Impact of cannibalism on predator-prey dynamics: size-structured interactions and apparent mutualism

Direct and indirect interactions between two prey species can strongly alter the dynamics of predator-prey systems. Most predators are cannibalistic, and as a consequence, even systems with only one predator and one prey include two prey types: conspecifics and heterospecifics. The effects of the complex direct and indirect interactions that emerge in such cannibalistic systems are still poorly understood. This study examined how the indirect interaction between conspecific and heterospecific prey affects cannibalism and predation rates and how the direct interactions between both species indirectly alter the effect of the cannibalistic predator. I tested for these effects using larvae of the stream salamanders Eurycea cirrigera (prey) and Pseudotriton ruber (cannibalistic predator) by manipulating the relative densities of the conspecific and heterospecific prey in the presence and absence of the predator in experimental streams. The rates of cannibalism and heterospecific predation were proportional to the respective densities and negatively correlated, indicating a positive indirect interaction between conspecific and heterospecific prey, similar to "apparent mutualism." Direct interactions between prey species did not alter the effect of the predator. Although both types of prey showed a similar 30% reduction in night activity and switch in microhabitat use in response to the presence of the predator, cannibalism rates were three times higher than heterospecific predation rates irrespective of the relative densities of the two types of prey. Cumulative predation risks differed even more due to the 48% lower growth rate of conspecific prey. Detailed laboratory experiments suggest that the 3:1 difference in cannibalism and predation rate was due to the higher efficiency of heterospecific prey in escaping immediate attacks. However, no difference was observed when the predator was a closely related salamander species, Gyrinophilus porphyriticus, indicating that this difference is species specific. This demonstrates that cannibalism can result in the coupling of predator and prey mortality rates that strongly determines the dynamics of predator-prey systems.     

Using the functional response of a consumer to predict biotic resistance to invasive prey

Predators sometimes provide biotic resistance against invasions by nonnative prey. Understanding and predicting the strength of biotic resistance remains a key challenge in invasion biology. A predator's functional response to nonnative prey may predict whether a predator can provide biotic resistance against nonnative prey at different prey densities. Surprisingly, functional responses have not been used to make quantitative predictions about biotic resistance. We parameterized the functional response of signal crayfish (Pacifastacus leniusculus) to invasive New Zealand mud snails (Potamopyrgus antipodarum; NZMS) and used this functional response and a simple model of NZMS population growth to predict the probability of biotic resistance at different predator and prey densities. Signal crayfish were effective predators of NZMS, consuming more than 900 NZMS per predator in a 12-h period, and Bayesian model fitting indicated their consumption rate followed a type 3 functional response to NZMS density. Based on this functional response and associated parameter uncertainty, we predict that NZMS will be able to invade new systems at low crayfish densities (< 0.2 crayfish/m2) regardless of NZMS density. At intermediate to high crayfish densities (> 0.2 crayfish/m2), we predict that low densities of NZMS will be able to establish in new communities; however, once NZMS reach a threshold density of -2000 NZMS/m2, predation by crayfish will drive negative NZMS population growth. Further, at very high densities, NZMS overwhelm predation by crayfish and invade. Thus, interacting thresholds of propagule pressure and predator densities define the probability of biotic resistance. Quantifying the shape and uncertainty of predator functional responses to nonnative prey may help predict the outcomes of invasions.     

The influence of size-specific indirect interactions in predator-prey systems

Nonlethal indirect interactions between predators often lead to nonadditive effects of predator number on prey survival and growth. Previous studies have focused on systems with at least two different predator species and one prey species. However, most predators undergo extreme ontological changes in phenotype such that interactions between different-sized cohorts of a predator and its prey could lead to nonadditive effects in systems with only two species. This may be important since different-sized individuals of the same species can differ more in their ecology than similar-sized individuals of different species. This study examined trait-mediated indirect effects in a two-species system including a cannibalistic predator with different-sized cohorts and its prey. I tested for these effects using larvae of two stream salamanders, Gyrinophilus porphyriticus (predator) and Eurycea cirrigera (prey), by altering the densities and combinations of predator size classes in experimental streams. Results showed that the presence of large individuals can significantly reduce the impact of density changes of smaller conspecifics on prey survival through nonlethal means. In the absence of large conspecifics, an increase in the relative frequency of small predators significantly increased predation rates, thereby reducing prey survival. However, with large conspecifics present, increasing the density of small predators did not decrease prey survival, resulting in a 14.3% lower prey mortality than predicted from the independent effects of both predator size classes. Small predators changed their microhabitat use in the presence of larger conspecifics. Prey individuals reduced activity in response to large predators but did not respond to small predators. Both predators reduced prey growth. These results demonstrate that the impact of a predator can be significantly altered by two different types of trait-mediated indirect effects in two-species systems: between different-sized cohorts and between different cohorts and prey. This study demonstrates that predictions based on simple numerical changes that assume independent effects of different size classes or ignore size structure can be strongly misleading. We need to account for the size structure within predator populations in order to predict how changes in predator abundance will affect predator-prey dynamics.     

The impact of parasite manipulation and predator foraging behavior on predator-prey communities

Parasites are known to directly affect their hosts at both the individual and population level. However, little is known about their more subtle, indirect effects and how these may affect population and community dynamics. In particular, trophically transmitted parasites may manipulate the behavior of intermediate hosts, fundamentally altering the pattern of contact between these individuals and their predators. Here, we develop a suite of population dynamic models to explore the impact of such behavioral modifications on the dynamics and structure of the predator-prey community. We show that, although such manipulations do not directly affect the persistence of the predator and prey populations, they can greatly alter the quantitative dynamics of the community, potentially resulting in high amplitude oscillations in abundance. We show that the precise impact of host manipulation depends greatly on the predator's functional response, which describes the predator's foraging efficiency under changing prey availabilities. Even if the parasite is rarely observed within the prey population, such manipulations extend beyond the direct impact on the intermediate host to affect the foraging success of the predator, with profound implications for the structure and stability of the predator-prey community.     

Harvest policies and nonmarket valuation in a predator — prey system

Although prey may not have commercial value, their economic value can be ascertained in a predator-prey model if the predator has a harvest value. The economic optimal (recovery) path of the predator and prey are carefully described when growth is quadratic in the predator (prey) and linear in prey (predator). Parameter values, in part, resembling Pacific halibut are used to provide numerical illustrations.     

Functional responses: a question of alternative prey and predator density

Throughout the study of ecology, there has been a growing realization that indirect effects among species cause complexity in food webs. Understanding and predicting the behavior of ecosystems consequently depends on our ability to identify indirect effects and their mechanisms. The present study experimentally investigates indirect interactions arising between two prey species that share a common predator. In a natural field experiment, we introduced different densities of mealworms (Tenebrio molitor), an alternative prey, to a previously studied predator-prey system in which paper wasps (Polistes dominulus) preyed on shield beetle larvae (Cassida rubiginosa). We tested if alternative prey affects predation on the first prey (i.e., the predator-dependent functional response of paper wasps) by modifying either interference among predators or the effective number of predators foraging on shield beetles. Presence of mealworms significantly reduced the effective number of predators, whereas predator interference was not affected. In this way, the experimentally introduced alternative prey altered the wasps' functional response and thereby indirectly influenced C. rubiginosa density. In all prey-density combinations offered, paper wasps constantly preferred T. molitor. This led to an asymmetrical, indirect interaction between both prey species: an increase in mealworm density significantly relaxed predation on C. rubiginosa, whereas an increase in C. rubiginosa density intensified predation on mealworms. Such asymmetrical outcomes of a fixed food preference can significantly affect the population dynamics of the species involved. In spite of the repeated finding of a Type III functional response in this system, our experiment did not reveal switching behavior in paper wasps. The variety of mechanisms underlying direct and indirect interactions within our study system exemplifies the importance of incorporating alternative prey when investigating the impact of a generalist predator on a focal prey population under realistic field conditions.     

Revisiting the classics: considering nonconsumptive effects in textbook examples of predator-prey interactions

Predator effects on prey dynamics are conventionally studied by measuring changes in prey abundance attributed to consumption by predators. We revisit four classic examples of predator-prey systems often cited in textbooks and incorporate subsequent studies of nonconsumptive effects of predators (NCE), defined as changes in prey traits (e.g., behavior, growth, development) measured on an ecological time scale. Our review revealed that NCE were integral to explaining lynx-hare population dynamics in boreal forests, cascading effects of top predators in Wisconsin lakes, and cascading effects of killer whales and sea otters on kelp forests in nearshore marine habitats. The relative roles of consumption and NCE of wolves on moose and consequent indirect effects on plant communities of Isle Royale depended on climate oscillations. Nonconsumptive effects have not been explicitly tested to explain the link between planktonic alewives and the size structure of the zooplankton, nor have they been invoked to attribute keystone predator status in intertidal communities or elsewhere. We argue that both consumption and intimidation contribute to the total effects of keystone predators, and that characteristics of keystone consumers may differ from those of predators having predominantly NCE. Nonconsumptive effects are often considered as an afterthought to explain observations inconsistent with consumption-based theory. Consequently, NCE with the same sign as consumptive effects may be overlooked, even though they can affect the magnitude, rate, or scale of a prey response to predation and can have important management or conservation implications. Nonconsumptive effects may underlie other classic paradigms in ecology, such as delayed density dependence and predator-mediated prey coexistence. Revisiting classic studies enriches our understanding of predator-prey dynamics and provides compelling rationale for ramping up efforts to consider how NCE affect traditional predator-prey models based on consumption, and to compare the relative magnitude of consumptive and NCE of predators.     

Can rare positive interactions become common when large carnivores consume livestock?

Livestock populations in protected areas are viewed negatively because of their interaction with native ungulates through direct competition for food resources. However, livestock and native prey can also interact indirectly through their shared predator. Indirect interactions between two prey species occur when one prey modifies either the functional or numerical responses of a shared predator. This interaction is often manifested as negative effects (apparent competition) on one or both prey species through increased predation risk. But indirect interactions can also yield positive effects on a focal prey if the shared predator modifies its functional response toward increased consumption of an abundant and higher-quality alternative prey. Such a phenomenon between two prey species is underappreciated and overlooked in nature. Positive indirect effects can be expected to occur in livestock-dominated wildlife reserves containing large carnivores. We searched for such positive effects in Acacia-Zizhypus forests of India's Gir sanctuary where livestock (Bubalus bubalis and Bos indicus) and a coexisting native prey (chital deer, Axis axis) are consumed by Asiatic lions (Panthera leo persica). Chital vigilance was higher in areas with low livestock density than in areas with high livestock density. This positive indirect effect occurred because lion predation rates on livestock were twice as great where livestock were abundant than where livestock density was low. Positive indirect interactions mediated by shared predators may be more common than generally thought with rather major consequences for ecological understanding and conservation. We encourage further studies to understand outcomes of indirect interactions on long-term predator-prey dynamics in livestock-dominated protected areas.     

Effects of density-dependent dispersal behaviours on the speed and spatial patterns of range expansion in predator-prey metapopulations

Dispersal can strongly affect the spatiotemporal dynamics of a species (its spread, spatial distribution and persistence). We investigated how two dispersal behaviours, namely prey evasion (PE) and predator pursuit (PP), affect the dynamics of a predator-prey system. PE portrays the tendency of prey avoiding predators by dispersing into adjacent patches with fewer predators, while PP describes the tendency of predators to pursue the prey by moving into patches with more prey. Based on the Beddington predation model, a spatially explicit metapopulation model was built to incorporate PE and PP. Numerical simulations were run to investigate the effects of PE and PP on the rate of spread, spatial synchrony and the persistence of populations. Results show that both PE and PP can alter spatial synchrony although PP has a weaker desynchronising effect than PE. The predator-prey system without PE and PP expanded in circular waves. The effect of PE can push the prey to distribute in a circular ring front, whereas the effect of PP can change the circular waves to anisotropic expansion. Furthermore, weak PE and PP can accelerate the spread of prey while strong and disproportionate intensities slow down the range expansion. The effects of PE and PP further enhance the population size, break down the spatial synchrony and promote the persistence of populations.     

How population dynamics shape the functional response in a one-predator-two-prey system

The type III functional response has historically been associated with switching predators; when there is a choice of prey the predator favors the more abundant prey type. Although this functional response has been found in experiments where both prey densities are manipulated, in real world studies the type II functional response is more commonly found. In modeling, the type III functional response is often used in systems where the second prey type is, implicitly, assumed to be constant. Here we define a functional response that takes into account both prey densities. This causes the functional response to show both type II and type III behavior, dependent on the interaction between the two prey densities. If we take into account population dynamics, we find a type II functional response in most cases, because predation regulates the relative prey densities. This explains why type III functional responses are found in experiments where both prey densities are manipulated, but type II functional responses occur when the feedback of population dynamics on the functional response is important. Furthermore, the results show that switching can have a stabilizing or destabilizing effect and can even lead to predator extinction.     

The growth-predation risk trade-off under a growing gape-limited predation threat

Growth is a critical ecological trait because it can determine population demography, evolution, and community interactions. Predation risk frequently induces decreased foraging and slow growth in prey. However, such strategies may not always be favored when prey can outgrow a predator's hunting ability. At the same time, a growing gape-limited predator broadens its hunting ability through time by expanding its gape and thereby creates a moving size refuge for susceptible prey. Here, I explore the ramifications of growing gape-limited predators for adaptive prey growth. A discrete demographic model for optimal foraging/growth strategies was derived under the realistic scenario of gape-limited and gape-unconstrained predation threats. Analytic and numerical results demonstrate a novel fitness minimum just above the growth rate of the gape-limited predator. This local fitness minimum separates a slow growth strategy that forages infrequently and accumulates low but constant predation risk from a fast growth strategy that forages frequently and experiences a high early predation risk in return for lower future predation risk and enhanced fecundity. Slow strategies generally were advantageous in communities dominated by gape-unconstrained predators whereas fast strategies were advantageous in gape-limited predator communities. Results were sensitive to the assumed relationships between prey size and fecundity and between prey growth and predation risk. Predator growth increased the parameter space favoring fast prey strategies. The model makes the testable predictions that prey should not grow at the same rate as their gape-limited predator and generally should grow faster than the fastest growing gape-limited predator. By focusing on predator constraints on prey capture, these results integrate the ecological and evolutionary implications of prey growth in diverse predator communities and offer an explanation for empirical growth patterns previously viewed to be anomalies.     

Consequences for predators of rescue and Allee effects on prey

The size of a population can be augmented by enriching the carrying capacity of its limiting resource, or by subsidising the renewal of the resource. The well known ‘paradox of enrichment’ models the first case, in which enrichment can force consumers and their limiting resource into destabilising limit cycles, whereas impoverishment stabilises the dynamics. In this paper we model the case of resource subsidy, where the resource is a limiting prey to predators. In contrast to enrichment, the system is stabilised by an influx of prey in the form of a rescue effect, and destabilised by an outflux of prey in the form of an Allee effect. Limit cycles are not sustained by the Allee effect; instead both populations collapse to zero over a large region of the predator-prey phase plane. The catastrophic extinction of prey requires the presence of both an Allee effect on prey and a predator with a type II functional response, though neither needs to contribute a large impact to prey dynamics. The novel implication is that consumers exaggerate the impact of Allee effects on a renewing resource. Conversely, an Allee effect in the form of a cull of resource, even of small value, can trigger local extinction of resource-dependent consumers.     

Local interactions between predators and prey call into question commonly used functional responses

Commonly used functional response models (Holling’s type I and type II models) assume that the encounter rate of a predator increases linearly with prey density, provided that the predator is searching for prey. In other other words, aN (a is the baseline encounter rate and N is prey density) describes the encounter rate. This study examined whether the models are adequate when predators and prey interact locally by using a spatially explicit individual based model because local interactions affect the spatial distribution of predators and prey, which also affects the encounter rate. Predators were assumed to possess a spatial perception range that influenced their foraging behavior (e.g., if a prey is in the perception range, the predator moves towards the prey). The effect of antipredator behavior by prey was also examined. The results suggest that prey and predator densities as well as handling time affect the baseline rate (i.e., parameter a) as opposed to the common assumption that the parameter is constant. The nature of model deviations depended on both the antipredator behavior and the predators’ perception range. Understanding these deviations is important as they qualitatively affect community dynamics.     

Harvesting creates ecological traps: consequences of invisible mortality risks in predator-prey metacommunities

Models of two-patch predator-prey metacommunities are used to explore how the global predator population changes in response to additional mortality in one of the patches. This could describe the dynamics of a predator in an environment that includes a refuge area where that predator is protected and a spatially distinct ("risky") area where it is harvested. The predator's movement is based on its perceived fitness in the two patches, but the risk from the additional mortality is potentially undetectable; this often occurs when the mortality is from human harvesting or from a novel type of top predator. Increases in undetected mortality in the risky area can produce an abrupt collapse of either the refuge population or of the entire predator population when the mortality rate exceeds a threshold level. This is due to the attraction of the risky patch, which has abundant prey due to its high predator mortality. Extinction of the refuge predator population does not occur when the refuge patch has a higher maximum per capita predator growth rate than the exploited patch because the refuge is then more attractive when the predator is rare. The possibility of abrupt extinction of one or both patches from high densities in response to a small increase in harvest is often associated with alternative states. In such cases, large reductions in mortality may be needed to avoid extinction in a collapsing predator population, or to reestablish an extinct population. Our analysis provides a potential explanation for sudden collapses of harvested populations, and it argues for more consideration of adaptive movement in designing protected areas.     

Prey-predator dynamics in rotifers: density-dependent consequences of spatial heterogeneity due to surface attachment

Classical models of prey-predator interactions assume that per capita prey consumption is dependent on prey density alone and that prey consumption (functional response) and consumer proliferation (numerical response) operate on the same timescales and without time lags. Several modifications have been proposed for resolving this timescale discrepancy, including variants where the functional response depends on both prey and predator densities. A microcosm system with the rotifer Brachionus 'Nevada' feeding on the prasinophyte Tetraselmis sp. showed significant (P < 0.0005) increases in steady-state biomasses of both prey and predators with increasing carrying capacity (represented by total phosphorus of the growth medium), which is inconsistent with predictions based on the traditional prey-only-dependent functional response. We provide data indicating that surfaces where the predator can attach provide a high-quality habitat for rotifers, which can result in a predator-dependent functional response. We also show that partitioning between the attached and free-swimming habitats was fast compared to the timescale of the numerical response. When attached to surfaces, rotifers maximized net energy gain by avoiding the high cost of swimming and by increased food capture due to reduced viscous drag. A mathematical model with prey-dependent functional response and wall-attached and free-swimming fractions of the population describes our data adequately. We discuss the implications of this finding for extrapolating microcosm experiments to systems with other surface-to-volume ratios, and to what extent our findings may apply to other popular model organisms for prey-predator interaction.     

Cost-Effective Suppression and Eradication of Invasive Predators

Abstract: Introduced predators can have pronounced effects on naïve prey species; thus, predator control is often essential for conservation of threatened native species. Complete eradication of the predator, although desirable, may be elusive in budget‐limited situations, whereas predator suppression is more feasible and may still achieve conservation goals. We used a stochastic predator–prey model based on a Lotka‐Volterra system to investigate the cost‐effectiveness of predator control to achieve prey conservation. We compared five control strategies: immediate eradication, removal of a constant number of predators (fixed‐number control), removal of a constant proportion of predators (fixed‐rate control), removal of predators that exceed a predetermined threshold (upper‐trigger harvest), and removal of predators whenever their population falls below a lower predetermined threshold (lower‐trigger harvest). We looked at the performance of these strategies when managers could always remove the full number of predators targeted by each strategy, subject to budget availability. Under this assumption immediate eradication reduced the threat to the prey population the most. We then examined the effect of reduced management success in meeting removal targets, assuming removal is more difficult at low predator densities. In this case there was a pronounced reduction in performance of the immediate eradication, fixed‐number, and lower‐trigger strategies. Although immediate eradication still yielded the highest expected minimum prey population size, upper‐trigger harvest yielded the lowest probability of prey extinction and the greatest return on investment (as measured by improvement in expected minimum population size per amount spent). Upper‐trigger harvest was relatively successful because it operated when predator density was highest, which is when predator removal targets can be more easily met and the effect of predators on the prey is most damaging. This suggests that controlling predators only when they are most abundant is the “best” strategy when financial resources are limited and eradication is unlikely.     

Four Factors Modifying the Effect of Competition on Carnivore Population Dynamics as Illustrated by African Wild Dogs

Abstract: Limitation of predator populations by prey availability and the effects of predators on prey populations are widely recognized as important ecological processes that affect carnivore conservation. Interspecific competition can also be a strong limiting factor for carnivore populations, and the effects of competition help explain why some carnivore species are prone to extinction. Competition among carnivores is unusual in some ways, so some predictions from traditional models of competition do not hold. For example, an increase in the density of prey can increase the effect of competition among carnivores, rather than weakening it. I used published data from African wild dogs (    Lycaon pictus ) to highlight four complexities that can modify the effects of competition on the population dynamics of carnivores: habitat fragmentation, counterintuitive effects of prey density, predator-prey size ratios, and habitat type.     

Individual- and population-level responses of a keystone predator to geographic variation in prey

Investigating how food supply regulates the behavior and population structure of predators remains a central focus of population and community ecology. These responses will determine the strength of bottom-up processes through the food web, which can potentially lead to coupled top-down regulation of local communities. However, characterizing the bottom-up effects of prey is difficult in the case of generalist predators and particularly with predators that have large dispersal scales, attributes that characterize most marine top predators. Here we use long-term data on mussel, barnacle, limpet, and other adult prey abundance and recruitment at sites spread over 970 km to investigate individual- and population-level responses of the keystone intertidal sunstar Heliaster helianthus on the coast of Chile. Our results show that this generalist predator responds to changes in the supply of an apparently preferred prey, the competitively dominant mussel Perumytilus purpuratus. Individual-level parameters (diet composition, per capita prey consumption, predator size) positively responded to increased mussel abundance and recruitment, whereas population-level parameters (density, biomass, size structure) did not respond to bottom-up prey variation among sites separated by a few kilometers. No other intertidal prey elicited positive individual predator responses in this species, even though a large number of other prey species was always included in the diet. Moreover, examining predator-prey correlations at approximately 80, 160, and 200 km did not change this pattern, suggesting that positive prey feedback could occur over even larger spatial scales or as a geographically unstructured process. Thus individual-level responses were not transferred to population changes over the range of spatial scales examined here, highlighting the need to examine community regulation processes over multiple spatial scales.     

Transient dynamics and the destabilizing effects of prey heterogeneity

The presence of prey heterogeneity and weakly interacting prey species is frequently viewed as a stabilizer of predator-prey dynamics, countering the destabilizing effects of enrichment and reducing the amplitude of population cycles. However, prior model explorations have largely focused on long-term, dynamic attractors rather than transient dynamics. Recent theoretical work shows that the presence of prey that are defended from predation can have strongly divergent effects on dynamics depending on time scale: prey heterogeneity can counteract the destabilizing effects of enrichment on predator-prey dynamics at long time scales but strongly destabilize systems during transient phases by creating long periods of low predator/prey abundance and increasing extinction probability (an effect that is amplified with increasing enrichment). We tested these general predictions using a planktonic system composed of a zooplankton predator and multiple algal prey. We first parameterized a model of our system to generate predictions and tested these experimentally. Our results qualitatively supported several model predictions. During transient phases, presence of defended algal prey increased predator extinctions at low and high enrichment levels compared to systems with only a single edible prey. This destabilizing effect was moderated at higher dilution rates, as predicted by our model. When examining dynamics beyond initial oscillations, presence of the defended prey increased predator-prey temporal variability at high nutrient enrichment but had no effect at low nutrient levels. Our results highlight the importance of considering transient dynamics when assessing the role of stabilizing factors on the dynamics of food webs.     

Prey–predator dynamics with predator switching regulated by a catabolic repression control mode

Since many predators can live under certain circumstances as saprophytes or consume more than one prey, and different enzymes are generally required for each prey or nutrient digestion, the predator must be sufficiently adaptive for effective utilization of the prey mass. Control modes as induction and repression, however, act at the level of genes and cause changes in the biosynthesis rate of these enzymes. In this work, an extension of the catabolic repression control mode from the level of genes to the level of the behavior of the predator is proposed, in order to model the balanced attack of the predator on the prey. It is demonstrated that, when the prey population has the competitive advantage over the predator (in using the common substrate), the catabolic repression mechanism favors the prey population, which dominates over the predator even at low specific dilution rate values, whereas, the stable steady or periodic coexistence state is not favored. When the predator has the competitive advantage at low substrate concentrations and the prey at high substrate concentrations, the introduction of the catabolic repression mechanism in the model again favors the stable steady state of the prey, while the coexistence region is dramatically reduced. Conversely, when the prey population has the competitive advantage at low and the predator at high substrate concentrations, dominance of prey and coexistence steady state could be favored by the catabolic repression mechanism. It is concluded that the catabolic repression control favors dominance of the prey population and, under certain circumstances, coexistence of both prey and predator populations.