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     

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.     

Stability in chemostat equations with delayed nutrient recycling

The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.Work performed within the activity of the research group Equazioni di evoluzione e applicazioni Fisico-Matematiche, MPI (Italy), and under the auspices of GNFM, CNR (Italy)     

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     

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.     

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     

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.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

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     

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 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     

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.     

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     

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.     

Predators buffer the effects of variation in prey nutrient content for nutrient deposition

Predator feeding behavior and digestion regulate the flow of nutrients through ecosystems by determining the fate of prey nutrients. Most predators feed on a diversity of prey items, which differ widely in traits including their nutrient content. Yet, relatively little is known of the mechanisms through which variation in prey nutrient content affects the form by which nutrients are deposited into the environment. The overall goal of this study was to test how variation in the nutrient content of prey affected the fate of nutrients following predation by an arthropod carnivore, the Carolina wolf spider Hogna carolinensis. We manipulated the macronutrient content of prey by varying the diet on which crickets were fed to produce prey treatments that differed in lipid and protein content. Nutrients were measured as both macronutrients and elements in prey and elements in excreta. We found that there was no effect of diet treatment on the amounts of elements or macronutrients in prey carcasses and excreta despite significant variation in the nutrient content of those prey. This is in contrast to studies of some aquatic systems where mass balance by consumers results in variation in excreta content depending on the nutrient content of food. Wolf spiders assimilated the majority of prey nutrients and deposited relatively small and similar amounts of nutrients following feeding. Hence, while prey can vary widely in nutrient content, our findings suggest that this variation has little effect on the amounts of nutrients deposited by predators.     

The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model

 A predator–prey interaction model based on a system of differential equations with temperature-dependent parameters chosen appropriately for a mite interaction on apple trees is analyzed to determine how the type of functional response influences bifurcation and stability behavior. Instances of type I, II, III, and IV functional responses are considered, the last of which incorporates prey interference with predation. It is shown that the model systems with the type I, II, and III functional responses exhibit qualitatively similar bifurcation and stability behavior over the interval of definition of the temperature parameter. Similar behavior is found in the system with the type IV functional response at low levels of prey interference. Higher levels of interference are destabilizing, as illustrated by the prevalence of bistability and by the presence of three attractors for some values of the model parameters. All four systems are capable of modeling population oscillations and outbreaks. Received 12 March 1996; received in revised form 25 October 1996     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.