Models of competition in the chemostat with instantaneous and delayed nutrient recycling

Two models for competition of two populations in a chemostat environment with nutrient recycling are considered. In the first model, the recycling is instantaneous, whereas in the second, the recycling is delayed. For each model an equilibrium analysis is carried out, and persistence criteria are obtained. This paper extends the work done by Beretta et al. (1990) for a single species.Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC A4823Research carried out at the University of Alberta while on a Canada-China Scholarly Exchange Program     

The population biology of bacterial plasmids: a hidden Markov model approach

Horizontal plasmid transfer plays a key role in bacterial adaptation. In harsh environments, bacterial populations adapt by sampling genetic material from a horizontal gene pool through self-transmissible plasmids, and that allows persistence of these mobile genetic elements. In the absence of selection for plasmid-encoded traits it is not well understood if and how plasmids persist in bacterial communities. Here we present three models of the dynamics of plasmid persistence in the absence of selection. The models consider plasmid loss (segregation), plasmid cost, conjugative plasmid transfer, and observation error. Also, we present a stochastic model in which the relative fitness of the plasmid-free cells was modeled as a random variable affected by an environmental process using a hidden Markov model (HMM). Extensive simulations showed that the estimates from the proposed model are nearly unbiased. Likelihood-ratio tests showed that the dynamics of plasmid persistence are strongly dependent on the host type. Accounting for stochasticity was necessary to explain four of seven time-series data sets, thus confirming that plasmid persistence needs to be understood as a stochastic process. This work can be viewed as a conceptual starting point under which new plasmid persistence hypotheses can be tested.     

Interactions leading to persistence in predator-prey systems with group defence

In previous work (Freedman and Wolkowicz, 1986;Bull. math. Biol. 48, 493–508) it was shown that in a predator-prey system where the prey population exhibits group defence, it is possible that enrichment of the environment could lead to extinction of the predator population. In this paper a third population is introduced and criteria are derived under which persistence of all populations will occur. In particular, criteria for a superpredator and for a competitor to stabilize the system in the sense of persistence are analyzed. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. This research was carried out while the author was a visiting scholar at the University of Alberta.     

Persistence despite perturbations for interacting populations

Two definitions of persistence despite perturbations in deterministic models are presented. The first definition, persistence despite frequent small perturbations, is shown to be equivalent to the existence of a positive attractor i.e. an attractor bounded away from extinction. The second definition, persistence despite rare large perturbations, is shown to be equivalent to permanence i.e. a positive attractor whose basin of attraction includes all positive states. Both definitions set up a natural dichotomy for classifying models of interacting populations. Namely, a model is either persistent despite perturbations or not. When it is not persistent, it follows that all initial conditions are prone to extinction due to perturbations of the appropriate type. For frequent small perturbations, this method of classification is shown to be generically robust: there is a dense set of models for which persistent (respectively, extinction prone) models lies within an open set of persistent (resp. extinction prone) models. For rare large perturbations, this method of classification is shown not to be generically robust. Namely, work of Josef Hofbauer and the author have shown there are open sets of ecological models containing a dense sets of permanent models and a dense set of extinction prone models. The merits and drawbacks of these different definitions are discussed.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

Structural sensitivity of grazing formulations in nutrient controlled plankton models

Summary Stability and persistence properties of a family of non-spatial plankton models, each differentiated by its herbivore grazing term, are analytically compared. The dynamic persistence function in the model is shown to operate uniformly even though stability configuration characteristics of the model may be topologically distinct. The persistence threshold for each model indicates that total nutrient is a fundamental biological control. In the parameter space, all of the models studied are structurally unstable; however, an important bifurcation mechanism associated with this instability governs persistence. While, topologically, model transfigurement through parameter modulation is non-continuous, the biological populations evolve in a continuous or a lower semicontinuous manner. A basic conclusion of the paper is that fundamental problems for these marine ecological models remain unresolved since each of the models is a structurally unstable system for a fixed dynamically persistent ecology.     

A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria

When the growth of bacteria in a chemostat is controlled by limiting the supply of a single essential nutrient, the growth rate is affected both by the concentration of this nutrient in the culture medium and by the amount of time that it takes for the chemical and physiological processes that result in the production of new biomass. Thus, although the uptake of nutrient by cells is an essentially instantaneous process, the addition of new biomass is delayed by the amount of time that it takes to metabolize the nutrient. Mathematical models that incorporate this "delayed growth response" (DGR) phenomenon have been developed and analysed. However, because they are formulated in terms of parameters that are difficult to measure directly, these models are of limited value to experimentalists. In this paper, we introduce a DGR model that is formulated in terms of measurable parameters. In addition, we provide for this model a complete set of criteria for determining persistence versus extinction of the bacterial culture in the chemostat. Specifically, we show that DGR plays a role in determining persistence versus extinction only under certain ranges of chemostat operating parameters. It is also shown, however, that DGR plays a role in determining the steady-state nutrient and bacteria concentrations in all instances of persistence. The steady state and transient behavior of solutions of our model is found to be in agreement with data that we obtained in growing Escherichia coli 23716 in a chemostat with glucose as a limiting nutrient. One of the theoretical predictions of our model that does not occur in other DGR models is that under certain conditions a large delay in growth response might actually have a positive effect on the bacteria's ability to persist.     

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.     

The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

Defining the requirements for immunological control of mycobacterial infections.

Tuberculosis remains a major public health problem throughout the world. Research into the immunological basis of the host-pathogen relationship has recently benefited from the fascinating convergence of genetic data from mouse models and from humans. Latency - the seemingly quiescent phase of bacterial persistence - is the central problem in controlling tuberculosis and will be the next frontier of research.     

Stability properties of 2-species models of mutualism: Simulation studies

Summary Most previous analyses of the stability properties of models of mutualism have emphasized the destabilizing effects of mutualism. However, these analyses can be shown to be based upon inappropriate assumptions, or to be applicable only for special cases of mutualism. In this paper three basic 2-species models of mutualism are presented and their six combinations are analyzed by computer simulation for their return time stability and persistence stability. Four out of six models show greater return time stability than an appropriate model without mutualism, and all models show higher persistence stability than the model without mutualism. It is argued that real biological systems can be related to the qualitative structure of each of the basic models of mutualism, and that therefore none of the basic models or their stability properties can be eliminated a priori as being inappropriate. The conclusion follows that while some kinds of mutualistic interactions may be relatively unstable, other mutualisms, probably representing the majority of cases, can be considered to be relatively stable. The limitations of these models and analyses are considered.     

Pathogen persistence in the environment and insect-baculovirus interactions: disease-density thresholds, epidemic burnout, and insect outbreaks

Classical epidemic theory focuses on directly transmitted pathogens, but many pathogens are instead transmitted when hosts encounter infectious particles. Theory has shown that for such diseases pathogen persistence time in the environment can strongly affect disease dynamics, but estimates of persistence time, and consequently tests of the theory, are extremely rare. We consider the consequences of persistence time for the dynamics of the gypsy moth baculovirus, a pathogen transmitted when larvae consume foliage contaminated with particles released from infectious cadavers. Using field-transmission experiments, we are able to estimate persistence time under natural conditions, and inserting our estimates into a standard epidemic model suggests that epidemics are often terminated by a combination of pupation and burnout rather than by burnout alone, as predicted by theory. Extending our models to allow for multiple generations, and including environmental transmission over the winter, suggests that the virus may survive over the long term even in the absence of complex persistence mechanisms, such as environmental reservoirs or covert infections. Our work suggests that estimates of persistence times can lead to a deeper understanding of environmentally transmitted pathogens and illustrates the usefulness of experiments that are closely tied to mathematical models.     

Individual behavior at habitat edges may help populations persist in moving habitats

Moving-habitat models aim to characterize conditions for population persistence under climate-change scenarios. Existing models do not incorporate individual-level movement behavior near habitat edges. These small-scale details have recently been shown to be crucially important for large-scale predictions of population spread and persistence in patchy landscapes. In this work, we extend previous moving-habitat models by including individual movement behavior. Our analysis shows that populations might be able to persist under faster climate change than previous models predicted. We also find that movement behavior at the trailing edge of the climatic niche is much more important for population persistence than at the leading edge.

     

Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling

We consider plankton-nutrient interaction models consisting of phytoplankton, herbivorous zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. For the model with constant nutrient input and different constant washout rates, conditions for boundedness of the solutions, existence and stability of non-negative equilibria, as well as persistence are given. We also consider the zooplankton-phytoplankton-nutrient interaction models with a fluctuating nutrient input and with a periodic washout rate, respectively. It is shown that coexistence of the zooplankton and phytoplankton may arise due to positive bifurcating periodic solutions.Research has been supported in part by a University of Alberta Ph.D. Scholarship and is in part based on the author's Ph.D. thesis under the supervision of Professor H. 1. Freedman, to whom the author owes a debt of appreciation and gratitude for his kind advice, helpful comments and continuous encouragement     

Modeling persistence and mutual interference among subpopulations of ecological communities

A model is proposed of an ecological community where some (or all) of the subpopulations exhibit mutual interference. Mutual interference introduces sublinearities which makes the persistence analysis of the community more complex since the model is no longer a dynamical system. A transformation is introduced which yields a dynamical system, thereby making a persistence analysis more tractable. The results are applied to determining top-predator persistence of a simple food chain and to the question of invasibility of a stable community by a new subpopulation. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Dynamical Adaptation of Parental Care

Two models are proposed to simulate population growth of species with mature stage and immature stage in which there are parental cares for immature. It is assumed that the protection of mature to their immature reduces mortality of immature at the cost of reduction of reproduction. Dynamical adaptation of parental care is incorporated into the models, one of which is described with the proportional transition rate from immature to mature (ODE model) and the other one is described with a transition rate from immature to mature according to a fixed age (DDE model). For the ODE model, it is shown that the adaptation of parental care enlarges the possibility of species survival in the sense that population is permanent under the influences of the adaptation, but becomes extinct in the absence of adaptation. It is proved that the outcome of the adaptation makes the population in an optimal state. It is also observed that there are parental care switches, from noncare strategy to care strategy, as the natural death rate of immature individuals increases. The analysis of the DDE model indicates that the adaptation also enlarges the opportunity of population persistence, but the stage delay has the tendency to hinder the movement of population evolution to the optimal state. It is found that the loss rate of immature in the absence of parental care can induce different patterns to disturb the adaptation of population to optimal state. However, it is shown that the adaptation of parental care approaches to the optimal state when parental care is required for the survival of the population, for example, when the loss rate of immature or competition among mature increases or the fecundity decreases. The research was supported by Heiwa–Nakajima Fund and National Science Fund of China (No. 10571143). The research was partly supported by the Sasakawa Scientific Research Grant from The Japan Science Society. The research was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ?(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Effects of temporal variability on rare plant persistence in annual systems

Traditional conservation biology regards environmental fluctuations as detrimental to persistence, reducing long-term average growth rates and increasing the probability of extinction. By contrast, coexistence models from community ecology suggest that for species with dormancy, environmental fluctuations may be essential for persistence in competitive communities. We used models based on California grasslands to examine the influence of interannual fluctuations in the environment on the persistence of rare forbs competing with exotic grasses. Despite grasses and forbs independently possessing high fecundity in the same types of years, interspecific differences in germination biology and dormancy caused the rare forb to benefit from variation in the environment. Owing to the buildup of grass competitors, consecutive favorable years proved highly detrimental to forb persistence. Consequently, negative temporal autocorrelation, a low probability of a favorable year, and high variation in year quality all benefited the forb. In addition, the litter produced by grasses in a previously favorable year benefited forb persistence by inhibiting its germination into highly competitive grass environments. We conclude that contrary to conventional predictions of conservation and population biology, yearly fluctuations in climate may be essential for the persistence of rare species in invaded habitats.