Identifying predator-prey processes from time-series

The functional response is a key element in predator-prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator-prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

Should "handled" prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory

Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered.     

Modeling changes in predator functional response to prey across spatial scales

Extrapolation of predator functional responses from laboratory observations to the field is often necessary to predict predation rates and predator-prey dynamics at spatial and temporal scales that are difficult to observe directly. We use a spatially explicit individual-based model to explore mechanisms behind changes in functional responses when the scale of observation is increased. Model parameters were estimated from a predator-prey system consisting of the predator Delphastus catalinae (Coleoptera: Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) on tomato plants. The model explicitly incorporates prey and predator distributions within single plants, the search behavior of predators within plants, and the functional response to prey at the smallest scale of interaction (within leaflets) observed in the laboratory. Validation revealed that the model is useful in scaling up from laboratory observations to predation in whole tomato plants of varying sizes. Comparing predicted predation at the leaflet scale, as observed in laboratory experiments, with predicted predation on whole plants revealed that the predator functional response switches from type II within leaflets to type III within whole plants. We found that the magnitude of predation rates and the type of functional response at the whole plant scale are modulated by (1) the degree of alignment between predator and prey distributions and (2) predator foraging behavior, particularly the effect of area-concentrated search within plants when prey population density is relatively low. The experimental and modeling techniques we present could be applied to other systems in which active predators prey upon sessile or slow-moving species.     

A functional response model of a predator population foraging in a patchy habitat

1. Functional response models (e.g. Holling's disc equation) that do not take the spatial distributions of prey and predators into account are likely to produce biased estimates of predation rates. 2. To investigate the consequences of ignoring prey distribution and predator aggregation, a general analytical model of a predator population occupying a patchy environment with a single species of prey is developed. 3. The model includes the density and the spatial distribution of the prey population, the aggregative response of the predators and their mutual interference. 4. The model provides explicit solutions to a number of scenarios that can be independently combined: the prey has an even, random or clumped distribution, and the predators show a convex, sigmoid, linear or no aggregative response. 5. The model is parameterized with data from an acarine predator-prey system consisting of Phytoseiulus persimis and Tetranychus urticae inhabiting greenhouse cucumbers. 6. The model fits empirical data quite well and much better than if prey and predators were assumed to be evenly distributed among patches, or if the predators were distributed independently of the prey. 7. The analyses show that if the predators do not show an aggregative response it will always be an advantage to the prey to adopt a patchy distribution. On the other hand, if the predators are capable of responding to the distribution of prey, then it will be an advantage to the prey to be evenly distributed when its density is low and switch to a more patchy distribution when its density increases. The effect of mutual interference is negligible unless predator density is very high. 8. The model shows that prey patchiness and predator aggregation in combination can change the functional response at the population level from type II to type III, indicating that these factors may contribute to stabilization of predator-prey dynamics.     

Impacts of Foraging Facilitation Among Predators on Predator-prey Dynamics

Whereas impacts of predator interference on predator-prey dynamics have received considerable attention, the “inverse” process—foraging facilitation among predators—have not been explored yet. Here we show, via mathematical models, that impacts of foraging facilitation on predator-prey dynamics depend on the way this process is modeled. In particular, foraging facilitation destabilizes predator-prey dynamics when it affects the encounter rate between predators and prey. By contrast, it might have a stabilizing effect if the predator handling time of prey is affected. Foraging facilitation is an Allee effect mechanism among predators and we show that for many parameters, it gives rise to a demographic Allee effect or a critical predator density in need to be crossed for predators to persist. We explore also the effects of predator interference, to make the picture “symmetric” and complete. Predator interference is shown to stabilize predator-prey dynamics once its strength is not too high, and thus corroborates results of others. On the other hand, there is a wide range of model parameters for which predator interference gives rise to three co-occurring co-existence equilibria. Such a multi-equilibrial regime is rather robust as we observe it for all the functional response types we explore. This is a previously unreported phenomenon which we show cannot occur for the Beddington–DeAngelis functional response. An interesting topic for future research thus might be to seek for general conditions on predator functional responses that would produce multiple co-existence equilibria in a predator-prey model.     

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 comparison of two predator-prey models with Holling's type I functional response

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.     

Predator functional response changed by induced defenses in prey

Functional responses play a central role in the nature and stability of predator-prey population dynamics. Here we investigate how induced defenses affect predator functional responses. In experimental communities, prey (Paramecium) expressed two previously undocumented inducible defenses--a speed reduction and a width increase--in response to nonlethal exposure to predatory Stenostomum. Nonlethal exposure also changed the shape of the predator's functional response from Type II to Type III, consistent with changes in the density dependence of attack rates. Handling times were also affected by prey defenses, increasing at least sixfold. These changes show that induced changes in prey have a real defensive function. At low prey densities, induction led to lower attack success; at high prey densities, attack rates were actually higher for induced prey. However, induction increased handling times sufficiently that consumption rates of defended prey were lower than those of undefended prey. Modification of attack rate and handling time has important potential consequences for population dynamics; Type III functional responses can increase the stability of population dynamics and persistence because predation on small populations is low, allowing a relict population to survive. Simulations of a predator-prey population dynamic model revealed the stabilizing potential of the Type III response.     

The nature of predation: prey dependent, ratio dependent or neither?

To describe a predator-prey relationship, it is necessary to specify the rate of prey consumption by an average predator. This functional response largely determines dynamic stability, responses to environmental influences and the nature of indirect effects in the food web containing the predator-prey pair. Nevertheless, measurements of functional responses in nature are quite rare. Recently, much work has been devoted to comparing two idealized forms of the functional response: prey dependent and ratio dependent. Although we agree that predator abundance often affects the consumption rate of individual predators, this phenomenon requires more attention. Disagreement remains over which of the two idealized responses serves as a better starting point in building models when data on predator dependence are absent.     

Global stability and periodic orbits for two-patch predator-prey diffusion-delay models

We consider a predator-prey system where the prey can diffuse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation. We assume a Volterra within-patch dynamics, and we assume further that the benefit for the predator comes also from predation in the past through an exponential-delay memory function. By homotopy techniques we prove that, if the prey diffusion is weak enough, then a nonzero globally stable equilibrium exists. This result essentially depends upon the self-regulating coefficient of the predator. If we put this coefficient equal to zero, assuming that the predator density is regulated only by predation, then we can prove the existence of a Hopf bifurcating orbit from the positive equilibrium. The main cause of periodic orbits is the time delay in the predator response functional. We prove that diffusion, lack of delay in the predator response, and increase in the rate of the exponential decay of the memory play stabilizing roles.     

A mechanistic model for interference and Allee effect in the predator population

We present the results of simulations in an individual-based model describing spatial movement and predator-prey interaction within a closed rectangular habitat. Movement of each individual animal is determined by local conditions only, so any collective behavior emerges owing to self-organization. It is shown that the pursuit of prey by predators entails predator interference, manifesting itself at the population level as the dependency of the trophic function (individual ration) on predator abundance. The stabilizing effect of predator interference on the dynamics of a predator-prey system is discussed. Inclusion of prey evasion induces apparent cooperation of predators and further alters the functional response, giving rise to a strong Allee effect, with extinction of the predator population upon dropping below critical numbers. Thus, we propose a simple mechanistic interpretation of important but still poorly understood behavioral phenomena that underlie the functioning of natural trophic systems.     

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.     

Density-dependent effects of prey defences

In this study, we show that the protective advantage of a defence depends on prey density. For our investigations, we used the predator-prey model system Chaoborus-Daphnia pulex. The prey, D. pulex, forms neckteeth as an inducible defence against chaoborid predators. This morphological response effectively reduces predator attack efficiency, i.e. number of successful attacks divided by total number of attacks. We found that neckteeth-defended prey suffered a distinctly lower predation rate (prey uptake per unit time) at low prey densities. The advantage of this defence decreased with increasing prey density. We expect this pattern to be general when a defence reduces predator success rate, i.e. when a defence reduces encounter rate, probability of detection, probability of attack, or efficiency of attack. In addition, we experimentally simulated the effects of defences which increase predator digestion time by using different sizes of Daphnia with equal vulnerabilities. This type of defence had opposite density-dependent effects: here, the relative advantage of defended prey increased with prey density. We expect this pattern to be general for defences which increase predator handling time, i.e. defences which increase attacking time, eating time, or digestion time. Many defences will have effects on both predator success rate and handling time. For these defences, the predator’s functional response should be decreased over the whole range of prey densities. Received: 15 September 1999 / Accepted: 23 December 1999     

Simulation model of a predator-prey system comprised ofPhytoseiulus persimilis andTetranychus urticae. II. Model sensitivity to variations in the life history parameters of both species and to variations in the functional response and components of the numerical response

A detailed sensitivity analysis of a model of a predator-prey system comprised of Tetranychus urticae and Phytoseiulus persimilis was performed. The aim was to assess the relative importance of the life history parameters of both species, the functional response, and the components of the numerical response. In addition, the impact of the initial predator-prey ratio and the timing of predator introduction were tested. Results indicated that the most important factors in the system were relative rates of predator and prey development, the time of onset of predator oviposition, and the mode of the predator's oviposition curve. The total oviposition of the predator, the effect of prey consumption on predator oviposition, and predator searching were important under some conditions. Factors of moderate importance were the adult female predator's functional response, total prey oviposition, the mode of the prey's oviposition curve, abiotic mortality of the pre-adult predator, and the effect of prey consumption on predator development and on the immature predator's mortality. Factors of least importance were the variances of the predator's and prey's oviposition curves, the abiotic mortality of the adult predator, the abiotic mortality of the pre-adult and adult prey, the functional response of the nymphal and adult male predators, and the effect of prey consumption on adult predator mortality. The sex ratios had little effect, except when the proportion of female predators was very low. The initial predator-prey ratio and time of predator introduction had significant impacts on system behavior, though the patterns of impact were different.     

Sigmoid functional responses and population stability

Sigmoid functional responses are known to stabilize the differential Lotka-Volterra predator-prey model. However, we have found that they have no such effect in a comparable discrete generation model. The difficulty in stabilizing this model results from the one-generation time delay between changes in predator population density and the level of prey mortality. By contrast, sigmoid functional responses can stabilize the system if the predator population remains relatively constant, as is more likely of generalist predators.     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

Effects of a disease affecting a predator on the dynamics of a predator-prey system

We study the effects of a disease affecting a predator on the dynamics of a predator-prey system. We couple an SIRS model applied to the predator population, to a Lotka-Volterra model. The SIRS model describes the spread of the disease in a predator population subdivided into susceptible, infected and removed individuals. The Lotka-Volterra model describes the predator-prey interactions. We consider two time scales, a fast one for the disease and a comparatively slow one for predator-prey interactions and for predator mortality. We use the classical “aggregation method” in order to obtain a reduced equivalent model. We show that there are two possible asymptotic behaviors: either the predator population dies out and the prey tends to its carrying capacity, or the predator and prey coexist. In this latter case, the predator population tends either to a “disease-free” or to a “disease-endemic” state. Moreover, the total predator density in the disease-endemic state is greater than the predator density in the “disease-free” equilibrium (DFE).     

A powerful truncated tail strength method for testing multiple null hypotheses in one dataset

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.     

Testing the validity of functional response models using molecular gut content analysis for prey choice in soil predators

Analysis of predator–prey interactions is a core concept of animal ecology, explaining structure and dynamics of animal food webs. Measuring the functional response, i.e. the intake rate of a consumer as a function of prey density, is a powerful method to predict the strength of trophic links and assess motives of prey choice, particularly in arthropod communities. However, due to their reductionist set‐up, functional responses, which are based on laboratory feeding experiments, may not display field conditions, possibly leading to skewed results. Here, we tested the validity of functional responses of centipede predators and their prey by comparing them with empirical gut content data from field‐collected predators. Our predator–prey system included lithobiid and geophilomorph centipedes, abundant and widespread predators of forest soils and their soil‐dwelling prey. First, we calculated the body size‐dependent functional responses of centipedes using a published functional response model in which we included natural prey abundances and animal body masses. This allowed us to calculate relative proportions of specific prey taxa in the centipede diet. In a second step, we screened field‐collected centipedes for DNA of eight abundant soil‐living prey taxa and estimated their body size‐dependent proportion of feeding events. We subsequently compared empirical data for each of the eight prey taxa, on proportional feeding events with functional response‐derived data on prey proportions expected in the gut, showing that both approaches significantly correlate in five out of eight predator–prey links for lithobiid centipedes but only in one case for geophilomorph centipedes. Our findings suggest that purely allometric functional response models, which are based on predator–prey body size ratios are too simple to explain predator–prey interactions in a complex system such as soil. We therefore stress that specific prey traits, such as defence mechanisms, must be considered for accurate predictions.     

Identifying Predator–Prey Processes from Time-Series

The functional response is a key element in predator–prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator–prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.