Models Of Predator-prey Interactions Research Articles

Natural History of Mass-action in Predator-prey Models: A Case Study from Wolf Spiders and Grasshoppers

Abstract Mass-action models of predator-prey interactions assume that predators encounter prey according to their relative densities as scaled by functional responses, although models seldom specify critical natural history and behavioral mechanisms that ensure that encounters actually occur. As a case study of this assumption, we assess the hypothesis that daily and seasonal activity and microhabitat use by wandering wolf spiders (Lycosidae: Schizocosa) searching for four common grasshopper species (Orthoptera: Acrididae) are coincident under natural conditions. There was great overlap in seasonal phenology and use of microhabitats between spiders and grasshoppers. Grasshoppers that were suitably sized (10–20 mm in length) as prey for spiders were relatively abundant from late spring through summer in this grassland. Three of the four common grasshopper species used microhabitat in a similar way, but differed from a fourth common species, Phoetaliotes nebrascensis. However, when they were active, spiders...

Fluctuations and correlations in lattice models for predator-prey interaction

Including spatial structure and stochastic noise invalidates the classical Lotka-Volterra picture of stable regular population cycles emerging in models for predator-prey interactions. Growth-limiting terms for the prey induce a continuous extinction threshold for the predator population whose critical properties are in the directed percolation universality class. We discuss the robustness of this scenario by considering an ecologically inspired stochastic lattice predator-prey model variant where the predation process includes next-nearest-neighbor interactions. We find that the corresponding stochastic model reproduces the above scenario in dimensions 1< d < or =4, in contrast with the mean-field theory, which predicts a first-order phase transition. However, the mean-field features are recovered upon allowing for nearest-neighbor particle exchange processes, provided these are sufficiently fast.

Comparison of predator–prey interaction models for fish assemblages from the neotropical region

Predation is one of the main fish assemblages structuring force in many aquatic ecosystems. Finding a functional relationship between predator and prey that improves our understanding of this process is a challenge to ecologists. In order to evaluate this interaction, several models have been created, each one of them representing a specific biological mechanism. Although, these models have not been intensively confronted against empirical data for some group of organisms, such as fish, they are the scientific base to predator–prey systems. In this paper, some models, with distinctive assumptions, were fitted to the same fish assemblage data set from an isolated lagoon of the upper Paraná River floodplain using non-linear procedure. Then, they were compared in order to explore the mechanisms (represented by model assumptions) that could be acting on that assemblage. We used the Lotka–Volterra model and its modifications to consider carrying capacity, saturation effect, ratio-dependence predator–prey trophic function, and environmental heterogeneity. To fit the models, all prey species were grouped as prey and the same was done to predators. Result of this approach allowed us to make four suggestions about predator–prey relationship for the environment studied: (i) predators were efficient in controlling prey populations; (ii) the best fitted model was found with the sigmoidal functional response; (iii) density-dependence presented better fit than ratio-dependence predator–prey trophic function; (iv) refuges did not present intense interference in the predator–prey relationship. Therefore, comparisons of models that represent different assumptions showed to be a good tool in evaluating main mechanisms acting on the relationship between predators and prey in a fish assemblage.

Macroecological analyses support an overkill scenario for late Pleistocene extinctions

The extinction of megafauna at the end of Pleistocene has been traditionally explained by environmental changes or overexploitation by human hunting (overkill). Despite difficulties in choosing between these alternative (and not mutually exclusive) scenarios, the plausibility of the overkill hypothesis can be established by ecological models of predator-prey interactions. In this paper, I have developed a macroecological model for the overkill hypothesis, in which prey population dynamic parameters, including abundance, geographic extent, and food supply for hunters, were derived from empirical allometric relationships with body mass. The last output correctly predicts the final destiny (survival or extinction) for 73% of the species considered, a value only slightly smaller than those obtained by more complex models based on detailed archaeological and ecological data for each species. This illustrates the high selectivity of Pleistocene extinction in relation to body mass and confers more plausibility on the overkill scenario.

Central assumptions of predator–prey models fail in a semi–natural experimental system

The relationship between the encounter rate of predators with prey and the density of this prey is fundamental to models of predator-prey interactions. The relationship determines, among other variables, the rate at which prey patches are depleted, and hence the impact of predator populations on their prey, and the optimal spatial distribution of foraging effort. Two central assumptions that are made in many models are that encounter rate is directly proportional to prey density and that it is independent of the proportion of prey already removed, other than via the decreased density. We show here, using captive great tits searching for winter moth caterpillars in their natural hiding positions, that neither of these assumptions hold. Encounter rate increased less than directly in proportion to prey density, and it depended not only on the current density of prey, but also on the proportion of prey already removed by previous foragers. Both of these effects are likely to have major consequences for the outcome of predator-prey interactions.

Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems

In this paper, we establish sufficient criteria for the existence of positive periodic solutions for a class of discrete time semi-ratio-dependent predator-prey interaction models based on systems of nonautonomous difference equations. The approach involves the coincidence degree and its related continuation theorem as well as some priori estimates.

Dynamics of a class of nonautonomous semi-ratio-dependent predator–prey systems with functional responses

In this paper, we investigate the dynamics of a class of the so-called semi-ratio-dependent predator–prey interaction models with functional responses based on systems of nonautonomous differential equations with time-dependent parameters. The functional responses are classified into five types and typical examples of each type are provided. Then we establish sufficient criteria for the boundedness of solutions, the permanence of system, and the existence, uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution. Some conclusive discussion is presented at the end of this paper.

A predator–prey interaction model with self and cross-diffusion

In this paper, a mathematical model for a predator–prey interaction with self and cross-diffusion is proposed and analysed. Criteria for local stability, instability and global stability are obtained. The effect of the critical wave length which can drive a system to instability is investigated. The effect of time-varying cross-diffusivity on the stability of the system is also examined.

Response of Great Horned Owls to Experimental "Hot Spots" of Snowshoe Hare Density

Predators that aggregate in of high prey density have been hy- pothesized to synchronize population cycles of small mammals. During a peak and decline in a snowshoe hare (Lepus americanus) cycle, we created artificial hot spots of increased hare abundance by adding food and excluding mammalian predators on three 1-km2 blocks and then recorded the response of radio-marked Great Horned Owls (Bubo virginianus) to these food additions. Territorial owls showed a decrease in home range size and patchiness of spatial use as hare densities peaked and declined, although this was better explained by smaller territory sizes due to a growing owl population rather than a direct behavioral re- sponse to changing food density. Experimental owls on food-enriched territories did not show a difference in conventional measurements of home-range size and patchiness of spa- tial use compared with controls, but the distances of owl locations to treatment blocks re- vealed concentrations of spatial use on experimental hot spots. At a larger scale, neither ter- ritorial owls nor nonterritorial floaters showed a tendency to leave poorer patches and move toward hot spots, and the territorial system of Great Horned Owls was largely resistant to extreme variations in prey density. The effect of social interference between predators has been assumed for several models of predator-prey interactions, but empirical evidence has rarely been demonstrated. Our results suggest that territorial behavior, in addition to lim- iting the growth of a predator population, also prevents large aggregations of predators at an intermediate spatial scale. Received 3 July 1997, accepted 29 December 1997.

Spatially induced speciation prevents extinction: the evolution of dispersal distance in oscillatory predator-prey models.

In a discrete-generation, individual-oriented model of predator-prey interactions that exhibits oscillations, we show that the self-structuring of the populations into spiral waves induces a selection pressure for ever-increasing dispersal distances in both populations. As the dispersal distances increase, the sizes of the spatial patterns increase, until they are too large to fit into the limited space. The patterns are then lost and the predators go extinct. This scenario, is, however, not the only outcome. A second selection pressure induced by the spatial boundary can cause reduction of the dispersal distances. Depending on the relative strengths of the two selection pressures, the predators and prey may speciate to give coexistence between short-dispersing boundary quasi-species and far-dispersing spiral quasi-species. Now, when pattern loss occurs, the predators switch to predating on the boundary prey quasi-species and do not go extinct. Also, if the populations reproduce sexually, local gene flow can inhibit the evolution of increasing dispersal distances, and hence the spatial patterns are not lost. Speciation and coexistence can also occur in the sexually reproducing species.

The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model

A predator–prey interaction model based on a system of differential equations with temperature-dependent parameters chosen appropriately for a mite interaction on apple trees is analyzed to determine how the type of functional response influences bifurcation and stability behavior. Instances of type I, II, III, and IV functional responses are considered, the last of which incorporates prey interference with predation. It is shown that the model systems with the type I, II, and III functional responses exhibit qualitatively similar bifurcation and stability behavior over the interval of definition of the temperature parameter. Similar behavior is found in the system with the type IV functional response at low levels of prey interference. Higher levels of interference are destabilizing, as illustrated by the prevalence of bistability and by the presence of three attractors for some values of the model parameters. All four systems are capable of modeling population oscillations and outbreaks.

Periodic travelling waves in a family of deterministic cellular automata

Abstract Reaction-diffusion equations whose kinetics contain a stable limit cycle are an established class of models for a range of biological and chemical systems. In this paper I construct a family of deterministic cellular automata, with nine states, which are qualitatively similar to oscillatory reaction-diffusion equations, in that their rules reflect both local oscillations and spatial diffusion. The automata can be crudely interpreted as models of predator-prey interactions, and I show that the behaviour following local perturbation of the prey-only state in one space dimension is very similar in the automata and in standard reaction-diffusion models for predator-prey systems. In particular, in many cases, invasion of prey by predators leaves behind periodic travelling waves in the wake of invasion. I study in detail these periodic plane waves in the automaton, by explicitly investigating periodic solutions of the difference equation governing travelling waves. I show that the automaton has many different periodic wave solutions, and I compare their properties with those of periodic wave solutions of reaction-diffusion systems. The basic conclusion is that included amongst the periodic waves in the automaton are a family of solutions which mimic quite closely the properties of reaction-diffusion periodic waves.

Modelling Seasonal Predator-Prey Interactions

We analyse an existing two variable model of predator-prey interactions. The analysis reveals problems in the model formulation. The main problem is related to the fact that the model is a mixture of empirically and causally based dynamic equations. An attempt to repair the model is made, but a completely satisfactory solution is not reached. We argue that model constructions, in which causal and empirical formulations are combined, contain fundamental problems.

How can honest signalling evolve? The role of handicap principle

Handicap principle is now widely accepted as a mechanism for maintaining honest signalling. However, little attention has been paid to how honest signalling evolves in the first place. Using a model of predatorprey interaction with two levels of prey's quality and signalling intensity, I examine the conditions for the evolution and maintenance of honest signalling and, in particular, elucidate the role of handicap principle. Major conclusions are as follows. 1. Prey's honest signalling can be maintained only if predators prefer prey of weak (or no) signalling. 2. Predator's preference of no-signalling prey is also prerequisite for the evolution of honest signalling from no signalling. 3. This predator's preference may evolve from no preference, by genetic drift or pleiotropy. 4. Once predator's no-signalling preference is in place, honest-signalling prey can invade and take over no-signalling ones under certain conditions. These conditions are more strict than those for honest-signalling maintenance. 5. Both conditions commonly imply that signalling pays (i.e. signalling cost should be smaller than the predation cost which must be paid for not signalling) for high escaping-ability prey whereas it doesn't for low escaping-ability one, articulating a form of handicap principle.

Great Horned Owls and Snowshoe Hares: What Causes the Time Lag in the Numerical Response of Predators to Cyclic Prey?

Predator populations often decline with a time log after the peak of prey cycles. Theoretical models of predator-prey interactions predict that this delay is caused by a higher rate of population growth in prey, which leaves predators with super-abundant food after the peak and buffers their decline. This situation is met when predator populations have a lower innate capacity for increase than their prey or when the increase is inhibited because of territorial behaviour. Here, I refer to this hypothesis as 'single prey hypothesis' (SPH) in contrast to the 'multiple prey hypothesis' (MPH), which predicts that the delayed decline is caused by high availability of other prey species. Results on population growth rates of great horned owls showed that the predictions of SPH were met, although the predicted difference was small when floaters were taken into account or social exclusion from breeding was removed in a population model. In their diet, great horned owls relied to a large degree on the main cyclic prey (snowshoe hares), and thus the results were not in agreement with the MPH. Inverse density-dependent growth rates in the territorial population, density-dependent accumulation of floaters, and replacements of territorial vacancies were consistent with the hypothesis that social behaviour limited the number of owl territories. Reproduction of resident owls was immediately affected by the prey decline, indicating that there was no buffering effect of super-abundant food. Therefore, neither MPH nor SPH were satisfactory explanations, and I propose a mechanism based on individual behaviour to explain delayed numerical responses : territorial predators monopolize a disproportionately large amount of resources for reproduction during the increase and peak of the cycle, and are then buffered against prey declines by adjusting their breeding activities. Non-territorial floaters have lower access to resources and their numbers are affected more immediately by declining prey.

Ecological chaos in the wake of invasion.

Irregularities in observed population densities have traditionally been attributed to discretization of the underlying dynamics. We propose an alternative explanation by demonstrating the evolution of spatiotemporal chaos in reaction-diffusion models for predator-prey interactions. The chaos is generated naturally in the wake of invasive waves of predators. We discuss in detail the mechanism by which the chaos is generated. By considering a mathematical caricature of the predator-prey models, we go on to explain the dynamical origin of the irregular behavior and to justify our assertion that the behavior we present is a genuine example of spatiotemporal chaos.

Spatial heterogeneity and anomalous kinetics: emergent patterns in diffusion-limited predatory-prey interaction

The Lotka-Volterra model of predator-prey interaction is based on the assumption of mass action, a concept borrowed from the traditional theory of chemical kinetics in which reactants are assumed to be homogeneously mixed. In order to explore the effect of spatial heterogeneity on predator-prey dynamics, we constructed a lattice-based reaction-diffusion model corresponding to the Lotka-Volterra equations. Spatial heterogeneity was imposed on the system using percolation maps, gradient percolation maps, and fractional Brownian surfaces. In all simulations where diffusion distances were short, anomalously low reaction orders and aggregated spatial patterns were observed, including traveling wave patterns. In general, the estimated reaction order decreased with increasing degrees of spatial heterogeneity. For simulations using percolation maps with p-values varying between 1.0 (all cells available) to 0.5 (50% available), order estimates varied from 1.27 to 0.47. Gradient percolation maps and fractional Brownian surfaces also resulted in anomalously low reaction orders. Increasing diffusion distances resulted in reaction order estimates approaching the expected value of 2. Analysis of the qualitative dynamics of the model showed little difference between simulations where individuals diffused locally and those where individuals moved to random locations, suggesting that global density dependence is an important determinant of the overall model dynamics. However, localized interactions did introduce time dependence in the system attractor owing to emergent spatial patterns. We conclude that individual-based spatially explicit models are important tools for modeling population dynamics as they allow one to incorporate fine-scale ecological data about localized interactions and then to observe emergent patterns through simulation. When heterogeneous patterns arise, it can lead to anomalies with respect to the predictions of traditional mathematical approaches using global state variables.

Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge

The non-linear behavior of a differential equations-based predator-prey model, incorporating a spatial refuge protecting a constant proportion of prey and wit temperature-dependent parameters chosen appropriately for a mite interaction on fruit trees, is examined using the numerical bifurcation code AUTO 86. The most significant result of this analysis is the existence of a temperature interval in which increasing the amount of refuge dynamically destabilizes the system; and on part of this interval the interaction is less likely to persist in that predator and prey minimum population densities are lower than when no refuge is available. It is also shown that increasing the amount of refuge can lead to population outbreaks due to the presence of multiple stable states. The ecological implications of a refuge are discussed with respect to the biological control of mite pests.

Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge

Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge

Diffusion driven instability in an inhomogeneous domain

Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.