Effects of perturbation on the predator–prey system in a heterogeneous landscape

Environmental perturbations occur in ecosystems as the result of disturbance, which is closely related to ecosystem stability and resilience. To understand how perturbations can affect ecosystems, we constructed a spatially explicit lattice model to simulate the integrative predator–prey–grass relationships. In this model, a predator (or prey) gives birth to offspring, according to a specific birth probability, when it is able to feed on prey (or grass). When a predator or prey animal was initially introduced or newly born, its health state was set at a given high value. This state decreased by 1 with each time step. When the state of an animal decreased to zero, the animal was considered dead and was removed from the system. In this model, the perturbation was defined as the sudden death of some portion of the population. The heterogeneous landscape was characterized by a parameter, H, which controlled the degree of heterogeneity. When H ≥ 0.6, the predator population size was positively influenced by the perturbation. However, the perturbation had little effect upon the population sizes of prey or grass, regardless of the value of H.     

Incorporating anthropogenic effects into trophic ecology: predator–prey interactions in a human-dominated landscape

Apex predators perform important functions that regulate ecosystems worldwide. However, little is known about how ecosystem regulation by predators is influenced by human activities. In particular, how important are top-down effects of predators relative to direct and indirect human-mediated bottom-up and top-down processes? Combining data on species'' occurrence from camera traps and hunting records, we aimed to quantify the relative effects of top-down and bottom-up processes in shaping predator and prey distributions in a human-dominated landscape in Transylvania, Romania. By global standards this system is diverse, including apex predators (brown bear and wolf), mesopredators (red fox) and large herbivores (roe and red deer). Humans and free-ranging dogs represent additional predators in the system. Using structural equation modelling, we found that apex predators suppress lower trophic levels, especially herbivores. However, direct and indirect top-down effects of humans affected the ecosystem more strongly, influencing species at all trophic levels. Our study highlights the need to explicitly embed humans and their influences within trophic cascade theory. This will greatly expand our understanding of species interactions in human-modified landscapes, which compose the majority of the Earth''s terrestrial surface.     

Viable populations in a prey–predator system

 Lotka–Volterra equations are considered a dynamical game, where the phenotypes of the predator and of the prey can vary. This differs from the usual procedure of specifying as a priori laws according to which strategies are supposed to change. The question at stake is the survival of each of the species, instead of the maximization of a given pay-off by each player, as it is commonly discussed in games. The predator needs the prey, while the prey can survive without the predator. These obvious and simplistic constraints are enough to shape the regulation of the system: notably, the largest closed set of initial conditions can be delineated, from which there exists at least one evolutionary path where the population can avoid extinction forever. To these so-called viable trajectories, viable strategies are associated, respectively for the prey or for the predator. A coexistence set can then be defined. Within this set and outside the boundary, strategies can vary arbitrarily within given bounds while remaining viable, whereas on the boundary, only specific strategies can guarantee the viability of the system. Thus, the largest set can be determined, outside of which strategies will never be flexible enough to avoid extinction. Received 2 May 1995; received in revised form 15 August 1995     

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.     

Revealing the role of predator interference in a predator–prey system with disease in prey population

Predation on a species subjected to an infectious disease can affect both the infection level and the population dynamics. There is an ongoing debate about the act of managing disease in natural populations through predation. Recent theoretical and empirical evidence shows that predation on infected populations can have both positive and negative influences on disease in prey populations. Here, we present a predator–prey system where the prey population is subjected to an infectious disease to explore the impact of predator on disease dynamics. Specifically, we investigate how the interference among predators affects the dynamics and structure of the predator–prey community. We perform a detailed numerical bifurcation analysis and find an unusually large variety of complex dynamics, such as, bistability, torus and chaos, in the presence of predators. We show that, depending on the strength of interference among predators, predators enhance or control disease outbreaks and population persistence. Moreover, the presence of multistable regimes makes the system very sensitive to perturbations and facilitates a number of regime shifts. Since, the habitat structure and the choice of predators deeply influence the interference among predators, thus before applying predators to control disease in prey populations or applying predator control strategy for wildlife management, it is essential to carefully investigate how these predators interact with each other in that specific habitat; otherwise it may lead to ecological disaster.     

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.     

Turing patterns in a predator–prey model with seasonality

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction–diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the ‘summer’ season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington–DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).

     

Cooperative hunting in a discrete predator–prey system II

ABSTRACT

We investigate a discrete-time predator–prey system with cooperative hunting in the predators proposed by Chow et al. by determining local stability of the interior steady states analytically in certain parameter regimes. The system can have either zero, one or two interior steady states. We provide criteria for the stability of interior steady states when the system has either one or two interior steady states. Numerical examples are presented to confirm our analytical findings. It is concluded that cooperative hunting of the predators can promote predator persistence but may also drive the predator to a sudden extinction.     

Reciprocity in predator–prey interactions: exposure to defended prey and predation risk affects intermediate predator life history and morphology

A vast body of literature exists documenting the morphological, behavioural and life history changes that predators induce in prey. However, little attention has been paid to how these induced changes feed back and affect the predators’ life history and morphology. Larvae of the phantom midge Chaoborus flavicans are intermediate predators in a food web with Daphnia pulex as the basal resource and planktivorous fish as the top predator. C. flavicans prey on D. pulex and are themselves prey for fish; as D. pulex induce morphological defences in the presence of C. flavicans this is an ideal system in which to evaluate the effects of defended prey and top predators on an intermediate consumer. We assessed the impact on C. flavicans life history and morphology of foraging on defended prey while also being exposed to the non-lethal presence of a top fish predator. We tested the basic hypothesis that the effects of defended prey will depend on the presence or absence of top predator predation risk. Feeding rate was significantly reduced and time to pupation was significantly increased by defended morph prey. Gut size, development time, fecundity, egg size and reproductive effort respond to fish chemical cues directly or significantly alter the relationship between a trait and body size. We found no significant interactions between prey morph and the non-lethal presence of a top predator, suggesting that the effects of these two biological factors were additive or singularly independent. Overall it appears that C. flavicans is able to substantially modify several aspects of its biology, and while some changes appear mere consequences of resource limitation others appear facultative in nature.     

Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

Numerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0–49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev's functional response for 0.1–14.3% of the parameter values. These two changes interact to create different model predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling's disc equation predicts a higher system resilience. Indeed, Ivlev's functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions?     

Remarks on the number of persistent states to a fragmented predator–prey model system

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.     

Asexual reproduction changes predator population dynamics in a life predator–prey system

Many organisms display oscillations in population size. Theory predicts that these fluctuations can be generated by predator–prey interactions, and empirical studies using life model systems, such as a rotifer-algae community consisting of Brachionus calyciflorus as predator and Chlorella vulgaris as prey, have been successfully used for studying such dynamics. B. calyciflorus is a cyclical parthenogen (CP) and clones often differ in their sexual propensity, that is, the degree to which they engage into sexual or asexual (clonal) reproduction. Since sexual propensities can affect growth rates and population sizes, we hypothesized that this might also affect population oscillations. Here, we studied the dynamical behaviour of B. calyciflorus clones representing either CPs (regularly inducing sex) or obligate parthenogens (OPs). We found that the amplitudes of population cycles to be increased in OPs at low nutrient levels. Several other population dynamic parameters seemed unaffected. This suggests that reproductive mode might be an important additional variable to be considered in future studies of population oscillations.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

Individual interaction data are required in community ecology: a conceptual review of the predator–prey mass ratio and more

Community ecology is traditionally species-based and assumes that species comprise identical individuals. However, intraspecific variation is ubiquitous in nature because of ontogenetic growth and critical in food-we dynamics. To understand individual interaction-mediated food webs, researchers have recently focused on body size as the most fundamental biological aspect and assessed a parameter called the predator–prey mass ratio (PPMR). Herein, I review the conceptual development of the PPMR and suggest four major concerns regarding its measurement: (1) PPMR should be measured at the individual level because species-averaged values distort actual feeding relationships, (2) individual-level PPMR data on gape-unconstrained predators (e.g., terrestrial carnivores) are limited because previous studies have targeted gape-limited fish predators, (3) predators’ prey size selectivity (preferred PPRM) is conceptually different from dietary prey size (realized PPMR) and should be distinguished by incorporating environmental prey abundance information, and (4) determinants of preferred PPMR, rather than those of realized PPMR, should be identified to describe size-dependent predation. Future studies are encouraged to explore not only predation but also other interaction types (e.g., competition, mutualism, and herbivory) at the individual level. However, this is not likely to occur while ecological communities are still considered to be interspecific interaction networks. To resolve this situation and more comprehensively understand biodiversity and ecosystem functioning, I suggest that community ecology requires a paradigm shift in the unit of interaction from species to individuals, similar to evolutionary biology, which revolutionized the unit of selection, because interactions occur between individuals.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Hatching in dabbling ducks and emergence in chironomids: a case of predator–prey synchrony?

It has been hypothesized that dabbling ducks (Anas spp.) time breeding to coincide with annual regional peaks in emerging dipterans, especially Chironomidae, which are important prey for newly hatched ducklings. However, this hypothesis has never been evaluated in a replicated lake-level study, including year effects in emergence patterns. We collected duck and invertebrate data from 12 lakes during the nesting seasons 1989–1994 in a watershed in southern Finland. The oligotrophic study lakes are typical of the boreal Holarctic, as are the three focal duck species: mallard Anas platyrhynchos L., widgeon Anas penelope L and teal Anas crecca L. Hatching of ducklings showed a clear peak in relation to ambient phenology (annual ice-out date of lakes), whereas chironomid emergence was more erratic and showed no clear peak at the lake level, although total watershed-level emergence was somewhat higher before and long after the duck hatching peak. Thus, we find no evidence that ducklings hatch in synchrony with abundance peaks of emerging chironomids. There was large within-year temporal variation in chironomid emergence among lakes, but this was not correlated with ambient temperature. The rank of individual lakes with respect to the abundance of emerging chironomids was consistent among as well as within years, a predictability that ought to make adaptive lake choice by ducks possible. On the lake level, there was a positive correlation between the total amount of emerging chironomids and brood use. We argue that emergence patterns of chironomids on typical boreal lakes are neither compressed nor predictable enough to be a major selective force on the timing of egg-laying and hatching in dabbling ducks. Despite spatial (among-lake) patterns of abundance of emerging chironomids being predictable within and among years, the observed pattern of brood use suggests that other factors, e.g. habitat structure, also affect lake choice.     

Functional responses and predator–prey models: a critique of ratio dependence

Arditi and Ginzburg (2012) propose ordinary differential equations (ODEs) with ratio-dependent functional responses as the new null model for predation, based on their earlier work on ratio-dependent food chains and a number of functional response measurements. Here, I discuss some of their claims, arguing for a flexible and problem-driven approach to predator–prey modeling. Models to understand population cycles and models to predict the effect of basal enrichment on food chains need not be the same. While ratio-dependent functional responses in ODE models might sometimes be useful as limit cases for food chains, they are not intrinsically more useful than prey-dependent models to understand the effect of a given predator on prey population dynamics—and sometimes less useful, given the small temporal scales considered in many models. “Instantism” is showed to be an invalid criticism when ODEs are interpreted as describing average trajectories of stochastic birth–death processes. Moreover, other modeling frameworks with strong ties to time series statistics, such as stochastic difference equations, should be promoted to improve the feedback loop between field and theoretical research. The main problems of current trophic ecology do not lie in a wrong null model, as ecologists have already several at their disposal. The loose connection of ODE models with empirical data and spatial/temporal scaling up of empirical measurements constitute more serious challenges to our understanding of trophic interactions and their consequences on ecosystem functioning.     

Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator–prey model

Many natural systems are subject to seasonal environmental change. As a consequence many species exhibit seasonal changes in their life history parameters—such as a peak in the birth rate in spring. It is important to understand how this seasonal forcing affects the population dynamics. The main way in which seasonal models have been studied is through a two dimensional bifurcation approach. We augment this bifurcation approach with extensive simulation in order to understand the potential solution behaviours for a predator–prey system with a seasonally forced prey growth rate. We consider separately how forcing influences the system when the unforced dynamics have either monotonic decay to the coexistence steady state, or oscillatory decay, or stable limit cycles. The range of behaviour the system can exhibit includes multi-year cycles of different periodicities, parameter ranges with coexisting multi-year cycles of the same or different period as well as quasi-periodicity and chaos. We show that the level of oscillation in the unforced system has a large effect on the range of behaviour when the system is seasonally forced. We discuss how the methods could be extended to understand the dynamics of a wide range of ecological and epidemiological systems that are subject to seasonal changes.