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.     

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.     

Endogenous metabolism and the stability of microbial prey-predator systems

We formulate a variant of the "double Monod model" which takes explicit account of endogenous metabolism. Using parameter values appropriate to carbohydrate-limited substrate, bacterial prey, and protozoan predator, we study the stability of steady states under chemostat conditions. We conclude that the predator's endogenous metabolism may have a stabilizing effect at low dilution rates.     

Complex dynamics in a model microbial system

The forced double-Monod model (for a chemostat with a predator, a prey and periodically forced inflowing substrate) displays quasiperiodicity, phase locking, period doubling and chaotic dynamics. Stroboscopic sections reveal circle maps for the quasiperiodic regimes and noninvertible maps of the interval for the chaotic regimes. Criticality in the circle maps sets the stage for chaos in the model. This criticality may arise with an increase in the period or amplitude of forcing.     

Effects of rapid prey evolution on predator–prey cycles

We study the qualitative properties of population cycles in a predator-prey system where genetic variability allows contemporary rapid evolution of the prey. Previous numerical studies have found that prey evolution in response to changing predation risk can have major quantitative and qualitative effects on predator-prey cycles, including: (1) large increases in cycle period, (2) changes in phase relations (so that predator and prey are cycling exactly out of phase, rather than the classical quarter-period phase lag), and (3) "cryptic" cycles in which total prey density remains nearly constant while predator density and prey traits cycle. Here we focus on a chemostat model motivated by our experimental system (Fussmann et al. in Science 290:1358-1360, 2000; Yoshida et al. in Proc roy Soc Lond B 424:303-306, 2003) with algae (prey) and rotifers (predators), in which the prey exhibit rapid evolution in their level of defense against predation. We show that the effects of rapid prey evolution are robust and general, and furthermore that they occur in a specific but biologically relevant region of parameter space: when traits that greatly reduce predation risk are relatively cheap (in terms of reductions in other fitness components), when there is coexistence between the two prey types and the predator, and when the interaction between predators and undefended prey alone would produce cycles. Because defense has been shown to be inexpensive, even cost-free, in a number of systems (Andersson et al. in Curr Opin Microbiol 2:489-493, 1999: Gagneux et al. in Science 312:1944-1946, 2006; Yoshida et al. in Proc Roy Soc Lond B 271:1947-1953, 2004), our discoveries may well be reproduced in other model systems, and in nature. Finally, some of our key results are extended to a general model in which functional forms for the predation rate and prey birth rate are not specified.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

On competition of predators and prey infection

A class of prey–predator models with infected prey is investigated. Predation terms are either of Holling type II or III, infection is either modelled by mass action or standard incidence. It is shown that the key for understanding the model behaviour is the competition of predators versus infection. In the presented models the predator is not susceptible to the infection and the infection of the prey has no influence on the ability of the predator of catching the prey. However, the prey population can be seen as a resource which both the predators and the infection depend on. The competition for this resource is strong—the principle of competitive exclusion holds for biologically meaningful choices of parameters as long as there is no destabilisation by a Hopf bifurcation. The representation of models in competition diagrams which are introduced in this article can be used for a wide range of competition models which seems to be a promising method with many potential applications.     

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.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003     

Density-dependent prey behaviours and mutable predator foraging modes induce Allee effects and over-prediction of prey mortality rates

1. Predator–prey models are often used to represent consumptive interactions between species but, typically, are derived using simple experimental systems with little plasticity in prey or predator behaviours. However, many prey and predators exhibit a broad suite of behaviours. Here, we experimentally tested the effect of density-dependent prey and predator behaviours on per capita relative mortality rates using Florida bass (Micropterus floridanus) consuming juvenile Bluegill (Lepomis macrochirus).
2. Experimental ponds were stocked with a factorial design of low, medium, and high prey and predator densities. Prey mortality, prey–predator behaviours, and predator stomach contents were recorded over or after 7 days. We assumed the mortality dynamics followed foraging arena theory. This pathologically flexible predator–prey model separates prey into invulnerable and vulnerable pools where predators can consume prey in the latter. As this approach can represent classic Lotka–Volterra and ratio-dependent dynamics, we fit a foraging arena predator–prey model to the number of surviving prey.
3. We found that prey exhibited density-dependent prey behaviours, hiding at low densities, shoaling at medium densities, and using a provided refuge at high densities. Predators exhibited ratio-dependent behaviours, using an ambush foraging mode when one predator was present, hiding in the shadows at low prey–high predator densities, and shoaling at medium and high prey–high predator densities. The foraging arena model predicted the mortality rates well until the high prey–high predator treatment where group vigilance prey behaviours occurred and predators probably interfered with one another resulting in the model predicting higher mortality than observed.
4. This is concerning given the ubiquity of predator–prey models in ecology and natural resource management. Furthermore, as Allee effects engender instability in population regulation, it could lead to inaccurate predictions of conservation status, population rebuilding or harvest rates.

     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

Density-dependent intraguild predation of an aphid parasitoid

A growing body of research has examined the effect of shared resource density on intraguild predation (IGP) over relatively short time frames. Most of this work has led to the conclusion that when the shared resource density is high, the strength of IGP should be lower, due to prey dilution. However, experiments addressing this topic have been done using micro- or mesocosms that excluded the possibility of intraguild predator aggregation. We examined the effect of shared resource density on IGP of an aphid parasitoid in an open field setting where the effects of prey dilution and predator aggregation could occur simultaneously. We brought potted soybean plants with 2, 20, or 200 soybean aphids (Aphis glycines) and 20 pupae (‘mummies’) of the soybean aphid parasitoid Binodoxys communis into soybean fields in Minnesota, USA. We monitored predator aggregation onto the potted plants, predation of parasitoid mummies, and successful adult emergence of B. communis. We found that predator aggregation was higher at the higher aphid densities on our experimental plants and that this coincided with lower adult emergence of B. communis, indicating that even if a prey dilution effect occurred in our study, it was overcome by short-term predator aggregation. Our results suggest that the effect of shared resource density on IGP may be more nuanced in a field setting than in microcosms due to predator aggregation.     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Dynamic analysis of a microbial process: A systems engineering approach

A general mathematical model of the chemostat system is developed in order to define an experimental program of dynamic testing. A glucose-limited culture ofSaccharomyces cerevisiae was grown in a chemostat using chemically defined medium. The chemostat was perturbed from an initial steady state by changes in input glucose concentration, dilution rate, pH, and temperature. Dynamic responses of cell mass, glucose, cell number, RNA, and protein concentrations were measured. A number of simulation techniques were used in developing a dynamic mathematical model and in comparing the developed model with experimental data as well as the Monod model. The resulting model was found to be quantitatively accurate and superior to the Monod model. The developed model was interpreted in the light of cell physiology. Adjustment of intracellular RNA fraction was found to be rate limiting in acceleration of cell specific growth rate.     

Predator density and competition modify the benefits of group formation in a shoaling reef fish

Synthesis Predation risk experienced by individuals living in groups depends on the balance between predator dilution, competition for refuges, and predator interference or synergy. These interactions operate between prey species as well: the benefits of group living decline in the presence of an alternative prey species. We apply a novel model‐fitting approach to data from field experiments to distinguish among competing hypotheses about shifts in predator foraging behavior across a range of predator and prey densities. Our study provides novel analytical tools for analyzing predator foraging behavior and offers insight into the processes driving the dynamics of coral reef fish. Studies of predator foraging behavior typically focus on single prey species and fixed predator densities, ignoring the potential importance of complexities such as predator dilution; predator‐mediated effects of alternative prey; heterospecific competition; or predator–predator interactions. Neglecting the effects of prey density is particularly problematic for prey species that live in mixed species groups, where the beneficial effects of predator dilution may swamp the negative effects of heterospecific competition. Here we use field experiments to investigate how the mortality rates of a shoaling coral reef fish (a wrasse: Thalassoma amblycephalum), change as a result of variation in: 1) conspecific density, 2) density of a predator (a hawkfish: Paracirrhites arcatus), and 3) presence of an alternative prey species that competes for space (a damselfish: Pomacentrus pavo). We quantify changes in prey mortality rates from the predator's perspective, examining the effects of added predators or a second prey species on the predator's functional response. Our analysis highlights a model‐fitting approach that discriminates amongst multiple hypotheses about predator foraging in a community context. Wrasse mortality decreased with increasing conspecific density (i.e. mortality was inversely density‐dependent). The addition of a second predator doubled prey mortality rates, without significantly changing attack rate or handling time – i.e. there was no evidence for predator interference. The presence of a second prey species increased wrasse mortality by 95%; we attribute this increase either to short‐term apparent competition (predator aggregation) or to a decrease in handling time of the predator (e.g. through decreased wrasse vigilance). In this system, 1) prey benefit from intraspecific group living though a reduced predation risk, and 2) the benefit of group living is reduced in the presence of an alternative prey species.     

Dynamics of Predator‐Prey Interactions in Continuous Cultures

This paper studies the behavior of a general unstructured kinetic model for continuous bioreactors involving interactions between predator, prey and a limiting substrate. The analysis carried out in this paper shows how closed analytical conditions for arbitrary growth rates can be derived that describe the conditions for the existence of the interacting species in an oscillatory behavior. It is also demonstrated how practical diagrams in terms of operating and kinetic parameters can be constructed that classify the different behavior predicted by the model. Applications of these general results to a number of experimentally validated models have revealed that the saturation model always predicts hard oscillations for a certain range of dilution rates, for any values of model parameters. Bifurcation diagrams in the operating parameter space permitted the delineation of regions of hard oscillations, regions of static coexistence, regions of predator washout and regions of total washout. The analysis of the double saturation model has proven its ability to predict two Hopf points. Hard oscillations are therefore expected within the dilution rates corresponding to the two Hopf points. Practical diagrams were also constructed to delineate the boundaries separating hard oscillations from static coexistence and washout conditions.     

The effect of growth factors on anoxic chemostat cultures of two Saccharomyces cerevisiae strains

In anoxic chemostat cultures of Saccharomyces cerevisiae ATCC 4126 and CBS 8066 grown in a medium containing yeast extract, a sharp increase in the steady-state residual glucose concentration occurred at relatively low dilution rates, contrary to the expected Monod kinetics. However, supplementation with vitamins and amino acids facilitated efficient glucose uptake. This enhanced requirement for growth factors under anoxic conditions and at high growth rates could explain the exceptionally high apparent k _s values for S. cerevisiae reported in the literature.     

Foraging facilitation among predators and its impact on the stability of predator–prey dynamics

Predator foraging facilitation may strongly influence the dynamics of a predator–prey system. This behavioral pattern is well-observed in real life interactions, but less is known about its possible impacts on the predator–prey dynamics. In this paper we analyze a modified Rosenzweig–MacArthur model, where a predator-dependent family of functions describing predator foraging facilitation is introduced into the Holling type II functional response. As the general assumption of foraging facilitation is that higher predator densities give rise to an increased foraging efficiency, we model predator facilitation with an increasing encounter rate function. Using the tools of bifurcation analysis we describe all the nonlinear phenomena that occur in the system provoked by foraging facilitation, these include the fold, Hopf, transcritial, homoclinic and Bogdanov–Takens bifurcation. We show that foraging facilitation can stabilize the coexistence in the predator–prey system for specific rates, but in most of the cases it can have fatal consequences for the predators themselves.     

Effect of resource subsidies on predator–prey population dynamics: a mathematical model

The influence of a resource subsidy on predator–prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator–prey-subsidy interactions also increases.