (1) above- and below-ground biomass at the time of Oxalis maximum biomass and (2) reproductive output of Oxalis and Lolium at the end of their respective growth cycles. Measurements were also taken for leaf N and P content. Soil samples were taken in the field from paired Oxalis-invaded and non-invaded plots located in Menorca (Balearic Islands) and available P was determined. KEY RESULTS: High nutrient availability increased Oxalis and Lolium vegetative biomass and reproductive output to a similar degree. Competition with Lolium had a much stronger negative effect on Oxalis bulb production than reduced nutrients. Lolium was a superior competitor than Oxalis; the latter did not affect Lolium maximum biomass and spike production. Significantly greater soil-P availability in Oxalis-invaded field soils relative to paired non-invaded soils suggest that Oxalis influences soil P cycling. CONCLUSIONS: Oxalis is a poor competitor. This is consistent with the preferential distribution of Oxalis in disturbed areas such as ruderal habitats, and might explain its low influence on the cover of native species in invaded sites. The results also suggest that certain disturbances (e.g. autumn ploughing) may greatly enhance Oxalis invasion.     

The gradostat: A model of competition along a nutrient gradient

The general mathematical theory of the gradostat is presented for two competitors. The gradostat provides a mechanism for studying competition along a nutrient gradient. In the two vessel case, the results are complete and the conditions are testable. In then-vessel case, the relevant conditions are stated in terms of the stability modulii of certain matrices and are testable for any specific case.     

Analysis of a model for random competition

A random competition model is reformulated as an urn model and its behavior is analysed.     

Soil nutrient heterogeneity interacts with elevated CO2 and nutrient availability to determine species and assemblage responses in a model grassland community

Interactive effects of atmospheric CO(2) concentration ([CO(2)]), soil nutrient availability and soil nutrient spatial distribution on the structure and function of plant assemblages remain largely unexplored. Here we conducted a microcosm experiment to evaluate these interactions using a grassland assemblage formed by Lolium perenne, Plantago lanceolata, Trifolium repens, Anthoxanthum odoratum and Holcus lanatus. Assemblages exhibited precise root foraging patterns, had higher total and below-ground biomass, and captured more nitrogen when nutrients were supplied heterogeneously. Root foraging responses were modified by nutrient availability, and the patterns of N capture by interactions between nutrient distribution, availability and [CO(2)]. Greater above-ground biomass was observed under elevated CO(2) only under homogeneous conditions of nutrient supply and at the highest availability level. CO(2) interacted with nutrient distribution and availability to determine foliar percentage N and below : above-ground biomass ratios, respectively. Interactions between nutrient distribution and CO(2) determined the relative contribution to above-ground biomass of four of the species. The responses of dominant and subordinate species to [CO(2)] were dependent on the availability and distribution of nutrients. Our results suggest that soil nutrient distribution has the potential to influence the response of plant species and assemblages to changes in [CO(2)] and nutrient availability.     

Differential herbivory tolerance of dominant and subordinate plant species along gradients of nutrient availability and competition

We tested whether differences in the herbivory tolerance of plant species is related to their abundance in grassland communities and how herbivory and nutrient availability affect competitive balances among plant species through changes in their tolerance. The experimental approach involved a simulated grazing treatment (clipping) of two competitive grass species (Arrhenatherum elatius and Holcus lanatus) and two subordinate forb species (Prunella vulgaris and Lotus corniculatus) along a gradient of nutrient availability and under conditions of competition. Total standing, aboveground, root, and regrowth biomass were evaluated at the end of the experiment as an estimate of the capacity to compensate for twice removing aboveground biomass at different nutrient levels (NPK). Although clipping had a more pronounced negative effect on dominant plant species (Arrhenatherum and Holcus) than on subordinate species, the negative effects on dominant species were offset by the application of fertilizer. The combined effect of fertilizer and competition had more negative effects on the performance of Lotus and Prunella than on the dominant species. In terms of competition, the regrowth ability of Arrhenatherum and Holcus increased with the application of fertilizer, while the opposite pattern was observed for Lotus and Prunella. The addition of fertilizer has a positive effect on both grass species in terms of growth in clipped pots and competition, while subordinate species did not respond to the addition of fertilizer to the clipped pots and were negatively affected by competition with both grass species. The results suggest (1) that species replacement towards subordinate species as a function of herbivory is partially dependent on the herbivory tolerance of that species, (2) competitive relations between competitive grass species and subordinate forb species change under different environmental conditions, and (3) although grazing disturbance significantly influences competitive relations in favor of less competitive species, increasing nutrient levels counteract the negative effect of grazing on dominant competitive plant species.     

Catalyst-induced growth with limited catalyst lifespan and competition

Spatially extended catalyst-induced growth processes are studied. This type of processes exists in all domains of biology, ranging from ecology (nutrients and growth), through immunology (antigens and lymphocytes) to molecular biology (signaling molecules initiating signaling cascades). The extinction-proliferation transition is considered for a system containing discrete catalysts (A) that induces the proliferation of a discrete reactant (B). The realization of this model on an infinite capacity d-dimensional discrete lattice for immortal catalysts has been previously considered (the AB model). It was shown that the adaptation of the reactants to the diffusive noise induced by stochastic fluctuations of catalyst density yields proliferation even if the average environmental conditions lead to extinction. This model is extended here to include more realistic situations, like finite lifespan of the catalysts and finite carrying capacity of the reactants. By using a combination of Monte Carlo simulation, percolation-theory-based estimations and an analytic perturbative analysis, the asymptotic behavior of these systems is studied. In both cases studied, it turns out that the overall survival of the reactant population at the long run is based on the size and shape of a typical single colony, related to the localized proliferation around spatio-temporal catalyst density fluctuations. If the density of these colonies (based on the lifetime of the spatial fluctuation and the carrying capacity of the medium) is large enough, i.e. above the percolation threshold, the reactant population survives even in (on average) hostile environment. This model provides a new insight on the population dynamics in chemical, biological and ecological systems.     

Incomplete mixing promotes species coexistence in a lottery model with permanent spatial heterogeneity

For many marine organisms, the population dynamics in multiple habitats are affected by migration of planktonic larvae. We herein examine the effect of incomplete larval mixing on the condition for species coexistence. The system consists of two heterogeneous habitats, each composed of a number of sites occupied by adults of two species. Larvae produced in a habitat form a pool and migrate to the pool of the other habitat. When an adult dies, the vacant site becomes occupied by an individual randomly chosen from the larval pool. We study (1). the invasibility of a inferior species which has no advantage in either habitats, (2). the dynamics when larval migration and competition among adults are symmetric between habitats, and (3). the case with unidirectional migration. The coexistence of competitors is more likely to occur when larval migration is weak.     

Species coexistence by permanent spatial heterogeneity in a lottery model

We studied the effect of permanent spatial heterogeneity in promoting species coexistence in a lottery model. The system consisted of multiple habitats, each composed of a number of sites occupied by adults of two species. Larvae produced from different habitats were mixed in a common pool. When an adult died, the vacant site became occupied by an individual randomly chosen from the larval pool. If there were n habitats, there could be up to n-1 internal equilibria with both species in addition to two single-species equilibria. These equilibria and their local stability can be calculated from a single function, indicating the difference among species in their average lifetime reproductive success. Our main result is that between-habitat variation in the ratio of mortalities of two species promotes coexistence, while that of reproductive rates does not. This conclusion is the opposite of the role of temporal variation in the standard lottery model, in which between-year variation in the reproductive rate, but not that in the mortalities, promotes coexistence.     

On a resource based ecological competition model with interference

A resource based ecological competition model with interference is proposed. The model is based on Lotka-Volterra dynamics with two predators competing for a single, limited prey. Interference effects are considered in this article. When the interference coefficient, expressing the damage effect from its rival, is small, the mathematical analysis shows that the winner in purely exploitative competition still outcompetes its rival. However, if the interference coefficient is large enough then the competition outcome will depend on initial population of predator species.     

Predation, competition, and nutrient recycling: a stoichiometric approach with multiple nutrients

A model for two competing prey species and one predator is formulated in which three essential nutrients can limit growth of all populations. Prey take up dissolved nutrients and predators ingest prey, assimilating a portion of ingested nutrients and recycling or respiring the balance. For all species, the nutrient contents of individuals vary and growth is coupled to increasing content of the limiting nutrient. This model was parameterized to describe a flagellate preying on two bacterial species, with carbon (C), nitrogen (N), and phosphorus (P) as nutrients. Parameters were chosen so that the two prey species would stably coexist without predators under some nutrient supply conditions. Using numerical simulations, the long-term outcomes of competition and predation were explored for a gradient of N:P supply ratios, varying C supply, and varying preference of the predator for the two prey. Coexistence and competitive exclusion both occurred under some conditions of nutrient supply and predator preference. As in simpler models of competition and predation these outcomes were largely governed by apparent competition mediated by the predator, and resource competition for nutrients whose effective supply was partly governed by nutrient recycling also mediated by the predator. For relatively small regions of parameter space, more complex outcomes with multiple attractors or three-species limit cycles occurred. The multiple constraints posed by multiple nutrients held the amplitudes of these cycles in check, limiting the influence of complex dynamics on competitive outcomes for the parameter ranges explored.     

Assessment of nutrient availability and limitation using macroalgae

Discs of the macroalga,Ulva lactuca L., were transplanted around an ocean outfall and at a reference site in Køge Bay, Denmark, to assess the influence of the outfall on the nutrient availability. At 2-wk intervals, samples were collected and analyzed for growth, nitrogen, and phosphorus content.The tissue concentrations of nitrogen and phosphorus decreased with distance to the outfall, showing that the tissue concentrations are suitable for monitoring nutrient availability in coastal areas and provide a time-integrated measure of the nutrient availability. The lowest tissue concentrations of nitrogen were recorded at the reference station, where the internal concentrations generally were below the critical concentration level, showing that nitrogen limited the growth. At the station located close to the outfall, the flux of nitrogen was sufficient to maintain the maximum growth rate. The tissue concentrations of phosphorus were only below the critical concentration level on one occasion, and the result showed a net uptake throughout the study period.It was concluded that in the Køge Bay, nitrogen was the main limiting factor for macroalgae growth during the summer. The applicability of tissue concentrations for assessment of nutrient availability is discussed and it is considered that the method, when evaluated against established critical concentrations, provides a valuable tool for assessing ecosystem health with regard to eutrophication.     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

Grazer control of nutrient availability in the periphyton

Summary Benthic algal assemblages are regulated by both abiotic (e.g., nutrient) and biotic (e.g., grazing) constraint. The objective of this study was to determine how changes in these two factors affected the structure of an algal assemblage in an ephemeral stream. Coverslips were incubated for 21 days in enclosures containing one of three nutrient environments (ambient, phosphorus-enriched, or phosphorus and nitrogen enriched) and one of four densities of the snail Gonibasis (0, 40, 80, or 120 snails/m²) and examined directly to enumerate the algal assemblage. The effect of grazing on algal biomass was dependent on the nutrient environment. An overstory of diatoms was susceptible to removal by grazing and was not strongly affected by nutrient enrichment. An understory of Stigeoclonium was more resistant to grazing and responded strongly to nutrient enrichment only in the presence of grazers. Snail grazers may mediate nutrient availability to the understory indirectly by removing overlying cells or by direct excretion of nutrients. Multiple interactions occur between benthic herbivores and algae, and, as shown here, some of them are positive and involve modifications of the nutrient environment.     

Obesity by choice: the powerful influence of nutrient availability on nutrient intake

The consumption of nutrients by rodents is markedly influenced by the number of containers of each nutrient provided. Most rats given a choice from separate sources of protein, carbohydrate, and fat thrived if given one cup of each but half failed to thrive if given one cup of each and three extra cups of carbohydrate or fat. Rats given five bottles of sucrose solution and one bottle of water became fatter than rats given five bottles of water and one of sucrose. These studies in rats may point to a model for human obesity, in which the availability of food can override physiological controls of ingestion.