Long run coexistence in the chemostat with multiple species

In this work we analyze the transient behavior of the dynamics of multiple species competing in a chemostat for a single resource, presenting slow/fast characteristics. We prove that coexistence among a subset of species, with growth functions close to each other, can last for a substantially long time. For these cases, we also show that the proportion of non-dominant species can be increasing before decreasing, under certain conditions on the initial distribution.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

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     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

The effectiveness of post-contact defenses in a prey with no pre-contact detection

Most empirical and theoretical papers on prey–predator interactions are for animals with long-range detection, animals that can detect and react to predators long before these touch the prey. Heavy-bodied and chemically defended harvestmen (Arachnida, Opiliones) are an exception to this general pattern and rely on contact to detect arthropod predators. We examined the interactions between the Brazilian wandering spider Ctenus ornatus with harvestmen (Mischonyx cuspidatus) or control prey (Gryllus sp. and M. cuspidatus immature, both with soft integuments). Considering a prey–predator system in which fleeing from or reacting to a predator at a distance is not possible, we predicted both a high survival value of near-range defense mechanisms and that mortality would be higher in the absence of such defense mechanisms. We also expected the predator to behave differently when interacting with harvestmen or with a control prey without such defense mechanisms. Our results from laboratory experiments partially matched our predictions: First of all, histological sections showed that the integument of adult harvestmen is thicker than that of immature harvestmen and that of crickets. Adult harvestmen were less preyed upon than the control prey; the heavy armature increases the survival rate but the secretions from the scent glands do not. The predator did behave differently when attacking harvestmen compared to crickets. Despite the large size difference between predator and harvestmen, the protection provided by the armature allowed some of the harvestmen to survive encounters without pre-contact detection, thus greatly reducing the reliance on long-range detection to survive encounters with predators. Harvestmen call for theoretical and empirical work on prey–predator interactions that take into account the possibility that prey may not detect the predator before contact is established.     

Dynamical behaviour of a two-predator model with prey refuge

A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models are relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local (using index arguments). This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the case of uninhibited specific growth rate.Research supported by the National Council of Science, Republic of ChinaResearch supported by National Science Foundation Grant, DMS-9204490Research supported by the Natural Science and Engineering Council of Canada. This author's contribution was made while on research leave visiting the Department of Ecology and Evolutionary Biology at Princeton University. She would especially like to thank Simon Levin for his guidance as well as for providing an exceptional working environment     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat with inhibitions

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models were relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local. The rigorous global analysis was done in a paper of Hsu, Waltman and Wolkowicz in the case of the uninhibited specific growth rates. This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the cases of combinations of inhibited and uninhibited specific growth rates.Research Supported by the National Council of Science, Republic of China     

Population dynamics and competition in chemostat models with adaptive nutrient uptake

 The standard Monod model for microbial population dynamics in the chemostat is modified to take into consideration that cells can adapt to the change of nutrient concentration in the chemostat by switching between fast and slow nutrient uptake and growing modes with asymmetric thresholds for transition from one mode to another. This is a generalization of a modified Monod model which considers adaptation by transition between active growing and quiescent cells. Global analysis of the model equations is obtained using the theory of asymptotically autonomous systems. Transient oscillatory population density and hysteresis growth pattern observed experimentally, which do not occur for the standard Monod model, can be explained by such adaptive mechanism of the cells. Competition between two species that can switch between fast and slow nutrient uptake and growing modes is also considered. It is shown that generically there is no coexistence steady state, and only one steady state, corresponding to the survival of at most one species in the chemostat, is a local attractor. Numerical simulations reproduce the qualitative feature of some experimental data which show that the population density of the winning species approaches a positive steady state via transient oscillations while that of the losing species approaches the zero steady state monotonically. Received 4 August 1995; received in revised form 15 December 1995     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

A new method for immunologically marking prey and its use in predation studies

We introduce a new method for immunologically examining predator gut contents. It differs from previously described gut content analyses because it does not require the development of prey-specific antibody probes. Instead, insect prey were marked with a readily available antigen, rabbit immunoglobulin G (IgG). We then assayed predators that had fed on IgG labeled prey with an enzyme-linked immunosorbent assay (ELISA) using goat anti-rabbit IgG. Of the predator species that fed on the IgG labeled prey, 98.8% of those with chewing mouthparts scored positive for IgG 1 h after feeding. Our prey-labeling ELISA was less efficient for detecting IgG prey remains in predators with piercing/sucking mouthparts. Only 29.5% of these individuals scored positive for rabbit IgG in their guts 1 h after feeding. An additional study was conducted to measure the retention time of IgG-labeled prey in the guts of two species of predators with chewing mouthparts. Results from this experiment showed that the retention time varied depending on the predator and prey species examined. Results from these studies indicate that this marking technique could have widespread use for analyzing the gut contents of predators with chewing mouthparts, but it has limited application for those predators with piercing/sucking mouthparts. This article presents the results of research only. Mention of a proprietary product does not constitute an endorsement or recommendation for its use by the USDA.     

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     

The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response

In this paper, the global asymptotic behavior of a chemostat model with Beddington-DeAngelis functional response is studied. The conditions for the global asymptotical stability of the model with time delays are obtained via monotone dynamical systems. Our results demonstrate that those time delays affect the competitive outcome of the organisms.     

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.     

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     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

The dynamics of a Lotka–Volterra predator–prey model with state dependent impulsive harvest for predator

According to the economic and biological aspects of renewable resources management, we propose a Lotka–Volterra predator–prey model with state dependent impulsive harvest. By using the Poincaré map, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three under some conditions. Numerical results are carried out to illustrate the feasibility of our main results. The bifurcation diagrams of periodic solutions are obtained by using the numerical simulations, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations, which implies that the presence of pulses makes the dynamic behavior more complex.     

环境污染下脉冲输入恒化器模型动力学性质的研究

研究了环境污染下脉冲输入恒化器模型的动力学性质,得到了微生物灭绝周期解的全局稳定性和系统的持续生存.