Pattern formation and individual-based models: The importance of understanding individual-based movement

This paper demonstrates that while pattern formation can stabilize individual-based models of predator–prey systems, the same individual-based models also allow for stabilization by alternate mechanisms, particularly localized consumption or diffusion limitation. The movement rules of the simulation are the critical feature which determines which of these mechanisms stabilizes any particular predator–prey individual-based model. In particular, systems from well-connected subpopulations, in each of which a predator can attack any prey, generally exhibit stabilization by pattern formation. In contrast, when restricted movement within a (sub-)population limits the ability of predators to consume prey, localized consumption or diffusion limitation can stabilize the system. Thus while the conclusions from differential equations on the role of pattern formation for stability may apply to discrete and noisy systems, it will take a detailed understanding of movement and scales of interaction to examine the role of pattern formation in real systems. Additionally, it will be important to link an understanding of both foraging and inter-patch movement, since by analogy to the models, both would be critical for understanding how real systems are stabilized by being discrete and spatial.     

Role of standard incidence in an eco-epidemiological system: A mathematical study

Ecology and epidemiology are two major fields of study in their own right, but they have some common features. [Chattopadhyay, J., Pal, S., El Abdllaoui, A., 2003. Classical predator–prey system with infection of prey population—a mathematical model. Math. Meth. Appl. Sci. 26, 1211–1222] considered a predator–prey model with disease in the prey population. They analyzed the system based on the assumption that horizontal incidence follows simple mass action incidence. Mass action incidence is appropriate for a constant population, but for a large population, standard incidence is more appropriate. The complicated dynamics around (0, 0, 0) arise because of standard incidence. The conditions under which the population reaches the origin either by following the axis or in a spiral pattern were determined. Numerical experiments were performed to observe the dynamics of the system with mass action incidence and standard incidence. This investigation showed that the infection rate plays a crucial role for predicting the behavior of the dynamics in the long run.     

Predator–prey fuzzy model

In this work we have used fuzzy rule-based systems to elaborate a predator–prey type of model to study the interaction between aphids (preys) and ladybugs (predators) in citriculture, where the aphids are considered as transmitter agents of the Citrus Sudden Death (CSD). Simulations were performed and a graph was drawn to show the prey population, the potentiality of the predators, and a phase-plane. From this phase-plane, a classic model of the Holling–Tanner type is fitted and its parameters were found. Finally, we have studied the stability of the critical points of the Holling–Tanner model.     

Modeling inverted biomass pyramids and refuges in ecosystems

Although the existence of robust inverted biomass pyramids (IBPs) seems paradoxical, they are well known to exist in planktonic communities, and have recently been discovered in pristine coral reefs and in a reef off the North Carolina coast. Understanding the underlying mechanisms which produce inverted biomass pyramids provides new ecological insights. Some ecologists hypothesize that “the high growth rate of prey and low death rate of predators” causes IBPs. However, we show this is not always the case (see Sections 3.1 and 4). We devise predator–prey models to describe three mechanisms that can lead to IBPs: (1) well-mixed populations with large prey turn-over rate, (2) well-mixed populations with prey immigration, and (3) non-mixed populations where the prey can hide in refuges. The three models are motivated by the three ecosystems where IBPs have been observed. We also devise three refuge mediated models, with explicit refuge size, which incorporate different prey responses in the refuge, and we discuss how these lead to IBPs.     

Modelling predation by transient leopard seals for an ecosystem-based management of Southern Ocean fisheries

Correctly quantifying the impacts of rare apex marine predators is essential to ecosystem-based approaches to fisheries management, where harvesting must be sustainable for targeted species and their dependent predators. This requires modelling the uncertainty in such processes as predator life history, seasonal abundance and movement, size-based predation, energetic requirements, and prey vulnerability. We combined these uncertainties to evaluate the predatory impact of transient leopard seals on a community of mesopredators (seals and penguins) and their prey at South Georgia, and assess the implications for an ecosystem-based management. The mesopredators are highly dependent on Antarctic krill and icefish, which are targeted by regional fisheries. We used a state-space formulation to combine (1) a mark-recapture open-population model and individual identification data to assess seasonally variable leopard seal arrival and departure dates, numbers, and residency times; (2) a size-based bioenergetic model; and (3) a size-based prey choice model from a diet analysis. Our models indicated that prey choice and consumption reflected seasonal changes in leopard seal population size and structure, size-selective predation and prey vulnerability. A population of 104 (90–125) leopard seals, of which 64% were juveniles, consumed less than 2% of the Antarctic fur seal pup production of the area (50% of total ingested energy, IE), but ca. 12–16% of the local gentoo penguin population (20% IE). Antarctic krill (28% IE) were the only observed food of leopard seal pups and supplemented the diet of older individuals. Direct impacts on krill and fish were negligible, but the “escapement” due to leopard seal predation on fur seal pups and penguins could be significant for the mackerel icefish fishery at South Georgia. These results suggest that: (1) rare apex predators like leopard seals may control, and may depend on, populations of mesopredators dependent on prey species targeted by fisheries; and (2) predatory impacts and community control may vary throughout the predator's geographic range, and differ across ecosystems and management areas, depending on the seasonal abundance of the prey and the predator's dispersal movements. This understanding is important to integrate the predator needs as natural mortality of its prey in models to set prey catch limits for fisheries. Reliable estimates of the variability of these needs are essential for a precautionary interpretation in the context of an ecosystem-based management.     

Food web structure affects the extinction risk of species in ecological communities

This paper studies the effect of food web structure on the extinction risk of species. We examine 793 different six-species food web structures with different number, position and strength of trophic links and expose them to stochasticity in a model with Lotka–Volterra predator–prey dynamics. The characteristics of species (intrinsic rates of increase as well as intraspecific density dependence) are held constant, but the interactions with other species and characteristics of the food web are varied.     

Deterministic approach to the study of the interaction predator-prey in a chemostat with predator mutual interference. Implications for the paradox of enrichment

Our understanding of predator-prey systems has progressed in recent decades mainly due to the ability to test models in chemostats. This study aimed to develop a deterministic model using differential equations to reproduce the dynamics of the interaction of a predator and a prey in a two stage chemostat focusing in the proposed previous prey dependent model of Fussmann et al. (2000) [Fussmann, G.F., Ellner, S.P., Shertzer, K.W., Hairston Jr., N.G., 2000. Crossing the Hopf bifurcation in a live predator-prey system. Science 290, 1358-1360]. The main problem with that model, but parameterized with the values obtained in this study (particularly the concentration of nutrient), was that the temporal trajectory of both the prey and the predator showed very high peaks that eventually led to the extinction of predator in all cases. In the same way the experimental time series obtained in this study does not exhibit the behavior predicted by the model of Fussman et al. On the contrary, as prey density increases, the system actually becomes more stable. Finally, the model that best explained the behavior of the predator and prey in the chemostat, at medium to high dilution rates, was the ratio dependent (algae-nitrogen) model with mutual interference measured in the chemostat (rotifer-alga) and that incorporated the age structure of the predator. Qualitative analysis of the dynamic behavior enabled evaluation of coexistence at equilibrium, coexistence on limit cycles, extinction of the predator or extinction of both populations.     

Behavioral response to predators reverses the outcome of competition between prey species

Summary In a laboratory experiment it was shown that piscivorous predators reversed the outcome of competitive interactions between two fish prey species, juveniles of roach (Rutilus rutilus) and perch (Perca fluviatilis), by behaviorally affecting their use of two available habitats, an open water habitat and a structurally complex refuge. The shift in the competitive relationship was the result of predators forcing the juvenile fishes into a prey refuge with high structural complexity. While roach was competitively superior in the unstructured habitat, perch was superior in the structurally complex prey refuge. The reversal in competitive relationship was demonstrated both with respect to foraging rate and growth rate and resulted from the high structural complexity in the prey refuge interfering with the roach's swimming performance. Because survival and growth patterns through the juvenile stages have profound effects on the population/community dynamics of size-structured populations such as those of fish, behaviorally induced changes in competitive ability should have significant implications also at the population and community levels.     

A resource-based approach to modelling the dynamics of interacting populations

The procedure for modelling the growth of single-species populations [Sakanoue, S., 2007. Extended logistic model for growth of single-species populations. Ecol. Model. 205, 159–168] is improved to be applicable to the study of the dynamics of interacting populations. The improved procedure is based on three assumptions: resource availability changes with population size as a variable, resource supply to populations and population demand for resources are defined as functions of resource availability and population size, and the variables of resource availability and population size shift in the supply function attracted to the demand function. These assumptions are organized into three equations. The equations can generate the dynamics models of plant, herbivore, and detritivore populations, and their own resources. The models can be used to describe prey–predator dynamics. They naturally contain nonlinear terms for the predator’s numerical and functional responses. Depending on the terms, the fluctuations in resource availability and population size stabilize. The three equations can also generate the dynamics models of different populations consuming the same resources. The analysis of zero isoclines of the models shows that a superior population can be simply defined as one with a higher intrinsic rate of natural increase, that a stable coexistence may be realized with the intraspecific interference or the interspecific facilitation of superiors, and that the interspecific interference or the intraspecific facilitation of inferiors may make the coexistence unstable and the inferiors winners depending on their initial population size.     

Predation affects the susceptibility of hard clam Meretrix lusoria to Hg-stressed birnavirus

Predator–prey interaction in aquatic ecosystem is one of the simplest drivers affecting the species population dynamics. Predation controls are recognized as important aspects of ecosystem husbandry and management. In this paper we investigated how predation control cause an increase in host growth in the abundance of hard clam (Meretrix lusoria) populations subject to mercury (Hg)-stressed birnavirus. Here we linked predator–prey relationships with a bioenergetic matrix population model (MPM) associated with a susceptible–infectious–mortality (SIM) model based on a host–pathogen–predator framework to quantify the predator effects on population dynamics of disease in hard clam populations. Our results indicated that relative high predation rates could promote the hard clam abundances in relation to predators that selectively captured the infected hard clam, by which the disease transmission was suppressed. The results also demonstrated that predator-induced modifications in host behavior could have potential negative or positive effects on host growth depending on relative species density and resource dynamics. The most immediate implication of this study for the management of aquatic ecosystem is that, beyond the potential for causing a growth in abundance, predation might provoke greater predictability in aquatic ecosystem species populations and thereby increase the safety of ecosystem production from stochastic environmental events.     

Chaos and pattern formation in a spatial tritrophic food chain

The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics. The model assumes that the populations are homogeneously mixed, so that the probability that any two individuals interact is uniform and space can be ignored. In this paper we propose a spatial version of the Hastings and Powell model in which predators seek their preys only in a finite neighborhood of their home location, breaking the mixing hypothesis. Treating both space and time as discrete variables we derive a set of coupled equations that describe the evolution of the populations at each site of the spatial domain. We show that the introduction of local predator–prey interactions result in qualitatively distinct dynamics of predator and prey populations. The evolution equations for the predators involve averages over the local density of preys, whereas the equations for the preys involve double averages, where the local density of both preys and predators appear. Our numerical simulations show that local predation also leads to spontaneous pattern formation and to qualitative changes in the global dynamics of the system. In particular, depending on the size of the predation neighborhoods, the chaotic strange attractor present in the original model of Hastings and Powell can be replaced by a stable fixed point or by an attractor of simpler topology.     

Dynamics of a two-prey–one-predator system with predator switching regulated by a catabolic repression control-like mode

The strategy of alternative prey (switching), regulated by a catabolic repression control-like (CRCL) mode, in which the desirable prey acts as repressor on the predator's attack on the alternative prey, was studied in a two-prey–one-predator chemostat system. In this system, alternative prey has to survive predation and competition for nutrients from the desirable prey. Therefore, when the alternative prey has no competitive advantage for the common substrate over the desirable prey, its survival depends on the protection offered by the desirable prey via CRCL. In this case, CRCL allows the coexistence of both desirable and alternative prey and predator populations. However, when the alternative prey has the competitive advantage over the desirable prey, CRCL negatively affects both the state of survival of the desirable prey and the coexistence state.     

Predator–prey models: an application for the plankton dynamics of lake Geneva

In this paper we present a hierarchical Bayesian analysis for a predator–prey model applied to ecology considering the use of Markov Chain Monte Carlo methods. We consider the introduction of a random effect in the model and the presence of a covariate vector. An application to ecology is considered using a data set related to the plankton dynamics of lake Geneva for the year 1990. We also discuss some aspects of discrimination of the proposed models.     

Sensitivity of system stability to model structure

A community is stable, and resilient, if the levels of all community variables can return to the original steady state following a perturbation. The stability properties of a community depend on its structure, which is the network of direct effects (interactions) among the variables within the community. These direct effects form feedback cycles (loops) that determine community stability. Although feedback cycles have an intuitive interpretation, identifying how they form the feedback properties of a particular community can be intractable. Furthermore, determining the role that any specific direct effect plays in the stability of a system is even more daunting. Such information, however, would identify important direct effects for targeted experimental and management manipulation even in complex communities for which quantitative information is lacking. We therefore provide a method that determines the sensitivity of community stability to model structure, and identifies the relative role of particular direct effects, indirect effects, and feedback cycles in determining stability. Structural sensitivities summarize the degree to which each direct effect contributes to stabilizing feedback or destabilizing feedback or both. Structural sensitivities prove useful in identifying ecologically important feedback cycles within the community structure and for detecting direct effects that have strong, or weak, influences on community stability. The approach may guide the development of management intervention and research design. We demonstrate its value with two theoretical models and two empirical examples of different levels of complexity.     

Transient dynamics and the destabilizing effects of prey heterogeneity

The presence of prey heterogeneity and weakly interacting prey species is frequently viewed as a stabilizer of predator-prey dynamics, countering the destabilizing effects of enrichment and reducing the amplitude of population cycles. However, prior model explorations have largely focused on long-term, dynamic attractors rather than transient dynamics. Recent theoretical work shows that the presence of prey that are defended from predation can have strongly divergent effects on dynamics depending on time scale: prey heterogeneity can counteract the destabilizing effects of enrichment on predator-prey dynamics at long time scales but strongly destabilize systems during transient phases by creating long periods of low predator/prey abundance and increasing extinction probability (an effect that is amplified with increasing enrichment). We tested these general predictions using a planktonic system composed of a zooplankton predator and multiple algal prey. We first parameterized a model of our system to generate predictions and tested these experimentally. Our results qualitatively supported several model predictions. During transient phases, presence of defended algal prey increased predator extinctions at low and high enrichment levels compared to systems with only a single edible prey. This destabilizing effect was moderated at higher dilution rates, as predicted by our model. When examining dynamics beyond initial oscillations, presence of the defended prey increased predator-prey temporal variability at high nutrient enrichment but had no effect at low nutrient levels. Our results highlight the importance of considering transient dynamics when assessing the role of stabilizing factors on the dynamics of food webs.     

Challenges of simulating complex environmental systems at the landscape scale: A controversial dialogue between two cups of espresso

With the advancement of computational systems and the development of model integration concepts, complexity of environmental model systems increased. In contrast to that, theory and knowledge about > environmental systems as well as the capability for environmental systems analyses remained, to a large extent, unchanged. As a consequence, model conceptualization, data gathering, and validation, have faced new challenges that hardly can be tackled by modellers alone. In this discourse-like review, we argue that modelling with reliable simulations of human-environmental interactions necessitate linking modelling and simulation research much stronger to science fields such as landscape ecology, community ecology, eco-hydrology, etc. It thus becomes more and more important to identify the adequate degree of complexity in environmental models (which is not only a technical or methodological question), to ensure data availability, and to test model performance. Even equally important, providing problem specific answers to environmental problems using simulation tools requires addressing end-user and stakeholder requirements during early stages of problem development. In doing so, we avoid modelling and simulation as an end of its own.     

Fine-scale observations of the predatory behaviour of the carnivorous copepod <Emphasis Type="Italic">Paraeuchaeta norvegica</Emphasis> and the escape responses of their ichthyoplankton prey,Atlantic cod (<Emphasis Type="Italic">Gadus morhua</Emphasis>)

Paraeuchaeta norvegica (8.5 mm total length) and yolk-sac stage Atlantic cod larvae (4 mm total length) (Gadus morhua) larvae were observed in aquaria (3 l of water) using silhouette video photography. This allowed direct observations (and quantitative measurement) of predator–prey interactions between these two species in 3-dimensions. Tail beats, used by cod larvae to propel themselves through the viscous fluid environment, also generate signals detectable by mechanoreceptive copepod predators. When the prey is close enough for detection and successful capture (approximately half a body-length), the copepod launches an extremely rapid high Reynolds number attack, grabbing the larva around its midsection. While capture itself takes place in milliseconds, minutes are required to subdue and completely ingest a cod larva. The behavioural observations are used to estimate the hydrodynamic signal strength of the cod larva’s tail beats and the copepod’s perceptive field for larval fish prey. Cod larvae are more sensitive to fluid velocity than P. norvegica and also appear capable of distinguishing between the signal generated by a swimming and an attacking copepod. However, the copepod can lunge at much faster velocities than a yolk-sac cod larva can escape, leading to the larva’s capture. These observations can serve as input to the predator–prey component of ecosystem models intended to assess the impact of P. norvegica on cod larvae.     

Identification of calanoid copepod prey species via molecular detection of carbon fixation genes

Zooplankton and their phytoplankton prey form the basis of the marine food web, yet historically it has been difficult to discern species-specific trophic interactions. Molecular techniques provide opportunities to obtain taxonomic data where the traditional methodologies for gut content analysis lack resolution. The large subunit gene of RubisC/O, rbcL, was utilized as a molecular marker for the identification of prey species in calanoid copepods. Clone libraries were generated from DNA extracted from seawater and whole copepods during a transect cruise on the northern Gulf of Mexico shelf. Sequence data analysis provided evidence of diatoms, nanoplankton-sized chlorophytes, and cyanobacteria in DNA extracted from whole copepods. These data demonstrate that rbcL can be a useful marker for the identification of copepod phytoplankton prey. Combining the described approach with quantitative techniques such as quantitative PCR will provide opportunities for the assessment of species-specific predator–prey interactions.     

Individual boldness affects interspecific interactions in sticklebacks

Within populations of many species, individuals that are otherwise similar to one another in age, size or sex can differ markedly in behaviours such as resource use, risk taking and competitive ability. There has been much research into the implications of such variation for intraspecific interactions, yet little investigation into its role in influencing interspecific interactions outside of a predator–prey context. In this study, we investigated the role of individual-level behavioural variation in determining the outcomes of interactions between two ecologically similar fishes, the threespine and ninespine sticklebacks (Gasterosteus aculeatus and Pungitius pungitius). Experiment 1 asked whether individuals of both species were consistent in their expression of two behaviours: activity in novel surroundings and latency to attack prey. For each behaviour, focal individuals were assayed twice, 10 days apart. Performances were positively correlated between exposures, suggesting behavioural consistency within individuals, at least over this timescale. Experiment 2 revealed not only differences in habitat use described both by species-level variation, with ninespines spending more time in vegetated areas, but also by individual differences, with more active individuals of both species spending more time in open water than in vegetation. Experiment 3 revealed that when heterospecific pairs competed for prey, bolder individuals consumed a greater share, irrespective of species. These findings suggest that individual-level variation can facilitate overlap in habitat use between heterospecifics and also determine the outcomes of resource contests when they meet. We discuss how this might vary between populations as a function of prevailing selection pressures and suggest approaches for testing our predictions.     

Vulnerability of the copepod Acartia tonsa to predation by the scyphomedusa Chrysaora quinquecirrha : effect of prey size and behavior

Although scyphomedusae have received increased attention in recent years as important predators in coastal and estuarine environments, the factors affecting zooplankton prey vulnerability to these jellyfish remain poorly understood. Current models predicting feeding patterns of cruising entangling predators, such as Chrysaora quinquecirrha (Desor, 1948), fail to account for the selection of fast-escaping prey such as copepods. Nevertheless, our analysis of gastric contents of field-collected medusae showed that this scyphomedusa fed selectively on the calanoid copepod Acartia tonsa (Dana, 1846) and preferentially ingested adult over copepodite stages. We measured feeding rates in a planktonkreisel while simultaneously videotaping predator–prey interactions. C. quinquecirrha consumed adult A. tonsa ten times faster than copepodites. Differences in prey behavior, in the form of predator–prey encounter rates or post-encounter escape responses, could not account for the elevated feeding rates on adults. Prey size, however, had a dramatic impact on the vulnerability of copepods. In experiments using heat-killed prey, feeding rates on adults (1.5 times longer than copepodites) were 11 times higher than on copepodites. In comparison, medusae ingested heat-killed prey at only two to three times the rate of live prey. These results suggest that during scyphomedusan–copepod interactions, prey escape ability is important, but ultimately small size is a more effective refuge from predation. Received: 26 September 1997 / Accepted: 20 May 1998