Dynamical behavior of two predators competing over a single prey

Dynamical behavior of a food web comprising two predators competing over a single prey has been investigated. The analysis of the food web model shows that the persistence is not possible for two competing predators sharing a single prey species in the cases when any one of the boundary prey–predator planes has a stable equilibrium point. The principle of competitive exclusion holds in such cases. However, numerical simulations exhibit persistence in the presence of periodic solutions in the boundary planes. The system exhibits quasi-periodic behavior in the positive octant. The co-existence in the form of a limit cycle is also possible in some cases.     

Dynamical behavior of two predators competing over a single prey

Dynamical behavior of a food web comprising two predators competing over a single prey has been investigated. The analysis of the food web model shows that the persistence is not possible for two competing predators sharing a single prey species in the cases when any one of the boundary prey–predator planes has a stable equilibrium point. The principle of competitive exclusion holds in such cases. However, numerical simulations exhibit persistence in the presence of periodic solutions in the boundary planes. The system exhibits quasi-periodic behavior in the positive octant. The co-existence in the form of a limit cycle is also possible in some cases.     

Coexistence of competing predators in a chemostat

An analysis is given of a mathematical model of two predators feeding on a single prey growing in the chemostat. In the case that one of the predators goes extinct, a global stability result is obtained. Under appropriate circumstances, a bifurcation theorem can be used to show that coexistence of the predators occurs in the form of a limit cycle.     

Chaotic dynamics of a three species prey-predator competition model with bionomic harvesting due to delayed environmental noise as external driving force

We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.     

The impact of variable stoichiometry on predator-prey interactions: a multinutrient approach

A model for prey and predators is formulated in which three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a fraction of ingested nutrients that depends on their current nutrient status, and recycle the balance. Although individuals are modeled as identical within populations, amounts of nutrients within individuals vary over time in both populations, with reproductive rates increasing with these amounts. Equilibria and their stability depend on nutrient supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon occurs if, with increasing nutrient, predators sequester rather than recycle nutrients. Furthermore, despite use of a linear functional response for predators, high nutrient supply can destabilize equilibria. Responses to nutrient supply depend on the balance between assimilation and recycling of nutrients by predators, which differs depending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon mineralization is reduced when supply of mineral nutrients is low and when equilibria are unstable. The extent to which predators recycle or sequester limiting nutrients for their prey is of critical importance for the stability of predator-prey systems and their response to enrichment.     

Ecoepidemic models with prey group defense and feeding saturation

In this paper we consider a model for the herd behavior of prey, that are subject to attacks by specialist predators. The latter are affected by a transmissible disease. With respect to other recently introduced models of the same nature, we focus here our attention to the possible feeding satiation phenomenon. The system dynamics is thoroughly investigated, to show the occurrence of several types of bifurcations. In addition to the transcritical and Hopf bifurcation that occur commonly in predator–prey system also a zero-Hopf and a global bifurcation occur. The Hopf and the global bifurcation occur only in the disease-free (so purely demographic) system. The latter is a heteroclinic connection for the between saddle equilibrium points where a stable limit cycle is disrupted and where the system disease-free collapses while in a parameter space region the endemic system exists stably.     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Diet choice and predator—prey dynamics

Summary We compare the dynamics of predator-prey systems with specialist predators or adaptive generalist predators that base diet choice on energy-maximizing criteria. Adaptive predator behaviour leads to functional responses that are influenced by the relative abundance of alternate prey. This results in the per capita predation risk being positively density-dependent near points of diet expansion. For a small set of parameter values, systems with adaptive predators can be locally stable whereas systems with specialist predators would be unstable. This occurs mainly when alternate prey have low enough profitability that predators cannot sustain themselves indefinitely when feeding on alternate prey. Local stability of systems with adaptive predator behaviour is inversely related to the goodness of fit to optimal diet choice criteria. Hence, typical patterns of partial prey preference are more stabilizing than perfect optimal diet selection. Locally stable systems with adaptive predators are often globally unstable, converging on limit cycles for many initial population densities. The small range of parameter combinations and initial population densities leading to stable equilibria suggest that adaptive diet selection is unlikely to be a ubiquitous stabilizing factor in trophic interactions.     

Prey preference of top predators manipulates the functioning and stability of multi-trophic ecosystems

The relationship between biodiversity and ecosystem functioning, and the mechanisms underpinning the food web stability, have been intensively investigated in ecological research. The ubiquities of generalists in natural food webs and its important role in dictating these ecosystem properties have been generally recognized. However, how competition between multiple top predators shape these ecosystem properties and determine the success of invasive predators remain largely unexplored. Here, we use a well-developed food web model to investigate the effects of prey preference of top predators on ecosystem functioning and food web stability in both local and invasive conditions. We design several modeling scenarios to mimic combinations of different types of top predators (specialist/generalist) and their origins (local/invasive). Our model theoretically shows that lower exploitation competition for prey between top predators (with distinct prey preferences featured by higher attack rates) would be beneficial for the ecosystem functioning and food web stability. We also demonstrate that the success of top predator invasion depends on the prey preference of both local and invasive top predators. Sensitivity analysis on the model further supports our findings. Our results highlight the importance of prey preference of multiple top predators in manipulating the properties of multi-trophic ecosystems. Our findings may have important implications because the current ongoing global changes profoundly change the phenology of many biological systems and create trophic mismatch, which may manipulate prey preference of top predators and in turn deteriorate ecosystem functioning and food web stability.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.     

Continuous cycling of grouped vs. solitary strategy frequencies in a predator-prey model

We present a model of predator and prey grouping strategies using game theory. As predators respond strategically to prey behavior and vice versa, the model is based on a co-evolution approach. Focusing on the "many eyes-many mouths" trade-off, this model considers the benefits and costs of being in a group for hunting predators and foraging prey: predators in a group have more hunting success than solitary predators but they have to share the prey captured; prey in a group face a lower risk of predation but greater competition for resources than lone prey. The analysis of the model shows that the intersections of four curves define distinct areas in the parameter space, corresponding to different strategies used by predators and prey at equilibrium. The model predictions are in accordance with empirical evidence that an open habitat encourages group living, and that low risks of predation favor lone prey. Under some conditions, continuous cycling of the relative frequencies of the different strategies may occur. In this situation, the proportions of grouped vs. solitary predators and prey oscillate over time.     

Is arthropod predation exclusively satiation‐driven?

Functional response models differ in which factors limit predation (e.g. searching efficiency, prey handling time, digestion) and whether predation behaviour is governed by an internal physiological state (e.g. satiation). There is now much evidence that satiation is a key factor in understanding changes in foraging behaviour, and that many predators are effectively digestion limited. Here, we ask if predation in a predatory arthropod can be explained from satiation-driven behaviour alone, or if behaviour is also influenced by the density of prey other than via the effect of prey ingestion on satiation. To address this question a satiation-based predation model is formulated, for which parameters are estimated on the basis of observations on digestion rate, satiation-related prey searching rate and prey capture behaviour, basically under high prey density conditions. The model predictions are subsequently tested against longer term predation experiments carried out at high and low prey densities. Since satiation can easily be linked with egg production, these tests are carried out both for predation and oviposition.
The predator–prey systems under study consist of females of two predatory mite species ( Neoseiulus barkeri and N. cucumeris ) and the larvae of two thrips species ( Thrips tabaci and Frankliniella occidentalis ) as their prey. For N. barkeri foraging on T. tabaci , the model gives good predictions at both high (4 larvae cm⁻¹) and low (0.1–1 larvae cm⁻²) prey densities. For N. cucumeris foraging on F. occidentalis , the predictions hold at the high prey density, but are too low at low prey densities. Thus our analysis indicates that we cannot fully explain density-dependent predation rates from satiation-driven behaviour alone. Different mechanisms are suggested on how prey density may affect foraging efficiency other than via satiation.     

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.     

Numerical exploration of the parameter plane in a discrete predator–prey model

We propose a variant of the discrete Lotka–Volterra model for predator–prey interactions. A detailed stability and numerical analysis of the model are presented to explore the long time behaviour as each of the control parameter is varied independently. We show how the condition for survival of the predator depends on the natural death rate of predator and the efficiency of predation. The model is found to support different dynamical regimes asymptotically including predator extinction, stable fixed point and limit cycle attractors for co-existence of predator and prey and more complex dynamics involving chaotic attractors. We are able to locate exactly the domain of chaos in the parameter plane using a dimensional analysis.     

Global analysis of a predator–prey model with variable predator search rate

We consider a modified Holling-type II predator–prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Holling-type II model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Holling-type II model. The main differences are that: 1. The paradox of enrichment which always occurs in the Holling-type II model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system.

     

Colour polymorphism and selective predation experiments

The significance of searching-image behaviour by predators in relation to visually polymorphic prey is outlined in relation to its possible role in maintaining these polymorphisms by frequency-dependent selection. A simple experimental design is described whereby innate preferences, selection for crypsis, and searching-image behaviour can be distinguished. Four experimental situations are described ranging from an artificial predator-prey relationship (humans as predators—dyed toothpicks as prey) to a semi-natural one (wild song thrushes as predators—artificial populations of live polymorphic snails as prey). The results suggest that “naive” predators may regularly form searching-images for a frequent prey type.     

Dynamical behaviour of a two-predator model with prey refuge

A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.     

Stable isotope analyses document intraguild predation in wolf spiders (Araneae: Lycosidae) and underline beneficial effects of alternative prey and microhabitat structure on intraguild prey survival

Intraguild predation (IGP) is common among generalist predators and an important issue in food web theory, because IGP may destabilise communities by increasing extinction of species. Also, IGP may interfere with the effectiveness of generalist predators as biological control agents. In general, occurrence of IGP in laboratory or field studies is inferred from abundance data or direct observations only. We investigated if tracing stable isotopes allows distinction between different types of predation and confirmation of IGP. Wolf spiders were chosen as model organisms for generalist predators; IGP of third instar A. cuneata on second instar P. palustris was investigated in a laboratory experiment. The availability of alternative prey and the complexity of the microhabitat were manipulated, since both factors are thought to facilitate coexistence of predators.
Stable isotope analysis documented predation of A. cuneata on P. palustris and predation on alternative prey by both juveniles. Both the presence of alternative prey and the availability of shelter reduced mortality of juvenile P. palustris during the first week. During the second week mortality increased in complex structure without alternative prey presumably due to enhanced activity and cannibalism among starving P. palustris . Thus, microhabitat complexity and prey abundance may foster coexistence of wolf spiders in the field.
In conclusion, stable isotope analysis was proven a powerful tool to investigate animal behaviour without direct observation. The method allowed disentangling predator feeding behaviour when more than one type of prey was present.     

Effect of hunting cooperation and fear in a predator-prey model

Dynamics of predator-prey systems under the influence of cooperative hunting among predators and the fear thus imposed on the prey population is of great importance from ecological point of view. The role of hunting cooperation and the fear effect in the predator-prey system is gaining considerable attention by the researchers recently. But the study on combined effect of hunting cooperation and fear in the predator-prey system is not yet studied. In the present paper, we investigate the impact of hunting cooperation among predators and predator induced fear in prey population by using the classical predator-prey model. We consider that predator populations cooperate during hunting. We also consider that hunting cooperation induces fear among prey, which has far richer and complex dynamics. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induced fear may destabilize the system and produce periodic solution via Hopf-bifurcation. The stability of the Hopf-bifurcating periodic solution is obtained by computing the Lyapunov coefficient. The limit cycles thus obtained may be supercritical or subcritical. We also observe that the system undergoes the Bogdanov-Takens bifurcation in two-parameter space. Further, we observe that the system exhibits backward bifurcation between predator-free equilibrium and coexisting equilibrium. The system also exhibits two different types of bi-stabilities due to subcritical Hopf-bifurcation (between interior equilibrium and stable limit cycle) and backward bifurcation (between predator-free and interior equilibrium points). Further, we observe strong demographic Allee phenomenon in the system. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MATCONT softwares.