Study of a tri-trophic prey-dependent food chain model of interacting populations

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf–Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.     

Cost of interacting with sexual partners in a facultative sexual microbe

The widespread occurrence of sexual organisms despite the high costs of sex has long intrigued biologists. The best-known costs are the twofold cost of producing males and the cost associated with producing traits to attract mates and to interact with mating partners, such as exaggerated sexual behaviors and morphological modifications. These costs have been inferred from studies of plants and animals but are thought to be absent in facultative sexual microbes. Here, using the facultative sexual fungus Cryptococcus neoformans, I provide experimental evidence showing that: (i) interactions with active sexual partners can be costly for vegetative fitness in a facultative sexual microbe; (ii) this cost is positively correlated to mating ability; (iii) this cost is composed of at least two distinct components, the cost of producing mating signals that exert effects on mating partners and that associated with responding to active mating partners; and (iv) extended asexual reproduction can reduce both components of the cost. This cost must have been compensated for by the production of zygotes and sexual spores to allow the initial evolution and spread of sexual reproduction in eukaryotes.     

Bi-trophic food chain dynamics with multiple component populations

Food web models describe the patterns of material and energy flow in communities. In classical food web models the state of each population is described by a single variable which represents, for instance, the biomass or the number of individuals that make up the population. However, in a number of models proposed recently in the literature the individual organisms consist of two components. In addition to the structural component there is an internal pool of nutrients, lipids or reserves. Consequently the population model for each trophic level is described by two state variables instead of one. As a result the classical predator-prey interaction formalisms have to be revised. In our model time budgets with actions as searching and handling provide the formulation of the functional response for both components. In the model, assimilation of the ingested two prey components is done in parallel and the extracted energy is added to a predators reserve pool. The reserves are used for vital processes; growth, reproduction and maintenance. We will explore the top-down modelling approach where the perspective is from the community. We will demonstrate that this approach facilitates a check on the balance equations for mass and energy at this level of organization. Here it will be shown that, if the individual is allowed to shrink when the energy reserves are in short to pay the maintenance costs, the growth process has to be 100% effective. This is unrealistic and some alternative model formulations are discussed. The long-term dynamics of a microbial food chain in the chemostat are studied using bifurcation analysis. The dilution rate and the concentration of nutrients in the reservoir are the bifurcation parameters. The studied microbial bi-trophic food chain with two-component populations shows chaotic behaviour.     

A selection mosaic in the facultative mutualism between ants and wild cotton

In protection mutualisms, one mutualist defends its partner against a natural enemy in exchange for a reward, usually food or shelter. For both partners, the costs and benefits of these interactions often vary considerably in space because the outcome (positive, negative or neutral) depends on the local abundance of at least three species: the protector, the beneficiary of protection and the beneficiary's natural enemy. In Gossypium thurberi (wild cotton), ants benefit nutritionally from the plant's extrafloral nectaries and guard plants from herbivores. Experimentally altering the availability of both ants and extrafloral nectar in three populations demonstrated that the mutualism is facultative, depending, in part, on the abundance of ants and the level of herbivore damage. The species composition of ants and a parasitic alga that clogs extrafloral nectaries were also implicated in altering the outcome of plant-ant interactions. Furthermore, experimental treatments that excluded ants (the putative selective agents) in combination with phenotypic selection analyses revealed that selection on extrafloral nectary traits was mediated by ants and, importantly, varied across populations. This work is some of the first to manipulate interactions experimentally across multiple sites and thereby document that geographically variable selection, mediated by a mutualist, can shape the evolution of plant traits.     

Oscillatory coexistence in a food chain model with competing predators

A food chain model with two predators feeding on a single prey in a chemostat is studied. Using a multiparameter bifurcation analysis, we find parameters values for which there is stable oscillatory coexistence of the predators. It is also shown how these coexistent states provide a transition between two possible states of competitive exclusion. It is shown that the competitive exclusion principle need not hold if one or more of the predators has oscillatory behavior in the absence of other predators.This work was partially supported by National Science Foundation Grant MCS 83-01881     

Simple mathematical models for interacting wild and transgenic mosquito populations

Two discrete-time models for interacting populations of wild and genetically altered mosquito are presented, where the genetically altered mosquitoes are grouped into a single population without distinguishing their zygosity. The birth and death rates for both populations are density-dependent, and the mating rates between the mosquitoes are assumed to be either constant or proportional to the total populations for the two models, respectively. The existence and stability of the boundary and positive equilibria are investigated. In particular, it is shown that bifurcations from both boundary and positive equilibria can appear for the model with proportional mating rates. Stable equilibria, periodic-doubling bifurcations, aperiodic oscillations, and chaotic behavior are all illustrated by numerical simulations.     

A mathematical model for the growth of mycelial pellet populations

In liquid culture, filamentous organisms often grow in the form of pellets. Growth result in an increase in radius, whereas shear forces result in release of hyphal fragments which act as centers for further pellet growth and development. A previously published model for pellet growth of filamentous microorganisms has been examined and is found to be unstable for certain parameter values. This instability has been identified as being due to inaccuracies in estimating the numbers of fragments which seed the pellet population. A revised model has been formulated, based on similar premises, but adopting a finite element approach. This considers the population of pellets to be distributed in a range of size classes. Growth results in movement to classes of increasing pellet size, while fragments enter the smallest size class, from which they grow to form further pellets. The revised model is stable and predicts changes in the distribution of pellet sizes within a population growing in liquid batch culture. It considers pellet growth and death, with fragmentation providing new centers of growth within the pellet population, and predicts the effects of shear forces on pellet growth and size distribution. Predictions of pellet size distributions are tested using previously published data on the growth of fungal pellets and further predictions are generated which are suitable for experimental testing using cultures of filamentous fungi or actinomycetes. (c) 1995 John Wiley & Sons, Inc.     

A mathematical model for plasmid replication and distribution in microbial populations

A mathematical model and a computer program for its implementation have been developed to predict the distribution of plasmid copy numbers in the individual cells of a microbial population. The kinetics of accumulation of plasmid-free cells. the copy number distribution within the population and the mean copy number can all be calculated using the computer program. The model has been shown to accurately predict these parameters for recombinant plasmids in yeast populations.     

Geographic variation in a facultative mutualism: consequences for local arthropod composition and diversity

Geographic variation in the outcome of interspecific interactions may influence not only the evolutionary trajectories of species but also the structure of local communities. We investigated this community consequence of geographic variation for a facultative mutualism between ants and wild cotton (Gossypium thurberi). Ants consume wild cotton extrafloral nectar and can protect plants from herbivores. We chose three sites that differed in interaction outcome, including a mutualism (ants provided the greatest benefits to plant fitness and responded to manipulations of extrafloral nectar), a potential commensalism (ants increased plant fitness but were unresponsive to extrafloral nectar), and a neutral interaction (ants neither affected plant fitness nor responded to extrafloral nectar). At all sites, we manipulated ants and extrafloral nectar in a factorial design and monitored the abundance, diversity, and composition of other arthropods occurring on wild cotton plants. We predicted that the effects of ants and extrafloral nectar on arthropods would be largest in the location with the mutualism and weakest where the interaction was neutral. A non-metric multidimensional scaling analysis revealed that the presence of ants altered arthropod composition, but only at the two sites in which ants increased plant fitness. At the site with the mutualism, ants also suppressed detritivore/scavenger abundance and increased aphids. The presence of extrafloral nectar increased arthropod abundance where mutual benefits were the strongest, whereas both arthropod abundance and morphospecies richness declined with extrafloral nectar availability at the site with the weakest ant–plant interaction. Some responses were geographically invariable: total arthropod richness and evenness declined by approximately 20% on plants with ants, and extrafloral nectar reduced carnivore abundance when ants were excluded from plants. These results demonstrate that a facultative ant–plant mutualism can alter the composition of arthropod assemblages on plants and that these community-level consequences vary across the landscape.     

Reduced recombination in gynodioecious populations of a facultative apomictic orchid

Many plants combine sexual reproduction with some form of asexual reproduction to different degrees, and lower genetic diversity is expected with asexuality. Moreover, the ratios of sexual morphs in species with gender dimorphism are expected to vary in proportion to the reproductive success of the sexual process. Hence, sex ratios can directly influence the genetic structure and diversity of a population. We investigated genotypic diversity in 23 populations of a facultative, apomictic gynodioecious orchid, Satyrium ciliatum, to examine the effect on genotypic diversity of variation in the frequency of females and in the amount of sexual reproduction. The study involved one pure female, seven gynodioecious (both females and hermaphrodites present) and 15 hermaphroditic populations. Pollinia receipt was higher in hermaphroditic than in gynodioecious populations. Analyses of variation in ISSRs demonstrated that genotypic diversity was high in all populations and was not significantly different between hermaphroditic and gynodioecious populations. We used character compatibility analysis to determine the extent to which recombination by sexual reproduction contributed to genotypic diversity. The results indicate that the contribution of recombination to genotypic diversity is higher in hermaphroditic than in gynodioecious populations, consistent with the finding that hermaphroditic populations received higher amounts of pollinia. Our finding of reduced recombination in gynodioecious populations suggests that maintenance of sex in hermaphrodites plays an important role in generating genotypic diversity in this apomictic orchid.     

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.     

Return time and vulnerability for a food chain model

Simulation studies have shown that the time it takes for a system of interacting species in a food chain to return to equilibrium after a disturbance increases as the number of trophic levels increase. It has been argued that this effect is important in limiting the length of food chains subject to perturbations of the real world. We show that for an asymptotically stable system a lower bound on the return time is directly proportional to the number of trophic levels in agreement with simulation studies. In addition, the lower bound on the return time is shown to be inversely proportional to the sum of products of the intraspecific competition coefficient and equilibrium population of the species. A new method for directly computing the vulnerability of a system to external perturbations is presented. Using this method we demonstrate that for a food chain where the number of species is equal to the number of trophic levels, the return time alone is not a proper measure of system vulnerability. Indeed, adding an additional trophic level may make the system less vulnerable to disturbances. Interspecific coupling between the trophic levels is shown to be an important factor in determining system vulnerability.     

A mathematical model with young predation

We generalise the model of [21] in which the author considered a predator-prey system with predators eating only the young ones (or eggs) of the prey species. The prime assumption of the present paper is that the birth rate (per unit individual per unit time) of predators depends not only on the current prey egg-level but also on all previous prey egg-levels. It is seen that under this assumption an otherwise stable system may be stable as well as unstable leading to the conclusion that young predation with time delay is less stable than without it. Finally for the model of [21] we prove a result which shows that large predation rates help in the co-existence of both predator and prey species.     

A model for the loss of memory potential assuming a requirement for two interacting cell populations with different functional half-lives

The biphasic exponential loss of memory potential in adoptively immunized rats cannot be explained by assuming that secondary antibody formation under these conditions requires the interaction of two different cell populations of different functional half-lives.     

A mathematical model to design a lignocellulosic biofuel supply chain system with a case study based on a region in Central Texas

This study formulates a model to maximize the profit of a lignocellulosic biofuel supply chain ranging from feedstock suppliers to biofuel customers. The model deals with a time-staged, multi-commodity, production/distribution system, prescribing facility locations and capacities, technologies, and material flows. A case study based on a region in Central Texas demonstrates application of the proposed model to design the most profitable biofuel supply chain under each of several scenarios. A sensitivity analysis identifies that ethanol (ETOH) price is the most significant factor in the economic viability of a lignocellulosic biofuel supply chain.     

A mathematical model of the food-consumer system I. A case without food replenishment

A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, and c from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance as x and new plant body as y, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

Coexistence in a simple food chain with diffusion

We show the global existence of classical positive solutions in each component of a Lotka-Volterra system with diffusion and logistic growing conditions. We are mainly interested in the search of coexistence states solving the associated elliptic problem under homogeneous Dirichlet boundary conditions.     

A prototype model structure for mixed microbial populations in homogeneous food products

An important factor which has not been included in many models in the field of predictive microbiology is the influence of a background of microflora in a food product. It is however generally known that the growth of a microorganism as a pure culture can be substantially different from its growth in a mixed culture, due to microbial interactions. Because of the importance of these interactions and the lack of suitable modeling techniques in the field of predictive microbiology to describe them, the potential of models in other research fields-namely ecology-to deal with interactions is explored in previous work of the authors. However, a model structure for microbial growth in food products cannot simply be copied from those elaborated in ecology. The structure of a predictive growth model is indeed typical, primarily due to the explicit modeling of a lag phase. The current paper proposes a prototype model structure for growth of mixed microbial populations in homogeneous food products. The model is able to describe a lag phase and reduces to a classical predictive growth model in the special case of single-species growth.