Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Stochastic analysis of predator–prey type ecosystems

A stochastic model for the predator–prey type ecosystems in a random environment is proposed and investigated. The model is a variation of the Lotka–Volterra type with an additional self-competition mechanism within the prey population. Two different situations are considered: (1) saturation of predators, and (2) competition among predators. Changes in the birth rate of the preys and the death rate of the predators are modeled as random processes. The stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distribution of the system state variables at the state of statistical stationarity. Asymptotic behaviors of the system are also investigated. Effects on the ecosystem behaviors are evaluated of (1) prey self-competition, (2) predator saturation and predator competition, (3) random variation in the prey birth rate, and (4) random variation in the predator death rate.     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Predator-prey interactions in a nonequilibrium context: the metapopulation approach to modeling "hide-and-seek" dynamics in a spatially explicit tri-trophic system

Predators and prey are usually heterogeneously distributed in space so that the ability of the predators to respond to the distribution of their prey may have a profound influence on the stability and persistence of a predator‐prey system. A special type of dynamics is “hide‐and‐seek” characterized by a high turnover rate of local populations of prey and predators, because once the predators have found a patch of prey they quickly overexploit it, whereupon the starving predators either should move to better places or die. Continued persistence of prey and predators thus hinges on a long‐term balance between local extinctions and founding of new subpopulations. The colonization rate depends on the rate of emigration from occupied patches and the likelihood of successfully arriving at a suitable new patch, while extinction rate depends on the local population dynamics. Since extinctions and colonizations are both discrete probabilistic events, these phenomena are most adequately modeled by means of a stochastic model. In order to demonstrate the qualitative differences between a deterministic and stochastic approach to population dynamics, a spatially explicit tritrophic predator‐prey model is developed in a deterministic and a stochastic version. The model is parameterized using data for the two‐spotted spider mite (Tetranychus urticae) and the phytoseiid mite predator Phytoseiulus persimilis inhabiting greenhouse cucumbers.
Simulations show that the deterministic and stochastic approaches yield different results. The deterministic version predicts that the populations will exhibit violent fluctuations, implying that the system is fundamentally unstable. In contrast, the stochastic version predicts that the two species will be able to coexist in spite of frequent local extinctions of both species, provided the system consists of a sufficiently large number of local populations. This finding is in agreement with experimental results. It is therefore concluded that demographic stochasticity in combination with dispersal is capable of producing and maintaining sufficient asynchrony between local populations to ensure long‐term regional (metapopulation) persistence.     

Optimal foraging in patches: A case for stochasticity

Like much mathematical modeling in biology, most optimal foraging theory is developed from deterministic analogs of basically stochastic processes. Unlike other models, however, it cannot depend on laws of large numbers to justify this simplification; ignoring stochasticity can lead to wrong answers. This is demonstrated for a predator searching spatially separated patches of prey; it is shown that the choice of an optimal procedure for deciding when to leave a patch must be based on a stochastic model—a predator whose procedure is based on a deterministic model can do arbitrarily badly by comparison with the stochastic optimizer. A general solution is given, and its complexity suggests some objections to standard optimality arguments, and some possible alternatives.     

A stochastic model for simulation of interactions between phytophagous spider mites and their phytoseiid predators

A stochastic predator/prey model describing the interaction betweenTetranychus urticae andPhytoseiulus persimilis in investigated via computer simulations and pilot experiments on Lima beans in a greenhouse. Most demographic events, including predation, death due to unknown causes, dispersal, and oviposition, are modelled as stochastic processes. Transitions from eggs to nymphs and from nymphs to adults are deterministic, as are management decisions (release of predators and application of miticide). Computer simulations provide adequate and realistic representations of biological processes, and the model shows stability over a range of inputs. Experimental validation of the model continues. Predictions of the model for optimal predator release or optimal timing of acaricide application have yet to be tested experimentally.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

Predator-Induced Fleeing Behaviors in Phytoplankton: A New Mechanism for Harmful Algal Bloom Formation?

In the plankton, heterotrophic microbes encounter and ingest phytoplankton prey, which effectively removes >50% of daily phytoplankton production in the ocean and influences global primary production and biochemical cycling rates. Factors such as size, shape, nutritional value, and presence of chemical deterrents are known to affect predation pressure. Effects of movement behaviors of either predator or prey on predation pressure, and particularly fleeing behaviors in phytoplankton are thus far unknown. Here, we quantified individual 3D movements, population distributions, and survival rates of the toxic phytoplankton species, Heterosigma akashiwo in response to a ciliate predator and predator-derived cues. We observed predator-induced defense behaviors previously unknown for phytoplankton. Modulation of individual phytoplankton movements during and after predator exposure resulted in an effective separation of predator and prey species. The strongest avoidance behaviors were observed when H. akashiwo co-occurred with an actively grazing predator. Predator-induced changes in phytoplankton movements resulted in a reduction in encounter rate and a 3-fold increase in net algal population growth rate. A spatially explicit population model predicted rapid phytoplankton bloom formation only when fleeing behaviors were incorporated. These model predictions reflected field observations of rapid H. akashiwo harmful algal bloom (HAB) formation in the coastal ocean. Our results document a novel behavior in phytoplankton that can significantly reduce predation pressure and suggests a new mechanism for HAB formation. Phytoplankton behaviors that minimize predatory losses, maximize resource acquisition, and alter community composition and distribution patterns could have major implications for our understanding and predictive capacity of marine primary production and biochemical cycling rates.     

Contemporary evolution and genetic change of prey as a response to predator removal

The ecological effects of predator removal and its consequence on prey behavior have been investigated widely; however, predator removal can also cause contemporary evolution of prey resulting in prey genetic change. Here we tested the role of predator removal on the contemporary evolution of prey traits such as movement, reproduction and foraging. We use EcoSim simulation which allows complex intra- and inter-specific interactions, based on individual evolving behavioral models, as well as complex predator–prey dynamics and coevolution in spatially homogenous and heterogeneous worlds. We model organisms' behavior using fuzzy cognitive maps (FCM) that are coded in their genomes which has a clear semantics making reasoning about causality of any evolved behavior possible. We show that the contemporary evolution of prey behavior owing to predator removal is also accompanied by prey genetic change. We employed machine learning methods, now recognized as holding great promise for the advancement of our understanding and prediction of ecological phenomena. A classification algorithm was used to demonstrate the difference between genomes belonging to prey coevolving with predators and prey evolving in the absence of predation pressure. We argue that predator introductions to naive prey might be destabilizing if prey have evolved and adapted to the absence of predators. Our results suggest that both predator introductions and predator removal from an ecosystem have widespread effects on the survival and evolution of prey by altering their genomes and behavior, even after relatively short time intervals. Our study highlights the need to consider both ecological and evolutionary time scales, as well as the complex interplay of behaviors between trophic levels, in determining the outcomes of predator–prey interactions.     

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.     

Allee effect makes possible patchy invasion in a predator–prey system

The dynamics of interacting ecological populations results from the interplay between various deterministic and stochastic factors and this is particularly the case for the phenomenon of biological invasion. Whereas the spread of invasive species via propagation of a population front was shown to appear as a result of deterministic processes, the spread via formation, interaction and movement of separate patches has been recently attributed to the influence of environmental stochasticity. An appropriate understanding of the comparative importance of deterministic and stochastic mechanisms is still lacking, however. In this paper, we show that the patchy invasion appears to be possible also in a fully deterministic predator–prey model as a result of the Allee effect.     

High reproductive rates result in high predation risks: a mechanism promoting the coexistence of competing prey in spatially structured populations

I tested the hypothesis that spatial structure provides a trade-off between reproduction and predation risk and thereby facilitates predator-mediated coexistence of competing prey species. I compared a cellular automata model to a mean-field model of two prey species and their common predator. In the mean-field model, the prey species with the higher reproductive rate (the superior competitor) always outcompeted the other species (the inferior competitor), both in the presence of and the absence of the predator. In the cellular automata model, both prey species, which differed only in their reproductive rates, coexisted for a long time in the presence of their common predator at intermediate levels of predation. At low predation rates, the superior competitor dominated, while high predation rates favored the inferior competitor. This discrepancy in the results of the different models was due to a trade-off that spontaneously emerged in spatially structured populations; that is, the more clustered distribution of the superior competitor made it more susceptible to predation. In addition, coexistence of competing prey species declined with increasing dispersal ranges of either prey or predator, which suggests that the trade-off that results from spatial structure becomes less important as either prey or predator disperse over a broader range.     

Individual base model of predator-prey system shows predator dominant dynamics

Individual base model of predator-prey system is constructed. Both predator and prey species have age structure and cohorts of early reproductive age have competitive advantage. The model has linear functional response in predation behavior and includes the effect of interference among predators and delay of population growth from resource intake, not by functional response but by calculation procedure. Each foraging action is calculated successively and surplus or scarce of acquired resources is interpreted into population size through individual birth and death. This model shows that biomass of prey killed by predator is dependent on demand of predator and that heterogeneity in predator population is essential in persistency and stability of predator-prey system. Heterogeneity of predator makes predator individuals of less competing ability die rapidly. Rapid death of weak individuals causes rapid decrease of total demand of predator and that makes enough room for survived predators. Therefore, the biomass of killed prey is dependent on predator's demand. As young or infant population of predator are the more vulnerable to shortage of prey, and when many of them cannot survive to reproductive age, they can stabilize the system by wasting excessive prey with only temporal numerical increase of predator population.     

Stochastic modelling of prey depletion processes

The modelling of prey-predator interactions is of major importance for the understanding of population dynamics. Classically, these interactions are modelled using ordinary differential equations, but this approach has the drawbacks of assuming continuous population variables and of being deterministic. We propose a general approach to stochastic modelling based on the concept of functional response for a prey depletion process with a constant number of predators. Our model could involve any kind of functional response, and permits a likelihood-based approach to statistical modelling and stable computation using matrix exponentials. To illustrate the method we use the Holling-Juliano functional response and compare the outcomes of our model with a deterministic counterpart considered by Schenk and Bacher [2002. Functional response of a generalist insect predator to one of its prey species in the field. Journal of Animal Ecology 71 (3), 524-531], who observed the depletion of Cassida rubiginosa due to its exclusive predator, Polistes dominulus. The predation was found to be Holling type III, reflecting the ability of the predator to regulate its prey. Our approach corroborates this result, but suggests that the prey depletion census should have been performed more often, and that predation features were significantly different between the two years for which data are available.     

Predator dispersal determines the effect of connectivity on prey diversity

Linking local communities to a metacommunity can positively affect diversity by enabling immigration of dispersal-limited species and maintenance of sink populations. However, connectivity can also negatively affect diversity by allowing the spread of strong competitors or predators. In a microcosm experiment with five ciliate species as prey and a copepod as an efficient generalist predator, we analysed the effect of connectivity on prey species richness in metacommunities that were either unconnected, connected for the prey, or connected for both prey and predator. Presence and absence of predator dispersal was cross-classified with low and high connectivity. The effect of connectivity on local and regional richness strongly depended on whether corridors were open for the predator. Local richness was initially positively affected by connectivity through rescue of species from stochastic extinctions. With predator dispersal, however, this positive effect soon turned negative as the predator spread over the metacommunity. Regional richness was unaffected by connectivity when local communities were connected only for the prey, while predator dispersal resulted in a pronounced decrease of regional richness. The level of connectivity influenced the speed of richness decline, with regional species extinctions being delayed for one week in weakly connected metacommunities. While connectivity enabled rescue of prey species from stochastic extinctions, deterministic extinctions due to predation were not overcome through reimmigration from predator-free refuges. Prey reimmigrating into these sink habitats appeared to be directly converted into increased predator abundance. Connectivity thus had a positive effect on the predator, even when the predator was not dispersing itself. Our study illustrates that dispersal of a species with strong negative effects on other community members shapes the dispersal-diversity relationship. When connections enable the spread of a generalist predator, positive effects of connectivity on prey species richness are outweighed by regional extinctions through predation.     

A stochastic foraging model with predator training effects. II. Optimal diets

Standard optimal diet models require that a predator's behavior while searching for food does not change in response to experiences with individual prey. There is evidence for rapid and reversible changes in feeding behavior caused by as few as one or two prey encounters. When these “training effects” occur, a given prey type is more likely to be captured next if it was the last type with which the predator had experience. This is not compatible with the standard foraging model. I present a stochastic model which incorporates predator training effects, and three types of training are explored: training in the ability to detect prey (search image formation), training in the probability of succeeding in an attempted capture, and training in the time to pursue, capture, and eat prey. The main result is that all three types of training can result in optimal diets which do not obey the standard optimal diet rules. Conditions under which these rules will suffice are discussed.     

Restricting Prey Dispersal Can Overestimate the Importance of Predation in Trophic Cascades

Predators can affect prey populations and, via trophic cascades, predators can indirectly impact resource populations (2 trophic levels below the predator) through consumption of prey (density-mediated indirect effects; DMIEs) and by inducing predator-avoidance behavior in prey (trait-mediated indirect effects; TMIEs). Prey often employ multiple predator-avoidance behaviors, such as dispersal or reduced foraging activity, but estimates of TMIEs are usually on individual behaviors. We assessed direct and indirect predator effects in a mesocosm experiment using a marine food chain consisting of a predator (toadfish – Opsanus tau), prey (mud crab - Panopeus herbstii) and resource (ribbed mussel – Geukensia demissa). We measured dispersal and foraging activity of prey separately by manipulating both the presence and absence of the predator, and whether prey could or could not disperse into a predator-free area. Consumption of prey was 9 times greater when prey could not disperse, probably because mesocosm boundaries increased predator capture success. Although predator presence did not significantly affect the number of crabs that emigrated, the presence of a predator decreased resource consumption by prey, which resulted in fewer resources consumed for each prey that emigrated in the presence of a predator, and reduced the overall TMIE. When prey were unable to disperse, TMIEs on mussel survival were 3 times higher than the DMIEs. When prey were allowed to disperse, the TMIEs on resource survival increased to 11-times the DMIEs. We found that restricting the ability of prey to disperse, or focusing on only one predator-avoidance behavior, may be underestimating TMIEs. Our results indicate that the relative contribution of behavior and consumption in food chain dynamics will depend on which predator-avoidance behaviors are allowed to occur and measured.     

Effects of deterministic and random refuge in a prey-predator model with parasite infection

Most natural ecosystem populations suffer from various infectious diseases and the resulting host-pathogen dynamics is dependent on host's characteristics. On the other hand, empirical evidences show that for most host pathogen systems, a part of the host population always forms a refuge. To study the role of refuge on the host-pathogen interaction, we study a predator-prey-pathogen model where the susceptible and the infected prey can undergo refugia of constant size to evade predator attack. The stability aspects of the model system is investigated from a local and global perspective. The study reveals that the refuge sizes for the susceptible and the infected prey are the key parameters that control possible predator extinction as well as species co-existence. Next we perform a global study of the model system using Lyapunov functions and show the existence of a global attractor. Finally we perform a stochastic extension of the basic model to study the phenomenon of random refuge arising from various intrinsic, habitat-related and environmental factors. The stochastic model is analyzed for exponential mean square stability. Numerical study of the stochastic model shows that increasing the refuge rates has a stabilizing effect on the stochastic dynamics.     

Food-signal theory: population regulation and the functional response

A fully stochastic food-signal model, a function of a patchy-preyencounter sequence, and a prey-processing function is described.The model shows how prey density and its second-order statisticalproperties can sequester prey from predators, questioning theuse of only numerical abundance of predator and prey organismsas a measure of preya — predator interactions. The modelhighlights the notion that patch structure can be generatedby relative velocity of predator and prey as well as by theirspatial distribution. The model extends ideas that include the‘biological pump’ and the downwelling of carbonfrom the upper ocean, the functional response, optimal-foragingtheory, and the connections between population dynamics andvariability in the physical environment.     

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.