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.     

Enrichment can damp population cycles: a balance of inflexible and flexible interactions

Destabilization of one predator–one prey systems with an increase in nutrient input has been viewed as a paradox. We report that enrichment can damp population cycles by a food‐web structure that balances inflexible and flexible interaction links (i.e. specialist and generalist predators). We modeled six predator–prey systems involving three or four species in which the predators practice optimal foraging based on prey profitability determined by handling time. In all models, the balance of interaction links simultaneously decreased the amplitude of population oscillations and increased the minimum density with increasing enrichment, leading to a potential theoretical resolution of the paradox of enrichment in non‐equilibrium dynamics. The stabilization mechanism was common to all of the models. Important previous studies on the stability of food webs have also demonstrated that a balance of interaction strengths stabilizes systems, suggesting a general rule of ecosystem stability.     

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.     

Towards resolving the paradox of enrichment: the impact of zooplankton vertical migrations on plankton systems stability

Eutrophication, often resulting from human activity, is a serious threat to aquatic communities. Theoretical analysis of this phenomenon, based on conceptual mathematical models, leads to controversial predictions known as Rosenzweig's paradox of enrichment. At the same time, field observations demonstrate that real plankton communities exhibit various mechanisms of self-regulation which can buffer negative effects of enrichment. In this paper, we study potential effects of zooplankton vertical migration on stability of plankton systems functioning. We consider an intrinsically unstable plankton model, which is characterized by an unlimited phytoplankton multiplication and population oscillations of increasing amplitude, and investigate whether vertical migrations of zooplankton can stabilize such a system at low plankton densities. By means of developing two different models accounting for different ecological situations, e.g. deep waters and shallow waters, we show that vertical migrations of zooplankton can result in stabilization of eutrophic plankton systems. Thus, we show that this mechanism, rarely taken into account in models of plankton dynamics, may be important for resolving the paradox of enrichment in plankton communities.     

Unpalatable prey resolves the paradox of enrichment

Enrichment is an increasingly serious trend in natural ecosystems. A theoretical model of a predator–prey system with a natural assumption of satiation in predation predicts that enrichment causes the populations to fluctuate to stochastic extinction. However, this ''paradox of enrichment'' does not always occur in experimental and natural communities. Here we present a theoretical model that describes a novel mechanism for resolving the paradox in the case of a predator with optimal selective feeding. Specifically, a less profitable but edible (thus `unpalatable'') prey species sharply reduces the amplitude of population oscillations and firmly prevents the minimum abundances of species from falling below certain values. The presence of such an unpalatable prey thus guarantees the robustness of the system against enrichment.     

The group-size paradox: effects of learning and patch departure rules

In many species, foraging in groups can enhance individual fitness.However, groups are often predicted to be larger than the sizethat maximizes individual fitness. This is because individualforagers are expected to continue joining a group until thefitness in the group falls to the level experienced by solitaryforagers. If such a process were pervasive, social foraging,paradoxically, would provide little evolutionary advantages.We propose a solution to the group-size paradox by allowingforagers to learn about habitat quality and leave food patcheswhen their current intake rate falls below that expected forthe whole habitat. By using a simulation model, we show thatunder a wide range of population sizes, foragers using suchrules abandon under- and overcrowded patches, ensuring thatgroup size remains close to the optimal value. The results holdin habitats with varying patch quality, but we note that thelack of food renewal in patches can disrupt the process of groupformation. We conclude that groups of optimal sizes can occurfrequently if fitness functions are peaked and resources patchilydistributed, without the need to invoke relatedness betweenjoiners and established group members, group defense againstjoiners, or other mechanisms that were proposed earlier to preventgroups from becoming too large.     

Paradoxes of enrichment: effects of increased light versus nutrient supply on pelagic producer-grazer systems

Energy-based plant-herbivore models produce the "paradox of enrichment," a destabilizing influence of enrichment on population dynamics. Because many plants change their carbon : nutrient stoichiometry in response to the light : nutrient supply ratio, enrichment with light can cause a mismatch between the elemental compositions of plants and their herbivores. Herbivore growth rates may then decrease with increased light supply, which is termed the "paradox of energy enrichment." I present a stoichiometric phytoplankton-grazer model that accounts for the dynamical vertical light gradient and explore how algal and grazer densities, mineral nutrient concentration, algal nutrient stoichiometry, and system stability respond to enrichment with light (through changes in irradiance, background turbidity, and water column depth) versus enrichment with nutrients. Parameterized for Daphnia, the model produces several "unusual" phenomena: multiple equilibria (with grazers extinct in spite of high algal biomass at one equilibrium), inconsistent light enrichment effects on stability (light enrichment first destabilizes and then stabilizes), and the paradox of energy enrichment. These phenomena are restricted to the low end of realistic nutrient supplies except in very shallow systems, where high sedimentation rates effectively deplete the water column of nutrients. At higher nutrient supplies, light enrichment produces the classical paradox of enrichment, leading first to an increase in grazers at a stable equilibrium and then to algae-grazer oscillations.     

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.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Note on optimizing environmental enrichment: a study of fennec fox and zoo guests

Environmental enrichment is widely used to stimulate animal time budgets that seem more natural and diverse than those of unenriched animals. Uncertainty of reward is a suggested means to maintain enrichment's efficacy. Foraging tasks are widely applied in zoo animal enrichment, yet few rely on the logic of optimal foraging theory to help maintain animal motivation. We applied a foraging strategy to zoo housed fennec foxes as enrichment. We varied only the probability of when and where food would occur in the animals' exhibit. Our methods increased behavioral diversity, animal activity, and stimulated guest interest in the exhibit.     

Stability of pitcher-plant microfaunal populations depends on food web structure

Enrichment (increasing K) destabilizes simple consumer–resource interactions, but increasing food web complexity in various ways can remove this paradox of enrichment. We varied resources and number of omnivorous predators (mosquitoes) and tested for effects on the stability (persistence and temporal variability) of microfaunal populations living in pitcher plants. Top-down (omnivorous) effects were destabilizing, decreasing the persistence time of a rotifer, Habrotrocha rosa, and perhaps a microflagellate, Bodo sp. Enrichment effects were more complex, in part due to effects of shredding midges on resource availability, and in part due to interactions with predation. The persistence of Bodo increased with resource availability (more bacteria due to shredding by midges; no paradox of enrichment). Increasing resources by adding ants decreased persistence of H. rosa when mosquitoes were rare (paradox of enrichment), but the effect was reversed in leaves with significant colonization by mosquitoes. Thus, in the microfaunal community of pitcher plants, omnivorous predation tends to be destabilizing, and also tends to remove the paradox of enrichment.     

The functional response predicts the effect of resource distribution on the optimal movement rate of consumers

Understanding how often individuals should move when foraging over patchy habitats is a central question in ecology. By combining optimality and functional response theories, we show analytically how the optimal movement rate varies with the average resource level (enrichment) and resource distribution (patch heterogeneity). We find that the type of functional response predicts the effect of enrichment in homogeneous habitats: enrichment should decrease movement for decelerating functional responses, but increase movement for accelerating responses. An intermediate resource level thus maximises movement for type‐III responses. Counterintuitively, greater movement costs favour an increase in movement. In heterogeneous habitats predictions further depend on how enrichment alters the variance of resource distribution. Greater patch variance always increases the optimal rate of movement, except for type‐IV functional responses. While the functional response is well established as a fundamental determinant of consumer–resource dynamics, our results indicate its importance extends to the understanding of individual movement strategies.     

富食悖论研究进展

Michael Rosenzweig于1971年首次提出＂富食悖论＂这一概念：在简单的被捕食者-捕食者系统中,随着营养物质供应的增加,系统变得不再稳定,并产生大振幅波动,最终导致系统内的物种灭亡。然而,许多实验结果并不支持Rosenzweig的理论。综述＂富食悖论＂的定义、理论和实验研究,同时综述各种解释实际与理论之间矛盾的机制,最后探讨＂富食悖论＂的研究前景。     

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

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

     

Stoichiometry in producer-grazer systems: Linking energy flow with element cycling

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator—prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer—grazer systems differs radically from nutrient enrichment. Hence, expressing producer—grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.     

Optimal foraging theory predicts effects of environmental enrichment in a group of adult golden lion tamarins

The success of environmental enrichment programs in effecting specific changes in the behavior of captive animals has not always been uniform. Separate studies demonstrated both an increase in food competition and a decrease in food competition among captive group-living primates upon introduction of foraging devices. The objectives of this study were to measure the effects of variation in resource distribution and availability on food competition in a group of captive adult golden lion tamarins (Leontopithecus rosalia). The resource variables chosen were suggested from optimal foraging theory. The energy invested to obtain an item influenced food transfer and aggressive behaviors while food abundance did not. All individuals obtained an equivalent number of items over the course of the experiment from the foraging device, even though some tamarins obtained most of their food rewards directly from the device while others received their food rewards primarily through food transfer from other group members. Because the monkeys appeared highly motivated to obtain food from the test apparatus and did not habituate to it, the foraging device used in this experiment could be used as regular environmental enrichment for golden lion tamarins. One way to circumvent potentially unacceptable rates of aggression, with this or any feeding protocol that increases foraging task complexity and search time, may be to provide more than one foraging device per group. Zoo Biol 17:231–244, 1998. © 1998 Wiley-Liss, Inc.     

Implications of resting eggs of zooplankton for the paradox of enrichment

In this study, we numerically investigated to what extent introducing resting-egg dynamics would stabilize simple Daphnia–algae consumer–resource models. In the models, the density of viable resting eggs was explicitly expressed, and we assumed that zooplankton produced resting eggs seasonally or in response to food deficiency and that resting eggs hatched seasonally. The models predicted that, although the paradox of enrichment was not completely resolved (i.e., the system was destabilized by eutrophication), we found the following conditions under which the stabilizing effects of resting eggs would be significantly large: (1) resting eggs are produced seasonally (rather than in response to food deficiency), (2) the annual average allocation ratio to resting eggs is large, and (3) the annual average hatching rate of resting eggs is low. The results suggest that resting-egg dynamics can significantly reduce the paradox of enrichment within the biologically meaningful parameter space and contribute to the stability of plankton community dynamics.     

Optimal foraging and predator-prey dynamics, II.

In this paper we consider one-predator-two-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predator-prey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types.     

Age structure in predator-prey systems: Intraspecific carnivore interaction,passive diffusion,and the paradox of enrichment

An existing arthropod predator-prey model incorporating age structure in the carnivore through the use of the von Foerster equation is extended to include the effects of intraspecific carnivore interaction and passive diffusion or migration. A linear stability analysis of the community equilibrium point of that differential-integral equation system is performed and the resulting secular equation analyzed by the method of D-partitions. These stability results are then compared to those obtained by employing an analogous differential equation model without age structure, in particular as they relate to the so-called paradox of enrichment. In the absence of passive diffusion, it is shown that, unlike for a differential equation model, the paradox of enrichment can occur even with a carnivore which exhibits intraspecific competition. This destabilizing effect of age structure is seen to occur most dramatically when interspecific interactions are large, while the effect of passive diffusion is to offset that tendency and restabilize the system. These predictions are in accordance with relevant experimental evidence involving mites.