Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

Aggregation and emergence in hierarchically organized systems: population dynamics

The aim of this work is to present aggregation methods of hierarchically organized systems allowing one to replace the initial micro-system by a macro-system described by a few global variables. We also study the relations between the fast micro-dynamics and the slow macro-dynamics which can produce global properties. Emergence corresponds to a bottom-up coupling that is the result effected by a micro-level at a macro-level. As an example, we present prey-predator models with different time scales in an heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to growth and interactions between the populations. Preys must go on spatial patches where resources are located and where predators can attack them. The efficiency of the predators to catch preys is patch dependent. Perturbation methods allow us to aggregate the initial system of differential equations for the patch sub-populations into a macro-system of two differential equations governing the total population densities. We study the case of density independent and density dependent migrations. In the latter case, we show that different functional responses can emerge in the macro prey-predator model as a result of the coupling between the slow and fast systems.     

Dynamics of predator and modular prey: effects of module consumption on stability of prey–predator system

In traditional models of predator–prey population dynamics, it is usually assumed that consumed prey are immediately removed from the population. However, in plant–herbivore interactions, damaged plants are generally alive after attacks by herbivores. This can result in successive or simultaneous attacks by multiple predators on a single prey item (here, the term prey is expanded to include plants). We constructed a mathematical model with two time scales, taking into account predation processes within a generation, considering post‐predation survival and the modularity of prey. We assumed that a prey item can be divided into modules and that it can be fed on by multiple predators or parasitized by multiple parasites at the same time. The model includes two essential factors: the modularity of prey for predators (n) and the detaching/attaching ratio of predators to prey (ε). Based on the formulae, we revealed a general property of realistic dynamics in plant–herbivore and host–parasite interactions. The analysis showed that the model could be approximated by models with the type I, type II or Beddington–DeAngelis functional responses by taking appropriate limits to the situations. When modularity is low or the detaching/attaching ratio is high, population dynamics tend to be stabilized. These stabilizing effects may be related to interference competition among predator individuals or increases in free prey modules and free predator individuals. In the limit of high modularity, the ratio of the attached prey modules to the total prey modules becomes negligible and the dynamics tend to be destabilized. However, if quantity and quality of prey modules are negatively correlated, the equilibrium is likely to be stabilized at high modularity as long as it remains feasible. These results suggest that considering post‐predation survival and modularity of prey is crucial to understand predator–prey interactions.     

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).     

Predator overcomes the Allee effect due to indirect prey–taxis

A mathematical model for spatiotemporal dynamics of prey–predator system was studied by means of linear analysis and numerical simulations. The model is a system of PDEs of taxis–diffusion–reaction type, accounting for the ability of predators to detect the locations of higher prey density, which is formalized as indirect prey–taxis, according to hypothesis that the taxis stimulus is a substance being continuously emitted by the prey, diffusing in space and decaying with constant rate in time (e.g. odour, pheromone, exometabolit). The local interactions of the prey and predators are described by the classical Rosenzweig – MacArthur system, which is modified in order to take into account the Allee effect in the predator population. The boundary conditions determine the absence of fluxes of population densities and stimulus concentration through the habitat boundaries. The obtained results suggest that the prey–taxis activity of the predator can destabilize both the stationary and periodic spatially-homogeneous regimes of the species coexistence, causing emergence of various heterogeneous patterns. In particular, it is demonstrated that formation of local dense aggregations induced by prey–taxis allows the predators to overcome the Allee effect in its population growth, avoiding the extinction that occurs in the model in the absence of spatial effects.     

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.     

Spatial variability in prey phenology determines predator movement patterns and prey survival

Species have phenological variation among local habitats that are located at relatively small spatial scales. However, less studies have tested how this spatial variability in phenology can mediate intra-/inter-specific interactions. When predators track phenological variation of prey among local habitats, survival of prey within a local habitat strongly influenced by phenological synchrony with their conspecifics in adjacent habitats. Theory predicts that phenological synchrony among local habitats increases prey survival in local habitat within spatially structured environments because the predators have to make a habitat choice for foraging. Consequently, total survival of prey at regional scale should be higher. By using a spatially explicit field experiment, we tested above hypothesis using a prey–predator interaction between tadpole (Rhacophorus arboreus) and newt (Cynops pyrrhogaster). We established enclosures (≈regional scale) consisting of two tanks (≈local habitat scale) with different degree of prey phenological synchrony. We found that phenological synchrony of prey between tanks within each enclosure decreased the mean residence time of the predator in each tank, which resulted in higher survival of prey at a local habitat scale, supporting the theoretical prediction. Furthermore, individual-level variation in predator residence time explained the between-tank variation in prey survival in enclosures with phenological synchrony, implying that movement patterns of the predator can mediate variation in local population dynamics of their prey. However, total survival at each enclosure was not higher under phenological synchrony. These results suggest the importance of relative timing of prey phenology, not absolute timing, among local habitats in determining prey–predator interactions.     

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.     

Complexity in a prey-predator model with prey refuge and diffusion

Prey-predator interaction is one of the most commonly observed relationships in ecosystem. In the study of prey-predator models, it is frequently assumed that the changes in population densities are only time-dependent and the dynamics is generally represented by coupled nonlinear ordinary differential equations. In natural system, however, either prey or predator or both move from one place to another for various reasons. In such a case, their dynamic interaction depends both on time and space and requires coupled nonlinear partial differential equations for its dynamic representation. It is also well documented that prey refuges affect the interaction between prey and predator significantly. In this paper, we studied the dynamics of a diffusive prey-predator interaction with prey refuge and type III response function. We have considered both one and two dimensional diffusivity in the model system and presented different stability results under the assumptions that one or both species may be mobile or sedentary. Our results showed that the system may exhibit different spatiotemporal (non-Turing) patterns, like spiral waves, patchy structures, spot pattern, or even spatiotemporal chaos depending on the refuge availability and diffusion rate of species. Another interesting finding was that the dynamic complexity in a prey-predator model increases in case of mobile predator and sedentary prey compare to mobile prey and sedentary predator while refuge availability is varied.     

The advantage of alternative tactics of prey and predators depends on the spatial pattern of prey and social interactions among predators

Individual variation in behavioral strategies is ubiquitous in nature. Yet, explaining how this variation is being maintained remains a challenging task. We use a spatially-explicit individual-based simulation model to evaluate the extent to which the efficiency of an alternative spacing tactic of prey and an alternative search tactic of predators are influenced by the spatial pattern of prey, social interactions among predators (i.e., interference and information sharing) and predator density. In response to predation risk, prey individuals can either spread out or aggregate. We demonstrate that if prey is extremely clumped, spreading out may help when predators share information regarding prey locations and when predators shift to area-restricted search following an encounter with prey. However, dispersion is counter-selected when predators interact by interference, especially under high predator density. When predators search for more randomly distributed prey, interference and information sharing similarly affect the relative advantage of spreading out. Under a clumped prey spatial pattern, predators benefit from shifting their search tactic to an area-restricted search following an encounter with prey. This advantage is moderated as predator density increases and when predators interact either by interference or information sharing. Under a more random prey pattern, information sharing may deteriorate the inferior search tactic even more, compared to interference or no interaction among predators. Our simulation clarifies how interactions among searching predators may affect aggregation behavior of prey, the relative success of alternative search tactics and their potential to invade established populations using some other search or spacing tactics.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Directed movement of predators and the emergence of density-dependence in predator-prey models.

We consider a bitrophic spatially distributed community consisting of prey and actively moving predators. The model is based on the assumption that the spatial and temporal variations of the predators' velocity are determined by the prey gradient. Locally, the populations follow the simple Lotka-Volterra interaction. We also assume predator reproduction and mortality to be negligible in comparison with the time scale of migration. The model demonstrates heterogeneous oscillating distributions of both species, which occur because of the active movements of predators. One consequence of this heterogeneity is increased viability of the prey population, compared to the equivalent homogeneous model, and increased consumption. Further numerical analysis shows that, on the spatially aggregated scale, the average predator density adversely affects the individual consumption, leading to a nonlinear predator-dependent trophic function, completely different from the Lotka-Volterra rule assumed at the local scale.     

Influence of Individual Aggressiveness on the Dynamics of Competitive Populations

Two populations are subdivided into two categories of individuals (hawks and doves). Individuals fight to have access to a resource which is necessary for their survival. Conflicts occur between individuals belonging to the same population and to different populations. We investigate the long term effects of the conflicts on the stability of the community. The modelis a set of ODE's with four variables corresponding to hawk and dove individuals of the two populations. Two time scales are considered. A fast time scale is used to describe frequent encounters and fightings between individuals trying to monopolize the resource. A slow time scale is used for the demography and the long term effects of encounters. We use aggregation methods in order to reduce this model into a system of two ODE's only for the total densities of the two populations which is found to be a classical Lotka-Volterra competition model. We study different cases of proportions of hawks and doves in both populations on the global coexistence and the mutal exclusion of the two populations. Pure dove tactics in both populations are unstable. In cases of mixed hawk and dove in both populations, there is coexistence. Pure dove or mixed hawk-dove tactics in one population can coexist with pure hawks in the other one when the costs of fightings between hawks are large enough. This revised version was published online in June 2006 with corrections to the Cover Date.     

Does prey capture induce area-restricted search? A fine-scale study using GPS in a marine predator, the wandering albatross

In a patchy environment, predators are expected to increase turning rate and start an area-restricted search (ARS) when prey have been encountered, but few empirical data exist for large predators. By using GPS loggers with devices measuring prey capture, we studied how a marine predator adjusts foraging movements at various scales in relation to prey capture. Wandering albatrosses use two tactics, sit and wait and foraging in flight, the former tactic being three times less efficient than the latter. During flight foraging, birds caught large isolated prey and used ARS at scales varying from 5 to 90 km, with large-scale ARS being used only by young animals. Birds did not show strong responses to prey capture at a large scale, few ARS events occurred after prey capture, and birds did not have high rates of prey capture in ARS. Only at small scales did birds increase sinuosity after prey captures for a limited time period, and this occurred only after they had caught a large prey item within an ARS zone. When this species searches over a large scale, the most effective search rule was to follow a nearly straight path. ARS may be used to restrict search to a particular environment where prey capture is more predictable and profitable.     

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.     

Functional responses modified by predator density

Realistic functional responses are required for accurate model predictions at the community level. However, controversy remains regarding which types of dependencies need to be included in functional response models. Several studies have shown an effect of very high predator densities on per capita predation rates, but it is unclear whether this predator dependence is also important at low predator densities. We fit integrated functional response models to predation data from 4-h experiments where we had varied both predator and prey densities. Using an information theoretic approach we show that the best-fit model includes moderate predator dependence, which was equally strong even at low predator densities. The best fits of Beddington–DeAngelis and Arditi–Akçakaya functional responses were closely followed by the fit of the Arditi–Ginzburg model. A Holling type III functional response did not describe the data well. In addition, independent behavioral observations revealed high encounter rates between predators. We quantified the number of encounters between predators and the time the focal predator spent interacting with other individuals per encounter. This time “wasted” on conspecifics reduced the total time available for foraging and may therefore account for lower predation rates at higher predator densities. Our findings imply that ecological theory needs to take realistic levels of predator dependence into account.     

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.     

Emergence of donor control in patchy predator—prey systems

We study a general predator—prey system in a spatially heterogeneous environment. The predation process, which occurs on a behavioural time-scale, is much faster than the other processes (reproduction, natural mortality and migrations) occurring on the population dynamics time-scale. We show that, taking account of this difference in time-scales, and assuming that the prey have a refuge, the dynamics of the system on a slow time-scale become donor-controlled. Even though predators may control the prey density locally and on a behavioural fast time-scale, nevertheless, both globally and on a slow time-scale, the prey dynamics are independent of predator density: the presence of predators generates a constant prey mortality. In other words, in heterogeneous environments, the prey population dynamics depend in a switch-like manner on the presence or absence of predators, not on their actual density.     

Predation across spatial scales in heterogeneous environments

A hierarchy of scales is introduced to the spatially heterogeneous Lotka-Volterra predator-prey diffusion model, and its effects on the model's spatial and temporal behavior are studied. When predators move on a large scale relative to prey, local coupling of the predator-prey interaction is replaced by global coupling. Prey with low dispersal ability become narrowly confined to the most productive habitats, strongly amplifying the underlying spatial pattern of the environment. As prey diffusion rate increases, the prey distribution spreads out and predator abundance declines. The model retains neutrally stable Lotka-Volterra temporal dynamics: different scales of predator and prey dispersal do not stabilize the interaction. The model predicts that, for prey populations that are limited by widely ranging predators, species with low dispersal ability should be restricted to discrete high density patches, and those with greater mobility should be more uniformly distributed at lower density.     

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.