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.     

Why ratio dependence is (still) a bad model of predation

The history of the idea that predation rates are functions of the ratio of prey density to predator density, known as ratio dependence, is reviewed and updated. When the term was introduced in 1989, it was already known that higher predator abundance often reduced an individual predator's average intake rate of prey. However, the idea that this effect was a universally applicable inverse proportionality was new. That idea was widely criticized in many articles in the early 1990s, and many of these criticisms have never been addressed. Nevertheless, ratio dependence seems to be gaining in popularity and is the subject of a recent monograph by the originators. This article revisits the most important objections to this theory, and assesses to what extent they have been answered by the theory's proponents. In this process, several new objections are raised. The counterarguments begin with the lack of a plausible, generally applicable mechanism that could produce ratio dependence. They include the fact that ratio dependence is a special case of predator‐density effects, which, in turn, are only one of many non‐prey species effects that influence the consumption rate of a particular prey. The proclaimed simplicity advantage of ratio dependence is at best small and is outweighed by its disadvantages; it predicts biologically implausible phenomena, and cannot easily be extended to describe multi‐species systems, trait‐mediated interactions, coevolution, and a number of other important ecological phenomena. Any potential small simplicity advantage disappears with corrections to remove unrealistic low‐density dynamics caused by ratio dependence. The frequent occurrence of strong predator dependence does not make ratio dependence a better ‘default’ model of predation than prey dependence, and empirical studies of the full range of non‐prey species effects on the consumption rates of predators are needed.     

A spatial theory for characterizing predator–multiprey interactions in heterogeneous landscapes

Trophic interactions in multiprey systems can be largely determined by prey distributions. Yet, classic predator–prey models assume spatially homogeneous interactions between predators and prey. We developed a spatially informed theory that predicts how habitat heterogeneity alters the landscape-scale distribution of mortality risk of prey from predation, and hence the nature of predator interactions in multiprey systems. The theoretical model is a spatially explicit, multiprey functional response in which species-specific advection–diffusion models account for the response of individual prey to habitat edges. The model demonstrates that distinct responses of alternative prey species can alter the consequences of conspecific aggregation, from increasing safety to increasing predation risk. Observations of threatened boreal caribou, moose and grey wolf interacting over 378 181 km² of human-managed boreal forest support this principle. This empirically supported theory demonstrates how distinct responses of apparent competitors to landscape heterogeneity, including to human disturbances, can reverse density dependence in fitness correlates.     

Individual variation in prey choice in a predator-prey community

One predator-two prey community models are studied with an emphasis on individual variation in predator behavior. The predator behaves according to a well-known prey choice model. The behavioral model predicts that predators should always attack the primary prey (more profitable prey of the two), but only attack the alternative prey (less profitable prey of the two) when the density of the primary prey is below a threshold density. The predator that accepts the alternative prey does not discriminate between the primary and alternative prey (all-or-nothing preference for the alternative prey). However, empirical studies do not result in clear all-or-nothing responses. Previous models examined the relaxation of the all-or-nothing response by assuming partial preference (e.g., predators preferentially forage on the primary prey even when they also attack the alternative prey). In this study, I consider individual variation in two predator traits (prey density perception and handling time) as the sources of the variation in the threshold density, which can make empirical data appear deviated from the expectation. I examine how community models with partial preference and individual variation differ in their dynamics and show that the differences can be substantial. For example, the dynamics of a model based on individual variation can be more stable (e.g., stable in a wider parameter region) than that of a model based on partial preference. As the general statistical property (Jensen’s inequality) is a main factor that causes the differences, the results of the study have general implications to the interpretation of models based on average per-capita rates.     

Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment

The Lotka-Volterra predator-prey model with prey density dependence shows the final prey density to be independent of its vital rates. This result assumes the community to be well mixed so that encounters between predators and prey occur as a product of the landscape densities, yet empirical evidence suggests that over small spatial scales this may not be the normal pattern. Starting from an individual-based model with neighborhood interactions and movements, a deterministic approximation is derived, and the effect of local spatial structure on equilibrium densities is investigated. Incorporating local movements and local interactions has important consequences for the community dynamics. Now the final prey density is very much dependent on its birth, death, and movement rates and in ways that seem counterintuitive. Increasing prey fecundity or mobility and decreasing the coefficient of competition can all lead to decreases in the final density of prey if the predator is also relatively immobile. However, analysis of the deterministic approximation makes the mechanism for these results clear; each of these changes subtly alters the emergent spatial structure, leading to an increase in the predator-prey spatial covariance at short distances and hence to a higher predation pressure on the prey.     

Mathematical model study of the dynamics of an exploited population of the common fox (Vulpes vulpes L.)

The observed dependence of intensity of hunting intensity and the share of reproducing females on fox population density was shown to cause the existence of several stable equilibrium values of fox numbers and incompatibility of requirements for both maximal prey and prey stability. Organization of forbid forests is an acceptable way to regulate hunting in this situation.     

Ambulatory dispersal behavior ofNeoseiulus fallacis (Acarina: Phytoseiidae) in relation to prey density and temperature

Ambulatory dispersal behavior ofNeoseiulus fallacis (Garman) was studied in the laboratory to evaluate within-plant movement in relation to temperature and prey density. Adult femaleN. fallacis were confined in 2.5-cm-diameter arenas on the abaxial surface of excised corn leaves. Four temperatures (23, 28, 33, and 39° C) and prey densities ranging from 0 to 55 spider mite eggs per cm² were used. The walking paths of these mites were traced, digitized and used to calculate turning angles, walking speeds and turning rates. A computer simulation of walking behavior used this information to model mite ambulatory behavior and predict dispersal rates.Neoseiulus fallacis behavior while on whole corn leaves was quantified to verify the results of the simulation. The results showed thatN. fallacis will follow a leaf or arena edge (edge-walking) at all temperatures and prey densities. In addition, this behavior was used to the exclusion of the other types of behavior such as resting, and random-walk type search when prey egg density was less than 4 eggs per cm². The exclusion of edge-walking behavior from the model caused the model to underestimate substantially the dispersal rates leaves. These data suggest that there are at least two recognizable types of ambulatory search used byN. fallacis—the random-walk type, which is used when prey density is high (searching within prey patches), and the edge-walking behavior, which is used when prey density is low. This behavior allows the mite to travel rapidly from leaf to leaf in search of new prey patches.     

Stabilizing effects of spatial heterogeneity in predator-prey systems

A general predator is assumed to divide its hunting time between two sub-habitats with different prey species, spending a larger fraction (φ) of search time in an area as the relative prey abundance there increases. This always causes switching in the model, and changes a functional response from one that imposes a risk on the average prey that decreases with prey density in the direction of one that imposes an increasing risk. I discuss the conditions for a response that is density dependent, and those predatory attributes that make such a response more likely. Transit time between subhabitats always increases the density dependent effect, and is necessary for “system stability” in a Lotka-Volterra model with two prey species. Experiments have confirmed the model's basic assumption. General predators do not fit easily into classical predator-prey models of simple “closed” communities, and then the degree of density dependence of the functional response becomes a useful measure of a predator's short-term stabilizing effect on a prey species. The model demonstrates how spatial heterogeneity can be stabilizing.     

Bacterial Predator-Prey Interaction at Low Prey Density

A bacterial predator-prey interaction was studied using Bdellovibrio and bioluminescent prey bacteria. The attacking bdellovibrio causes decay of bioluminescence, which is correlated with bdellovibrio penetration into the prey. The behavior of the prey and predator populations over time was found to be well described by a Lotka-Volterra model. By using this model, the probability of bdellovibrio penetration after encountering a prey cell was found to be approximately 3.0%. The prey density required to give the bdellovibrios a 50% chance of survival was calculated to be at least 3.0 × 10⁶ cells per ml, and the density required for population equilibria was calculated to be about 7 × 10⁵ prey bacteria per ml. These values, not generally characteristic of natural habitats, suggest that the existence of Bdellovibrio in nature is limited to special ecological niches.     

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.     

Age structure effects in predator-prey interactions

A mathematical model of predator-prey interactions is proposed which incorporates both age structure in the predators and density dependence in the prey. The properties of the model are investigated by a linearized analysis, which enables the conditions for stability to be formulated. The analysis indicates that for a substantial domain of parameter space, a stable equilibrium is possible with the prey well below its carrying capacity. The effect of violating the stability conditions on the behaviour of the model was investigated by computer simulation. Two further types of behaviour are noted in which coexistence is possible. The first is a two point limit cycle in which young and old predators occur in alternate time periods. The second involves a limit cycle in which the component population trajectories lie on closed curves in phase space.     

Predator–prey encounter rates in freshwater piscivores: effects of prey density and water transparency

One of the most fundamental components of predator–prey models is encounter rate, modelled as the product of prey density and search efficiency. Encounter rates have, however, rarely been measured in empirical studies. In this study, we used a video system approach to estimate how encounter rates between piscivorous fish that use a sit-and-wait foraging strategy and their prey depend on prey density and environmental factors such as turbidity. We first manipulated prey density in a controlled pool and field enclosure experiments where environmental factors were held constant. In a correlative study of 15 freshwater lakes we then estimated encounter rates in natural habitats and related the results to both prey fish density and environmental factors. We found the expected positive dependence of individual encounter rates on prey density in our pool and enclosure experiments, whereas the relation between school encounter rate and prey density was less clear. In the field survey, encounter rates did not correlate with prey density but instead correlated positively with water transparency. Water transparency decreases with increasing prey density along the productivity gradient and will reduce prey detection distance and thus predator search efficiency. Therefore, visual predator–prey encounter rates do not increase, and may even decrease, with increasing productivity despite increasing prey densities.     

Density dependence in resource exploitation: empirical test of Levins' metapopulation model

Levins' model of metapopulation dynamics is modified to incorporate variable degrees of density dependence in the per capita exploitation of resource patches. We demonstrate a simple means of testing for this density dependence in a sample of metapopulations, each at its equilibrium balance of local colonization to extinction. The fraction of habitable unoccupied patches equilibrates to a constant number under the null model of density independent colonization, and to a constant proportion under strong density dependence. We compare the null model to two density dependent alternatives, using data on exploitation of nest boxes by collared flycatchers Ficedula albicollis . The analysis shows how predicted trends in the equilibrium unoccupied fraction are similar for both spatial interference and net immigration. This needs to be recognized, since the null hypothesis of a constant unused resource applies also to the dynamics of consumable resources, where it is expressed in a constant stock of uneaten prey at the dynamic equilibrium of predators to prey.     

Emergent Allee effects in top predators feeding on structured prey populations

Top predators that forage in a purely exploitative manner on smaller stages of a size-structured prey population have been shown to exhibit an Allee effect. This Allee effect emerges from the changes that predators induce in the prey-population size distribution and represents a feedback of predator density on its own performance, in which the feedback operates through and is modified by the life history of the prey. We demonstrate that these emergent Allee effects will occur only if the prey, in the absence of predators, is regulated by density dependence in development through one of its juvenile stages, as opposed to regulation through adult fecundity. In particular, for an emergent Allee effect to occur, over-compensation is required in the maturation rate out of the regulating juvenile stage, such that a decrease in juvenile density will increase the total maturation rate to larger/older stages. If this condition is satisfied, predators with negative size selection, which forage on small prey, exhibit an emergent Allee effect, as do predators with positive size selection, which forage on large adult prey. By contrast, predators that forage on juveniles in the regulating stage never exhibit emergent Allee effects. We conclude that the basic life-history characteristics of many species make them prone to exhibiting emergent Allee effects, resulting in an increased likelihood that communities possess alternative stable states or exhibit catastrophic shifts in structure and dynamics.     

Influence of density dependence on predator-prey seabird interactions at large spatio-temporal scales

Theoretical investigations of competitive dynamics have noted that numbers of predator and prey influence each other. However, few empirical studies have demonstrated how a life-history trait of the prey (such as fecundity) can be affected simultaneously by its own density and the density of predators. For instance, density dependence can reduce fecundity with increasing number of prey, while inverse density dependence or Allee effects may occur especially when the prey is a social organism. Here we analysed an intraguild predator-prey system of two seabird species at a large spatio-temporal scale. As expected, we found that fecundity of prey was negatively affected by predator density. Nevertheless, fecundity of prey also increased nonlinearly with its own density and strikingly with the prey-predator ratio. Small groups of prey were probably not able to defend their nests especially against large number of predators. At the highest prey densities (i.e. when anti-predator strategies should be most efficient), prey fecundity also lowered, suggesting the appearance of density dependence mediated by food competition. Allee effects and density dependence occurred across a broad range of population sizes of both the prey and the predator at several local populations facing different ecological environments.     

Size and scaling of predator-prey dynamics

We propose a scaled version of the Rosenzweig–MacArthur model using both Type I and Type II functional responses that incorporates the size dependence of interaction rates. Our aim is to link the energetic needs of organisms with the dynamics of interacting populations, for which survival is a result of a game-theoretic struggle for existence. We solve the scaled model of predator–prey dynamics and predict population level characteristics such as the scaling of coexistence size ranges and the optimal predator–prey size ratio. For a broad class of such models, the optimal predator–prey size ratio given available prey of a fixed size is constant. We also demonstrate how scaling predictions of prey density differ under resource limitation vs. predator drawdown. Finally, we show how evolution of predator size can destabilize population dynamics, compare scaling of predator–prey cycles to previous work, as well as discuss possible extensions of the model to multispecies communities.     

Predator‐mediated interactions between preferred,alternative and incidental prey in the arctic tundra

Apparent competition between prey is hypothesized to occur more frequently in environments with low densities of preferred prey, where predators are forced to forage for multiple prey items. In the arctic tundra, numerical and functional responses of predators to preferred prey (lemmings) affect the predation pressure on alternative prey (goose eggs) and predators aggregate in areas of high alternative prey density. Therefore, we hypothesized that predation risk on incidental prey (shorebird eggs) would increase in patches of high goose nest density when lemmings were scarce. To test this hypothesis, we measured predation risk on artificial shorebird nests in quadrats varying in goose nest density on Bylot Island (Nunavut, Canada) across three summers with variable lemming abundance. Predation risk on artificial shorebird nests was positively related to goose nest density, and this relationship was strongest at low lemming abundance when predation risk increased by 600% as goose nest density increased from 0 to 12 nests ha^?1. Camera monitoring showed that activity of arctic foxes, the most important predator, increased with goose nest density. Our data support our incidental prey hypothesis; when preferred prey decrease in abundance, predator mediated apparent competition via aggregative response occurs between the alternative and incidental prey items.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

A discrete time stochastic model of a two prey, one predator species interaction

A stochastic discrete time model of a two prey, one predator interaction, an extension of one and two species models proposed by Leslie (1958) and Leslie and Gower, 1958, Leslie and Gower, 1960, is studied. Monte Carlo simulations and the stability properties of the analogous continuous time deterministic model suggest the following hypotheses. (1) The two prey, one predator interaction is in general unstable. The range of parameters allowing coexistence of all three species is small. (2) Deterministically the predator always survives. (3) If the parameters defining the effects of density on the rates of population growth are large, the simulations lead to the rapid extinction of all three species or all but one of the prey species even if the interaction is deterministically stable. (4) The outcome of this three species interaction is largely probabilistic over a wide range of parameters. (5) A prey species with a competitive advantage over a second prey species may still find it difficult to invade and displace the second prey species if the density of the second prey species is high. Increasing the density of the predator offsets this numerical advantage somewhat. (6) The introduction of a predator common to two noncompeting species of prey usually leads to the extinction of one of the prey species. (7) In a stable two prey, one predator interaction the fluctuations of the two prey species are nonperiodic and erratic. The fluctuations of the rarer prey species are damped relative to the commoner species and the fluctuations of the rarer prey species behave as if the series has no fixed mean abundance. The predator population fluctuates with a remarkably constant period. The relevance of these hypotheses to the problem of relating population stability and persistence with the number of species in a community is discussed.     

Population dynamical consequences of reduced predator switching at low total prey densities

Several models of rapid switching by a predator in a two-prey environment are analyzed. The goal is to determine how the dynamics of the system and the potential indirect effects between prey are affected by the dependence of switching on total prey density. In exploring this question, the difference between the population-level consequences of switching in stable and cycling predator-prey systems is also examined. We concentrate on reduced switching at low densities, a feature that is likely because of the difficulty of distinguishing between two very low densities. The main findings are: (1) switching in unstable systems can produce positive indirect effects of one prey species on the other; and (2) reduced switching at low densities can greatly alter the dynamics of the system and the indirect effects between prey. Both of the possibilities are only evident in cycling systems. Reduced switching at low total prey densities leads to heavier predation on the slower-growing prey when both prey species are rare. As a consequence, there is a lag in the recovery of the slower-growing prey species after predator densities fall, and the dynamics of the two prey become desynchronized. The net result is increased indirect interactions between prey, and a greater likelihood of exclusion of the slower growing prey. The analysis of these models suggests a need for more empirical work to determine whether switching is reduced by very low total prey densities, and to study the long-term dynamics that occur in systems with switching predators.