Successful invasion of a food web in a chemostat

A food web in a chemostat is considered in which an arbitrary number of competitor populations compete for a single, essential, nonreproducing, growth-limiting substrate, and an arbitrary number of predator populations prey on some or all of the competitor populations. Although any number of predator populations may prey on the same competitor population, each predator population preys on only one competitor population. The dynamics of substrate uptake is modeled by Lotka-Volterra or Michaelis-Menten (Holling type I or II), but the dynamics of competitor uptake is restricted to Lotka-Volterra. Based on certain parameters, the model predicts the asymptotic survival or extinction of each of the different populations and suggests how competitor and/or predator populations could successfully invade the chemostat with or without causing a diverse ecosystem to crash. Similarly, it suggests how the elimination of certain populations could result in a more diverse or less diverse system.     

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.     

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     

Effects of spatial heterogeneity on the dynamics of a microbial feeding interaction

A mathematical model for an ideal chemostat in which one microbial population feeds on another and where Monod's model is used for the specific growth rates of both populations predicts a less stable behavior for the system than the one observed experimentally. Various factors have been proposed as being the reason for the increased stability of such systems. In this work, the effect of spatial heterogeneity on the dynamics of the microbial feeding interaction is studied. It is concluded that spatial heterogeneity has a stabilizing effect on the system. This effect combined with other factors could be the reason for the increased stability observed in systems where a microbial feeding interaction occurs.     

Obligate mutualism with a predator: Stability and persistence of three-species models

We present a general model for three interacting populations, where one population, called a mutualist, benefits a predator in its interaction with the prey. Biologically, there are four different ways in which the mutualist could benefit the predator: by enhancing prey growth rate, by enhancing the rate of prey capture, by providing an alternative food supply for the predator, and by enhancing the efficiency of utilization of prey, once they are ingested. We discuss examples of each type of interaction. We restrict our model to those situations in which the predator cannot survive on the prey in the absence of the mutualist. Therefore, if mutualism exists, it is obligate for the predator. Other conditions of the model include the dynamics of the prey and the mutualist alone and together in the absence of the predator. Given additional reasonable restrictions on the model, we determine the conditions for persistence, where persistence is defined as the continued existence of all three populations without any of them going extinct. There are two ways in which survival may arise in these models. Under one set of conditions, which is equivalent to the predator being able to invade a prey-mutualist system when rare, persistence will occur for any set of positive critical population sizes. Alternatively, survival will occur if there is an asymptotically stable interior equilibrium. However, the conditions for this are complex, and survival may occur only for initial populations in a limited region around the equilibrium.     

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.     

A synthetic Escherichia coli predator–prey ecosystem

We have constructed a synthetic ecosystem consisting of two Escherichia coli populations, which communicate bi‐directionally through quorum sensing and regulate each other's gene expression and survival via engineered gene circuits. Our synthetic ecosystem resembles canonical predator–prey systems in terms of logic and dynamics. The predator cells kill the prey by inducing expression of a killer protein in the prey, while the prey rescue the predators by eliciting expression of an antidote protein in the predator. Extinction, coexistence and oscillatory dynamics of the predator and prey populations are possible depending on the operating conditions as experimentally validated by long‐term culturing of the system in microchemostats. A simple mathematical model is developed to capture these system dynamics. Coherent interplay between experiments and mathematical analysis enables exploration of the dynamics of interacting populations in a predictable manner.     

Quantifying predator dependence in the functional response of generalist predators

A long‐standing debate concerns how functional responses are best described. Theory suggests that ratio dependence is consistent with many food web patterns left unexplained by the simplest prey‐dependent models. However, for logistical reasons, ratio dependence and predator dependence more generally have seen infrequent empirical evaluation and then only so in specialist predators, which are rare in nature. Here we develop an approach to simultaneously estimate the prey‐specific attack rates and predator‐specific interference (facilitation) rates of predators interacting with arbitrary numbers of prey and predator species in the field. We apply the approach to surveys and experiments involving two intertidal whelks and their full suite of potential prey. Our study provides strong evidence for predator dependence that is poorly described by the ratio dependent model over manipulated and natural ranges of species abundances. It also indicates how, for generalist predators, even the qualitative nature of predator dependence can be prey‐specific.     

Prey life-history and bioenergetic responses across a predation gradient

To evaluate the importance of non-consumptive effects of predators on prey life histories under natural conditions, an index of predator abundance was developed for naturally occurring populations of a common prey fish, the yellow perch Perca flavescens, and compared to life-history variables and rates of prey energy acquisition and allocation as estimated from mass balance models. The predation index was positively related to maximum size and size at maturity in both male and female P. flavescens, but not with life span or reproductive investment. The predation index was positively related to size-adjusted specific growth rates and growth efficiencies but negatively related to model estimates of size-adjusted specific consumption and activity rates in both vulnerable (small) and invulnerable (large) size classes of P. flavescens. These observations suggest a trade-off between growth and activity rates, mediated by reduced activity in response to increasing predator densities. Lower growth rates and growth efficiencies in populations with fewer predators, despite increased consumption suggests either 1) a reduction in prey resources at lower predator densities or 2) an intrinsic cost of rapid prey growth that makes it unfavourable unless offset by a perceived threat of predation. This study provides evidence of trade-offs between growth and activity rates induced by predation risk in natural prey fish populations and illustrates how behavioural modification induced through predation can shape the life histories of prey fish species.     

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.     

Application of a model of scale dependence to quantify scale domains in open predation experiments

We use a model of open predation experiments to define scale domains that differ in terms of the controlling processes and scale dependence of predator impacts. For experimental arenas that are small compared to the movements of the prey (small scale domain) the model predicts that predator impacts are scale independent and controlled by prey movements. For arenas of intermediate scale we predict that predator impacts are scale dependent and controlled by both prey movements and direct predation, and for the largest scale domain we predict weak scale dependence and predation control.
We propose that the scale‐domain concept is useful when designing and interpreting field experiments. As an illustration we apply the concept to experiments examining predator effects on the stream benthos. First, we test two key assumptions of the underlying model: that area‐specific prey migration rates decrease with increasing size of experimental arenas and that predation rates are independent of arena size. For this purpose we used published estimates of prey emigration and predator consumption rates for nine studies examining the effects of stream predators on benthic prey. We found that prey per capita emigration rates but not predation rates decreased with increasing arena length.
Second, we demonstrate a method for identifying the scale domain of real experiments. The model of predation experiments was parameterized using experimental data and the expected spatial and temporal scale dependence of predator impacts on prey in these experiments was simulated. The simulations suggest that the studies conducted in the largest arenas (length 15–35 m) should be classified as large‐scale, consumption‐controlled experiments, whereas the experiments conducted in smaller arenas (length 1.5–6 m) should be classified as small or intermediate‐scale. We also attempted to determine the scale domain of the experiments in a large data set, including results from most published stream predation experiments. The majority of arenas used in these experiments (73%) were smaller than 1 m in length. Our data on the scale dependence of predation and prey migration rate suggest that experiments in this scale range (<1 m) should be classified as small‐scale, movement‐controlled experiments for most prey taxa.     

Protozoan feeding and bacterial wall growth.

Monod's model is often assumed to describe the kinetics of feeding of a protozoan population on a bacterial population in a chemostat. An earlier study (J. L. Jost et al., J. Bacteriol., 113, 84 (1973)) of the feeding of Tetrahymena pyriformis on either Escherichia coli or Azotobacter vinelandii found that this model correctly predicted the occurrence of sustained oscillations of population densities but made predictions of minimum bacterial population densities that were much smaller than those observed. The earlier study removed the discrepancy between the model and data by replacing Monod's model with a different model. It is shown in the present study that the discrepancy can be explained equally as well if Monod's model for the feed relation is retained and if, in addition, growth of bacteria on the chemostat walls is allowed for in the model equations.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.     

Introduced species provide a novel temporal resource that facilitates native predator population growth

Non-native species are recognized as important components of change to food web structure. Non-native prey may increase native predator populations by providing an additional food source and simultaneously decrease native prey populations by outcompeting them for a limited resource. This pattern of apparent competition may be important for plants and sessile marine invertebrate suspension feeders as they often compete for space and their immobile state make them readily accessible to predators. Reported studies on apparent competition have rarely been examined in biological invasions and no study has linked seasonal patterns of native and non-native prey abundance to increasing native predator populations. Here, we evaluate the effects of non-native colonial ascidians (Diplosoma listerianum and Didemnum vexillum) on population growth of a native predator (bloodstar, Henricia sanguinolenta) and native sponges through long-term surveys of abundance, prey choice and growth experiments. We show non-native species facilitate native predator population growth by providing a novel temporal resource that prevents loss of predator biomass when its native prey species are rare. We expect that by incorporating native and non-native prey seasonal abundance patterns, ecologists will gain a more comprehensive understanding of the mechanisms underlying the effects of non-native prey species on native predator and prey population dynamics.     

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     

Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

Predators reverse the direction of density dependence for juvenile salmon mortality

The effect of predators on prey populations depends on how predator-caused mortality changes with prey population density. Predators can enforce density-dependent prey mortality and contribute to population stability, but only if they have a positive numerical or behavioral response to increased prey density. Otherwise, predator saturation can result in inversely density-dependent mortality, destabilizing prey populations and increasing extinction risk. Juvenile salmon and trout provide some of the clearest empirical examples of density-dependent mortality in animal populations. However, although juvenile salmon are very vulnerable to predators, the demographic effects of predators on juvenile salmon are unknown. We tested the interactive effects of predators and population density on the mortality of juvenile Atlantic salmon (Salmo salar) using controlled releases of salmon in natural streams. We introduced newly hatched juvenile salmon at three population density treatments in six study streams, half of which contained slimy sculpin (Cottus cognatus), a common generalist predator (18 release sites in total, repeated over two summers). Sculpin reversed the direction of density dependence for juvenile salmon mortality. Salmon mortality was density dependent in streams with no sculpin, but inversely density dependent in streams where sculpin were abundant. Such predator-mediated inverse density dependence is especially problematic for prey populations suppressed by other factors, thereby presenting a fundamental challenge to persistence of rare populations and restoration of extirpated populations.