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.     

A resolution of the paradox of enrichment

Theoretical studies have shown a paradoxical destabilizing response of predator-prey ecosystems to enrichment, but there is the gap between the intuitive view of nature and this theoretical prediction. We studied a minimal predator-prey system (a two predator-two prey system) in which the paradox of enrichment pattern can vanish; the destabilization with enrichment is reversed, leading to stabilization (a decrease in the amplitude of oscillation of population densities). For resolution of the paradox, two conditions must be met: (1) the same prey species must be preferred as a dietary item by both predator species, creating the potential for high exploitative competition between the predator species, and (2), while both predators are assumed to select their diet in accordance with optimal diet utilization theory, one predator must be a specialist and the other a generalist. In this system, the presence of a less profitable prey species can cause the increase in population oscillation amplitudes associated with increasing enrichment to be suppressed via the optimal diet utilization of the generalist predator. The resulting stabilization is explained by the mitigating effect of the less profitable prey showing better population growth with increasing enrichment on the destabilization underlying the specialist predator and prey relation, thus resolving the paradox of enrichment.     

The stability of ecosystems: A brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system,thereby collapsing the trophic interaction,a phenomenon referred to as "the paradox of enrichment". After it was first pro posed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator -prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form,inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that,although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms,which may differ among systems,need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models.     

The paradox of enrichment in an adaptive world

Paradoxically, enrichment can destabilize a predator-prey food web. While adaptive dynamics can greatly influence the stability of interaction systems, few theoretical studies have examined the effect of the adaptive dynamics of interaction-related traits on the possibility of resolution of the paradox of enrichment. We consider the evolution of attack and defence traits of a predator and two prey species in a one predator-two prey system in which the predator practises optimal diet use. The results showed that optimal foraging alone cannot eliminate a pattern of destabilization with enrichment, but trait evolution of the predator or prey can change the pattern to one of stabilization, implying a possible resolution of the paradox of enrichment. Furthermore, trait evolution in all species can broaden the parameter range of stabilization. Importantly, rapid evolution can stabilize this system, but weaken its stability in the face of enrichment.     

Not attackable or not crackable—How pre‐ and post‐attack defenses with different competition costs affect prey coexistence and population dynamics

It is well‐known that prey species often face trade‐offs between defense against predation and competitiveness, enabling predator‐mediated coexistence. However, we lack an understanding of how the large variety of different defense traits with different competition costs affects coexistence and population dynamics. Our study focusses on two general defense mechanisms, that is, pre‐attack (e.g., camouflage) and post‐attack defenses (e.g., weaponry) that act at different phases of the predator—prey interaction. We consider a food web model with one predator, two prey types and one resource. One prey type is undefended, while the other one is pre‐ or post‐attack defended paying costs either by a higher half‐saturation constant for resource uptake or a lower maximum growth rate. We show that post‐attack defenses promote prey coexistence and stabilize the population dynamics more strongly than pre‐attack defenses by interfering with the predator's functional response: Because the predator spends time handling “noncrackable” prey, the undefended prey is indirectly facilitated. A high half‐saturation constant as defense costs promotes coexistence more and stabilizes the dynamics less than a low maximum growth rate. The former imposes high costs at low resource concentrations but allows for temporally high growth rates at predator‐induced resource peaks preventing the extinction of the defended prey. We evaluate the effects of the different defense mechanisms and costs on coexistence under different enrichment levels in order to vary the importance of bottom‐up and top‐down control of the prey community.     

Trophic supplements to intraguild predation

Intraguild predation (IGP) is a dominant community module in terrestrial food webs that occurs when multiple consumers feed both on each other and on a shared prey. This specific form of omnivory is common in terrestrial communities and is of particular interest for conservation biology and biological control given its potential to disrupt management of threatened or pest species. Extensive theory exists to describe the dynamics of three-species IGP, but these models have largely overlooked the potential for other, exterior interactions, to alter the dynamics within the IGP module. We investigated how three forms of feeding outside of the IGP module by intraguild predators (i.e. trophic supplementation) affect the dynamics of the predators (both IG predator and IG prey) and their shared resource. Specifically, we examined how the provision of a constant donor-controlled resource, the availability of an alternative prey species, and predator plant-feeding affect the dynamics of IGP models. All three forms of trophic supplements modified the basic expectations of IGP theory in two important ways, and their effects were similar. First, coexistence was possible without the IG prey being a superior competitor for the original shared resource if the IG prey could effectively exploit one of the types of trophic supplements. However, supplements to the IG predator restricted the potential for coexistence. Second, supplements to the IG prey ameliorated the disruptive effects of the IG predator on the suppression of the shared resource, promoting effective control of the resource in the presence of both predators. Consideration of these three forms of trophic supplementation, all well documented in natural communities, adds substantial realism and predictive power to intraguild predation theory.     

The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs

This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.     

The interaction between predation risk and food ration on behavior and morphology of Eurasian perch

The risk of both predation and food level has been shown to affect phenotypic development of organisms. However, these two factors also influence animal behavior that in turn may influence phenotypic development. Hence, it might be difficult to disentangle the behavioral effect from the predator or resource‐level effects. This is because the presence of predators and high resource levels usually results in a lower activity, which in turn affects energy expenditure that is used for development and growth. It is therefore necessary to study how behavior interacts with changes in body shape with regard to resource density and predators. Here, we use the classic predator‐induced morphological defense in fish to study the interaction between predator cues, resource availability, and behavioral activity with the aim to determine their relative contribution to changes in body shape. We show that all three variables, the presence of a predator, food level, and activity, both additively and interactively, affected the body shape of perch. In general, the presence of predators, lower swimming activity, and higher food levels induced a deep body shape, with predation and behavior having similar effect and food treatment the smallest effect. The shape changes seemed to be mediated by changes in growth rate as body condition showed a similar effect as shape with regard to food‐level and predator treatments. Our results suggests that shape changes in animals to one environmental factor, for example, predation risk, can be context dependent, and depend on food levels or behavioral responses. Theoretical and empirical studies should further explore how this context dependence affects fitness components such as resource gain and mortality and their implications for population dynamics.     

The effect of predator limitation on the dynamics of simple food chains

We investigate the influence of competition between predators on the dynamics of bitrophic predator–prey systems and of tritrophic food chains. Competition between predators is implemented either as interference competition, or as a density-dependent mortality rate. With interference competition, the paradox of enrichment is reduced or completely suppressed, but otherwise, the dynamical behavior of the systems is not fundamentally different from that of the Rosenzweig–MacArthur model, which contains no predator competition and shows only continuous transitions between fixed points or periodic oscillations. In contrast, with density-dependent predator mortality, the system shows a surprisingly rich dynamical behavior. In particular, decreasing the density regulation of the predator can induce catastrophic shifts from a stable fixed point to a large oscillation where the predator chases the prey through a cycle that brings both species close to the threshold of extinction. Other catastrophic bifurcations, such as subcritical Hopf bifurcations and saddle-node bifurcations of limit cycles, do also occur. In tritrophic food chains, we find again that fixed points in the model with predator interference become unstable only through Hopf bifurcations, which can also be subcritical, in contrast to the bitrophic situation. The model with a density limitation shows again catastrophic destabilization of fixed points and various nonlocal bifurcations. In addition, chaos occurs for both models in appropriate parameter ranges.     

Restricting Prey Dispersal Can Overestimate the Importance of Predation in Trophic Cascades

Predators can affect prey populations and, via trophic cascades, predators can indirectly impact resource populations (2 trophic levels below the predator) through consumption of prey (density-mediated indirect effects; DMIEs) and by inducing predator-avoidance behavior in prey (trait-mediated indirect effects; TMIEs). Prey often employ multiple predator-avoidance behaviors, such as dispersal or reduced foraging activity, but estimates of TMIEs are usually on individual behaviors. We assessed direct and indirect predator effects in a mesocosm experiment using a marine food chain consisting of a predator (toadfish – Opsanus tau), prey (mud crab - Panopeus herbstii) and resource (ribbed mussel – Geukensia demissa). We measured dispersal and foraging activity of prey separately by manipulating both the presence and absence of the predator, and whether prey could or could not disperse into a predator-free area. Consumption of prey was 9 times greater when prey could not disperse, probably because mesocosm boundaries increased predator capture success. Although predator presence did not significantly affect the number of crabs that emigrated, the presence of a predator decreased resource consumption by prey, which resulted in fewer resources consumed for each prey that emigrated in the presence of a predator, and reduced the overall TMIE. When prey were unable to disperse, TMIEs on mussel survival were 3 times higher than the DMIEs. When prey were allowed to disperse, the TMIEs on resource survival increased to 11-times the DMIEs. We found that restricting the ability of prey to disperse, or focusing on only one predator-avoidance behavior, may be underestimating TMIEs. Our results indicate that the relative contribution of behavior and consumption in food chain dynamics will depend on which predator-avoidance behaviors are allowed to occur and measured.     

Confronting the Paradox of Enrichment to the Metacommunity Perspective

Resource enrichment can potentially destabilize predator-prey dynamics. This phenomenon historically referred as the "paradox of enrichment" has mostly been explored in spatially homogenous environments. However, many predator-prey communities exchange organisms within spatially heterogeneous networks called metacommunities. This heterogeneity can result from uneven distribution of resources among communities and thus can lead to the spreading of local enrichment within metacommunities. Here, we adapted the original Rosenzweig-MacArthur predator-prey model, built to study the paradox of enrichment, to investigate the effect of regional enrichment and of its spatial distribution on predator-prey dynamics in metacommunities. We found that the potential for destabilization was depending on the connectivity among communities and the spatial distribution of enrichment. In one hand, we found that at low dispersal regional enrichment led to the destabilization of predator-prey dynamics. This destabilizing effect was more pronounced when the enrichment was uneven among communities. In the other hand, we found that high dispersal could stabilize the predator-prey dynamics when the enrichment was spatially heterogeneous. Our results illustrate that the destabilizing effect of enrichment can be dampened when the spatial scale of resource enrichment is lower than that of organismss movements (heterogeneous enrichment). From a conservation perspective, our results illustrate that spatial heterogeneity could decrease the regional extinction risk of species involved in specialized trophic interactions. From the perspective of biological control, our results show that the heterogeneous distribution of pest resource could favor or dampen outbreaks of pests and of their natural enemies, depending on the spatial scale of heterogeneity.     

The effects of enrichment on the dynamics of apparent competitive interactions in stage-structured systems

In the absence of other limiting factors, assemblages in which species share a common, effective natural enemy are not expected to persist. Although a variety of mechanisms have been postulated to explain the coexistence of species that share natural enemies, the role of productivity gradients has not been explored in detail. Here, we examine how enrichment can affect the outcome of apparent competition. We develop a structured resource/consumer/natural enemy model in which the prey are exposed to attacks during a vulnerable life phase, the length of which depends on resource availability. With a single prey species, the model exhibits the "paradox of enrichment," with unstable dynamics at high levels of resource productivity. We extend this model to consider two prey species linked by a shared predator, each with their own distinct resource base. We derive invasion and stability conditions and examine how enrichment influences prey species exclusion and coexistence. Contrary to expectations from simpler, prey-dependent models, apparent competition is not necessarily strong at high productivity, and prey species coexistence may thus be more likely in enriched environments. Further, the coexistence of apparent competitors may be facilitated by unstable dynamics. These results contrast with the standard theory that apparent competition in productive environments leads to nonpersistent interactions and that coexistence of multispecies interactions is more likely under equilibrial conditions.     

Effect of scavenging on predation in a food web

Scavenging can have important consequences for food web dynamics, for example, it may support additional consumer species and affect predation on live prey. Still, few food web models include scavenging. We develop a dynamic model that includes two facultative scavenger species, which we refer to as the predator or scavenger species according to their natural scavenging propensity, as well as live prey, and a carrion pool to show ramifications of scavenging for predation in simple food webs. Our modeling suggests that the presence of scavengers can both increase and decrease predator kill rates and overall predation in model food webs and the impact varies (in magnitude and direction) with context. In particular, we explore the impact of the amount of dynamics (exploitative competition) allowed in the predator, scavenger, and prey populations as well as the direction and magnitude of interference competition between predators and scavengers. One fundamental prediction is that scavengers most likely increase predator kill rates, especially if there are exploitative feedback effects on the prey or carrion resources like is normally observed in natural systems. Scavengers only have minimal effects on predator kill rate when predator, scavenger, and prey abundances are kept constant by management. In such controlled systems, interference competition can greatly affect the interactions in contrast to more natural systems, with an increase in interference competition leading to a decrease in predator kill rate. Our study adds to studies that show that the presence of predators affects scavenger behavior, vital rates, and food web structure, by showing that scavengers impact predator kill rates through multiple mechanisms, and therefore indicating that scavenging and predation patterns are tightly intertwined. We provide a road map to the different theoretical outcomes and their support from different empirical studies on vertebrate guilds to provide guidance in wildlife management.     

The nature of predation: prey dependent, ratio dependent or neither?

To describe a predator-prey relationship, it is necessary to specify the rate of prey consumption by an average predator. This functional response largely determines dynamic stability, responses to environmental influences and the nature of indirect effects in the food web containing the predator-prey pair. Nevertheless, measurements of functional responses in nature are quite rare. Recently, much work has been devoted to comparing two idealized forms of the functional response: prey dependent and ratio dependent. Although we agree that predator abundance often affects the consumption rate of individual predators, this phenomenon requires more attention. Disagreement remains over which of the two idealized responses serves as a better starting point in building models when data on predator dependence are absent.     

Impacts of Foraging Facilitation Among Predators on Predator-prey Dynamics

Whereas impacts of predator interference on predator-prey dynamics have received considerable attention, the “inverse” process—foraging facilitation among predators—have not been explored yet. Here we show, via mathematical models, that impacts of foraging facilitation on predator-prey dynamics depend on the way this process is modeled. In particular, foraging facilitation destabilizes predator-prey dynamics when it affects the encounter rate between predators and prey. By contrast, it might have a stabilizing effect if the predator handling time of prey is affected. Foraging facilitation is an Allee effect mechanism among predators and we show that for many parameters, it gives rise to a demographic Allee effect or a critical predator density in need to be crossed for predators to persist. We explore also the effects of predator interference, to make the picture “symmetric” and complete. Predator interference is shown to stabilize predator-prey dynamics once its strength is not too high, and thus corroborates results of others. On the other hand, there is a wide range of model parameters for which predator interference gives rise to three co-occurring co-existence equilibria. Such a multi-equilibrial regime is rather robust as we observe it for all the functional response types we explore. This is a previously unreported phenomenon which we show cannot occur for the Beddington–DeAngelis functional response. An interesting topic for future research thus might be to seek for general conditions on predator functional responses that would produce multiple co-existence equilibria in a predator-prey model.     

Native and Non-Native Plants Provide Similar Refuge to Invertebrate Prey,but Less than Artificial Plants

Non-native species introductions are widespread and can affect ecosystem functioning by altering the structure of food webs. Invading plants often modify habitat structure, which may affect the suitability of vegetation as refuge and could thus impact predator-prey dynamics. Yet little is known about how the replacement of native by non-native vegetation affects predator-prey dynamics. We hypothesize that plant refuge provisioning depends on (1) the plant’s native status, (2) plant structural complexity and morphology, (3) predator identity, and (4) prey identity, as well as that (5) structurally similar living and artificial plants provide similar refuge. We used aquatic communities as a model system and compared the refuge provided by plants to macroinvertebrates (Daphnia pulex, Gammarus pulex and damselfly larvae) in three short-term laboratory predation experiments. Plant refuge provisioning differed between plant species, but was generally similar for native (Myriophyllum spicatum, Ceratophyllum demersum, Potamogeton perfoliatus) and non-native plants (Vallisneria spiralis, Myriophyllum heterophyllum, Cabomba caroliniana). However, plant refuge provisioning to macroinvertebrate prey depended primarily on predator (mirror carp: Cyprinus carpio carpio and dragonfly larvae: Anax imperator) and prey identity, while the effects of plant structural complexity were only minor. Contrary to living plants, artificial plant analogues did improve prey survival, particularly with increasing structural complexity and shoot density. As such, plant rigidity, which was high for artificial plants and one of the living plant species evaluated in this study (Ceratophyllum demersum), may interact with structural complexity to play a key role in refuge provisioning to specific prey (Gammarus pulex). Our results demonstrate that replacement of native by structurally similar non-native vegetation is unlikely to greatly affect predator-prey dynamics. We propose that modification of predator-prey interactions through plant invasions only occurs when invading plants radically differ in growth form, density and rigidity compared to native plants.     

Impact of vigilance on the density variations in a food chain model

Anti-predator behavior in the form of vigilance greatly influences the dynamics of a predator-prey system. In the present work, we investigate the impact of prey vigilance in a three-species food chain model where both prey and middle predator use vigilance as a survival strategy in the presence of their respective predators. We present basic mathematical results such as local and global stability, bifurcation behavior of the system. We explore the variation of the densities of the populations in different bi-parameter spaces and observe that vigilance plays a crucial role in the survival and extinction of the populations.     

PREY ADAPTATION AS A CAUSE OF PREDATOR-PREY CYCLES

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.     

Influence of density dependence on predator-prey seabird interactions at large spatio-temporal scales

Theoretical investigations of competitive dynamics have noted that numbers of predator and prey influence each other. However, few empirical studies have demonstrated how a life-history trait of the prey (such as fecundity) can be affected simultaneously by its own density and the density of predators. For instance, density dependence can reduce fecundity with increasing number of prey, while inverse density dependence or Allee effects may occur especially when the prey is a social organism. Here we analysed an intraguild predator-prey system of two seabird species at a large spatio-temporal scale. As expected, we found that fecundity of prey was negatively affected by predator density. Nevertheless, fecundity of prey also increased nonlinearly with its own density and strikingly with the prey-predator ratio. Small groups of prey were probably not able to defend their nests especially against large number of predators. At the highest prey densities (i.e. when anti-predator strategies should be most efficient), prey fecundity also lowered, suggesting the appearance of density dependence mediated by food competition. Allee effects and density dependence occurred across a broad range of population sizes of both the prey and the predator at several local populations facing different ecological environments.     

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.