Dynamics of a mechanistically derived stoichiometric producer-grazer model

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137–1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers.     

Enrichment and stability: a phenomenological coupling of energy value and carrying capacity

Simple predator-prey models with a prey-dependent functional response predict that enrichment (increased carrying capacity) destabilizes community dynamics: this is the 'paradox of enrichment'. However, the energy value of prey is very important in this context. The intraspecific chemical composition of prey species determines its energy value as a food for the potential predator. Theoretical and experimental studies establish that variable chemical composition of prey affects the predator-prey dynamics. Recently, experimental and theoretical approaches have been made to incorporate explicitly the stoichiometric heterogeneity of simple predator-prey systems. Following the results of the previous experimental and theoretical advances, in this article we propose a simple phenomenological formulation of the variation of energy value at increased level of carrying capacity. Results of our study demonstrate that coupling the parameters representing the phenomenological energy value and carrying capacity in a realistic way, may avoid destabilization of community dynamics following enrichment. Additionally, under such coupling the producer-grazer system persists for only an intermediate zone of production--a result consistent with recent studies. We suggest that, while addressing the issue of enrichment in a general predator-prey model, the phenomenological relationship that we propose here might be applicable to avoid Rosenzweig's paradox.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Carrying capacity models should not use fixed prey density thresholds: a plea for using more tools of behavioural ecology

Earlier studies have developed models of carrying capacity to predict the number of animals a certain area can support. These models assume that resources are not renewed after consumption ('standing stock' models), and that the initial number of prey and the rate of prey consumption determine the time a population of foragers can live in an area. Within such areas, foragers give up feeding at a sub-site or patch when intake rates no longer cover energy expenditure. To improve the success rate of the models' predictions, we here change the existing rate-maximising models into fitness-maximising models, and include dynamics in the availability of patches. These new (conceptual) models show that the approaches used so far may over- as well as underestimate carrying capacity. We review empirical studies that have aimed to estimate carrying capacity, and discuss how concepts have been confused. We make explicit suggestions on how to proceed in predicting carrying capacities in future studies.     

Two predator-prey difference equations considering delayed population growth and starvation

A predator-prey model with overlapping generations is proposed, assuming delayed population growth and a new estimate of the number of predators starving to death. Differences among simulation results caused by manipulation of values of coefficients are discussed. The main results are: (i) although conceptually predator density is not associated so closely with prey density as in several other models (mainly by mortality of predators and not by their natality), the model shows only a few changes in its dynamic properties. (ii) Both the capacity of predators to resist starvation and the carrying capacity of prey have an important influence on the stability of the simulation results. (iii) Stable limit cycles were found without assuming influences of carrying capacity.     

Self-organization, scale and stability in a spatial predator-prey interaction

Simple predator-prey models often predict extreme instability in interactions where the prey are depressed well below their carrying capacity. Although the behaviour of some laboratory systems conforms to this pattern, field and mesocosm studies generally show prolonged co-existence of prey and predator. Prominent among the possible causes of this discrepancy are the effects of spatial heterogeneity. In this paper we show that both discrete and continuous representations of the spatial Rosenzweig-McArthur model with immobile prey can be stabilized by self-organized prey heterogeneity. This concordance of behaviour closely parallels that which we have previously established in the context of invasion waves. We use the continuous model variant to calculate the characteristic spatial scales of the self-organized structures. The discrete variant forms the basis of a simulation study demonstrating the variety of stable structures and elucidating their relation to the history of the system. We note that all stable prey distributions take the form of a network of occupied patches separated by prey-free regions, and liken the process which generates such assemblages to the formation of a landscape mozaic.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.     

Seasonal subsidy stabilizes food web dynamics: Balance in a heterogeneous landscape

Resource subsidies from external habitats can substantially affect the food web dynamics of local habitats. In this paper, we explore a mathematical model that is tailored for a stream food web, studied by Nakano and colleagues, in which consumers, in situ prey and subsidies all show seasonal fluctuation. The model reveals that the food web dynamics are stabilized if subsidies increase in summer when in situ productivity is low. Consumer dynamics are stabilized because subsidies complement seasonal resource deficiency. In situ prey dynamics are stabilized because subsidies indirectly balance the predation pressure by consumers, with seasonal change in prey carrying capacity. In summer when prey carrying capacity is low, seasonally abundant subsidies indirectly decrease predation pressure, whereas in winter, with high prey carrying capacity, scarce subsidies increase the predation pressure. Our results suggest that temporal productivity differences between spatially linked habitats are important to promote the stability of food web dynamics in a landscape context.     

Determination of Foraging Thresholds and Effects of Application on Energetic Carrying Capacity for Waterfowl

Energetic carrying capacity of habitats for wildlife is a fundamental concept used to better understand population ecology and prioritize conservation efforts. However, carrying capacity can be difficult to estimate accurately and simplified models often depend on many assumptions and few estimated parameters. We demonstrate the complex nature of parameterizing energetic carrying capacity models and use an experimental approach to describe a necessary parameter, a foraging threshold (i.e., density of food at which animals no longer can efficiently forage and acquire energy), for a guild of migratory birds. We created foraging patches with different fixed prey densities and monitored the numerical and behavioral responses of waterfowl (Anatidae) and depletion of foods during winter. Dabbling ducks (Anatini) fed extensively in plots and all initial densities of supplemented seed were rapidly reduced to 10 kg/ha and other natural seeds and tubers combined to 170 kg/ha, despite different starting densities. However, ducks did not abandon or stop foraging in wetlands when seed reduction ceased approximately two weeks into the winter-long experiment nor did they consistently distribute according to ideal-free predictions during this period. Dabbling duck use of experimental plots was not related to initial seed density, and residual seed and tuber densities varied among plant taxa and wetlands but not plots. Herein, we reached several conclusions: 1) foraging effort and numerical responses of dabbling ducks in winter were likely influenced by factors other than total food densities (e.g., predation risk, opportunity costs, forager condition), 2) foraging thresholds may vary among foraging locations, and 3) the numerical response of dabbling ducks may be an inconsistent predictor of habitat quality relative to seed and tuber density. We describe implications on habitat conservation objectives of using different foraging thresholds in energetic carrying capacity models and suggest scientists reevaluate assumptions of these models used to guide habitat conservation.     

Predator search pattern and the strength of interference through prey depression

We develop a model of predators foraging within a single patch,on prey that become temporarily immune to predation (depressed)after detecting a predator. Interference through prey depressionoccurs because the proportion of vulnerable prey (and henceintake rate) decreases as predator density increases. Predatorsin our model are not forced to move randomly within the patch,as is the case in other similar models, but can avoid areasof depressed prey and so preferentially forage over vulnerableprey. We compare the extent to which different avoidance rules(e.g., move more quickly over depressed prey or turn if approachingdepressed prey) influence the amount of time spent foragingover depressed and vulnerable prey, and how this influencesthe strength of interference. Although based on a different mechanism, our model produces two similar general predictionsto interference models based on direct interactions betweenpredators: the strength of interference increases with (1)increased competitor density and (2) decreased prey encounterrate. This suggests that there are underlying similarities in the nature of interference even when it arises through differentprocesses. Not surprisingly, avoidance of depressed prey cansubstantially reduce the strength of interference comparedwith random foraging. However, we identify the region of themodel's parameter space in which this reduction is particularlylarge and show that the only system for which suitable dataare available, redshank Tringa totanus feeding on Corophium volutator, falls within this region. The model shows that, byadjusting its search path to avoid areas of depressed prey,a predator can substantially reduce the amount of the interferenceit experiences and that this applies over a wide range of parameterspace, including the region occupied by a real system. Thissuggests that behavior-based interference models should consider predator search pattern if they are to accurately predict thestrength of the interference.     

Effect of prey mass and selection on predator carrying capacity estimates

The ability to determine the prey-specific biomass intake of large predators is fundamental to their conservation. In the absence of actual prey data, researchers generally use a “unit mass” method (estimated as 3/4 adult female mass) to calculate the biomass intake of predators. However, differences in prey preference and range across geographic regions are likely to have an influence on biomass calculations. Here we investigate the influence of estimated prey mass on leopard biomass calculations, and subsequent carrying capacity estimates, in an understudied mountain population. Potential leopard feeding sites were identified using global positioning system (GPS) location clusters obtained from GPS collars. We investigated 200 potential leopard feeding sites, of which 96 were actual feeding sites. Jaw bones, horns, hooves, and other indicative bones were used to determine gender and age of prey items, which were subsequently used to calculate mass of each prey item based on previously published values. There were significant differences in the biomass values calculated using the traditional unit mass method and the calculated prey masses obtained from leopard feeding sites. However, there were no considerable differences in the carrying capacity estimates using the preferred prey species model and leopard density estimates calculated using a non-biased spatial approach, which suggests that estimating carnivore carrying capacity based on 3/4 adult female masses is a reliable method also for the mountain population in this study.     

How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

In Rosenzweig-MacArthur models of predator-prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator's functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator-prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents.     

Functional response,numerical response,and stability in arthropod predator-prey ecosystems involving age structure

Summary A general model of arthropod predator-prey systems incorporating age structure in the predator is employed to study the role of functional and numerical responses on stability and the paradox of enrichment. The destabilizing effect of age structure leads to both qualitatively and quantitatively new results for an environment which has an infinite prey carrying capacity, including a lower bound to prey density for a stable equilibrium, a feature not present in models without age structure. When applied to an environment with finite prey carrying capacity, the effect of age structure is to reinforce the arguments implicit to the paradox of enrichment originally developed for traditional models lacking age structure.     

The ecology of fear and inverted biomass pyramids

In inverted biomass pyramids (IBPs) prey are outnumbered by their predators when measured by biomass. We investigate how prey should behave in the face of danger from higher predator biomass, and how anti-predator behavior (in the form of vigilance) can, in turn, affect the predator–prey system. In this study, we incorporate anti-predator behaviors into a Lotka–Volterra predator–prey model in the form of fixed and facultative vigilance. Facultative vigilance models behavior as a dynamic foraging game, allowing us to assess optimal behavioral responses in the context of IBPs using a dynamical fitness optimization approach. We model vigilance as a tradeoff between safety and either the prey's maximum growth rate or its carrying capacity. We assess the population dynamics of predators and prey with fear responses, and investigate the role fear plays on trophic structure. We found that the ecology of fear plays an important role in predator–prey systems, impacting trophic structure and the occurrence of IBPs. Fixed vigilance works against IBP structure by always reducing the predator–prey biomass ratio at equilibrium with increasing levels of vigilance. Facultative vigilance can actually promote IBPs, as prey can now adjust their vigilance levels to cope with increased predation and the costs associated with vigilance. This is especially true when the effectiveness of vigilance is low and predators are very lethal. In general, these trends are true whether the costs of vigilance are felt on the prey's maximum growth rate or its carrying capacity. Just as the ecology of fear, when first introduced, was used to explain why top carnivores are rare in terrestrial systems, it can also be used to understand how big fierce predators can be common in IBPs.     

How to model the local interaction in the predator–prey system at slow diffusion in a heterogeneous environment?

We examine the nonlinear reaction–diffusion–advection equations to modeling of the predator–prey system under heterogeneous carrying capacity of the prey, and Holling type II functional response. When advection and diffusion fluxes are absent or small, we detect the discrepancy between the resource (carrying capacity) and species distributions. The large diffusion eliminates this effect. We propose a modification of the functional response coefficients to provide the correlation between species distribution and resource in both cases. The numerical simulation of several models both under small and moderate advection–diffusion fluxes is carried out.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Coexistence and displacement in consumer-resource systems with local and shared resources

Competition for local and shared resources is widespread. For example, colonial waterbirds consume local prey in the immediate vicinity of their colony, as well as shared prey across multiple colonies. However, there is little understanding of conditions facilitating coexistence vs. displacement in such systems. Extending traditional models based on type I and type II functional responses, we simulate consumer-resource systems in which resources are “substitutable,” “essential,” or “complementary.” It is shown that when resources are complementary or essential, a small increase in carrying capacity or decrease in handling time of a local resource may displace a spatially separate consumer species, even when the effect on shared resources is small. This work underscores the importance of determining both the nature of resource competition (substitutable, essential, or complementary) and appropriate scale-dependencies when studying metacommunities. We discuss model applicability to complex systems, e.g., urban wildlife that consume natural and anthropogenic resources which may displace rural competitors by depleting shared prey.     

A derivative of the CRMP2 binding compound lanthionine ketimine provides neuroprotection in a mouse model of cerebral ischemia

Lanthionines are novel neurotrophic and neuroprotective small molecules that show promise for the treatment of neurodegenerative diseases. In particular, a recently developed, cell permeable lanthionine derivative known as LKE (lanthionine ketimine 5-ethyl ester) promotes neurite growth at low nanomolar concentrations. LKE also has neuroprotective, anti-apoptotic, and anti-inflammatory properties. Its therapeutic potential in cerebral ischemia and its mechanisms of neurotrophic action remain to be fully elucidated. Here, we hypothesize that the neuroprotective actions of LKE could result from induction or modulation of CRMP2. We found that treating primary cultured mouse neurons with LKE provided significant protection against t-butyl hydroperoxide-induced neuronal death possibly through CRMP2 upregulation. Similarly, in vivo studies showed that LKE pre and/or post-treatment protects mice against permanent distal middle cerebral artery occlusion (p-MCAO) as evidenced by lower stroke lesions and improved functional outcomes in terms of rotarod, grip strength and neurologic deficit scores in treated groups. Protein expression levels of CRMP2 were higher in brain cortices of LKE pretreated mice, suggesting that LKE’s neuroprotective activity may be CRMP2 dependent. Lower activity of cleaved PARP-1 and higher activity of SIRT-1 was also observed in LKE treated group suggesting its anti-apoptotic properties. Our results suggest that LKE has potential as a therapeutic intervention in cerebral ischemia and that part of its protective mechanism may be attributed to CRMP2 mediated action and PARP-1/SIRT-1 modulation.     

Estimating Carrying Capacity for Sea Otters in British Columbia

ABSTRACT We estimated carrying capacity for sea otters (Enhydra lutris) in the coastal waters of British Columbia, Canada, by characterizing habitat according to the complexity of nearshore intertidal and sub-tidal contours. We modeled the total area of complex habitat on the west coast of Vancouver Island by first calculating the complexity of the Checleset Bay-Kyuquot Sound (CB-KS) region, where sea otters have been at equilibrium since the mid-1990s. We then identified similarly complex areas on the west coast of Vancouver Island (WCVI model), and adapted the model to identify areas of similar complexity along the entire British Columbia coast (BC model). Using survey data from the CB-KS region, we calculated otter densities for the habitat predicted by the 2 models. The density estimates for CB-KS were 3.93 otters/km² and 2.53 otters/km² for the WCVI and BC models, respectively, and the resulting 2 estimates of west coast of Vancouver Island complex habitat carrying capacity were not significantly different (WCVI model: 5,123, 95% CI = 3,337–7,104; BC model: 4,883, 95% CI = 3,223–6,832). The BC model identified the region presently occupied by otters on the central British Columbia coast, but the amount of coast-wide habitat it predicted (5,862 km²) was relatively small, and the associated carrying capacity estimate (14,831, 95% CI = 9,790–20,751) was low compared to historical accounts. We suggest that our model captured a type of high-quality or optimum habitat prevalent on the west coast of Vancouver Island, typified by the CB-KS region, and that suitable sea otter habitat elsewhere on the coast must include other habitat characteristics. We therefore calculated a linear, coast-wide carrying capacity of 52,459 sea otters (95% CI = 34,264–73,489)—a more realistic upper limit to sea otters in British Columbia. Our carrying capacity estimates are helping set population recovery targets for sea otters in Canada, and our habitat predictions represent a first step in Critical Habitat identification. This habitat-based approach to estimating carrying capacity is likely suitable for other nonmigratory, density-dependent species.     

Integrating stomach content and stable isotope analyses to quantify the diets of pygoscelid penguins

Stomach content analysis (SCA) and more recently stable isotope analysis (SIA) integrated with isotopic mixing models have become common methods for dietary studies and provide insight into the foraging ecology of seabirds. However, both methods have drawbacks and biases that may result in difficulties in quantifying inter-annual and species-specific differences in diets. We used these two methods to simultaneously quantify the chick-rearing diet of Chinstrap (Pygoscelis antarctica) and Gentoo (P. papua) penguins and highlight methods of integrating SCA data to increase accuracy of diet composition estimates using SIA. SCA biomass estimates were highly variable and underestimated the importance of soft-bodied prey such as fish. Two-source, isotopic mixing model predictions were less variable and identified inter-annual and species-specific differences in the relative amounts of fish and krill in penguin diets not readily apparent using SCA. In contrast, multi-source isotopic mixing models had difficulty estimating the dietary contribution of fish species occupying similar trophic levels without refinement using SCA-derived otolith data. Overall, our ability to track inter-annual and species-specific differences in penguin diets using SIA was enhanced by integrating SCA data to isotopic mixing modes in three ways: 1) selecting appropriate prey sources, 2) weighting combinations of isotopically similar prey in two-source mixing models and 3) refining predicted contributions of isotopically similar prey in multi-source models.