Enrichment and ecosystem stability: effect of toxic food

Enrichment in resource availability theoretically destabilizes predator-prey dynamics (the paradox of enrichment). However, a minor change in the resource stoichiometry may make a prey toxic for the predator, and the presence of toxic prey affects the dynamics significantly. Here, theoretically we explore how, at increased carrying capacity, a toxic prey affects the oscillation or destabilization of predator-prey dynamics, and how its presence influences the growth of the predator as well as that of a palatable prey. Mathematical analysis determines the bounds on the food toxicity that allow the coexistence of a predator along with a palatable and a toxic prey. The overall results demonstrate that toxic food counteracts oscillation (destabilization) arising from enrichment of resource availability. Moreover, our results show that, at increased resource availability, toxic food that acts as a source of extra mortality may increase the abundance of the predator as well as that of the palatable prey.     

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.     

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.     

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 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.     

Predation risk tradeoffs in prey: effects on energy and behaviour

The complexity of behavioural interactions in predator-prey systems has recently begun to capture trait-effects, or non-lethal effects, of predators on prey via induced behavioural changes. Non-lethal predation effects play crucial roles in shaping population and community dynamics, particularly by inducing changes to foraging, movement and reproductive behaviours of prey. Prey exhibit trade-offs in behaviours while minimizing predation risk. We use a novel evolutionary ecosystem simulation EcoSim to study such behavioural interactions and their effects on prey populations, thereby addressing the need for integrating multiple layers of complexity in behavioural ecology. EcoSim allows complex intra- and inter-specific interactions between behaviourally and genetically unique individuals called predators and prey, as well as complex predator-prey dynamics and coevolution in a tri-trophic and spatially heterogeneous world. We investigated the effects of predation risk on prey energy budgets and fitness. Results revealed that energy budgets, life history traits, allocation of energy to movements and fitness-related actions differed greatly between prey subjected to low-predation risk and high-predation risk. High-predation risk suppressed prey foraging activity, increased total movement and decreased reproduction relative to low-risk. We show that predation risk alone induces behavioural changes in prey which drastically affect population and community dynamics, and when interpreted within the evolutionary context of our simulation indicate that genetic changes accompanying coevolution have long-term effects on prey adaptability to the absence of predators.     

Predator and Prey Models with Flexible Individual Behavior and Imperfect Information

To begin identifying what behavioral details might be needed to characterize community dynamics and stability, we examined the effect of prey behavioral responses to predation risk on community dynamics and stability. We considered the case of prey altering their foraging effort to trade off energy gain and predation risk. We used state-dependent dynamic optimization to calculate the optimal trade-off for four models of prey behaviorally responding to predation risk. We consider a fixed behavior model in which prey use constant levels of foraging effort and three flexible behavior models in which prey change their foraging effort according to their physiological state and their perceived level of predation risk. Flexible behavior was destabilizing at the community level as evidenced by higher predator-prey oscillations and lower community persistence times. The mechanisms by which prey estimated predation risk also affected community stability. We found that community dynamics resulting from prey with flexible behavior and fixed perception of risk approximated community dynamics resulting from prey with flexible behavior and perfect information about predation risk, however neither approximated the community dynamics resulting from prey with flexible behavior and flexible perception of risk. Thus, whether it might be possible to abstract complex behavior with simpler rules when modeling community dynamics depends on the prey's behavioral mechanisms, which are empirically poorly known.     

Predator-prey dynamics driven by feedback between functionally diverse trophic levels

Neglecting the naturally existing functional diversity of communities and the resulting potential to respond to altered conditions may strongly reduce the realism and predictive power of ecological models. We therefore propose and study a predator-prey model that describes mutual feedback via species shifts in both predator and prey, using a dynamic trait approach. Species compositions of the two trophic levels were described by mean functional traits--prey edibility and predator food-selectivity--and functional diversities by the variances. Altered edibility triggered shifts in food-selectivity so that consumers continuously respond to the present prey composition, and vice versa. This trait-mediated feedback mechanism resulted in a complex dynamic behavior with ongoing oscillations in the mean trait values, reflecting continuous reorganization of the trophic levels. The feedback was only possible if sufficient functional diversity was present in both trophic levels. Functional diversity was internally maintained on the prey level as no niche existed in our system, which was ideal under any composition of the predator level due to the trade-offs between edibility, growth and carrying capacity. The predators were only subject to one trade-off between food-selectivity and grazing ability and in the absence of immigration, one predator type became abundant, i.e., functional diversity declined to zero. In the lack of functional diversity the system showed the same dynamics as conventional models of predator-prey interactions ignoring the potential for shifts in species composition. This way, our study identified the crucial role of trade-offs and their shape in physiological and ecological traits for preserving diversity.     

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.     

Predator-prey coevolution driven by size selective predation can cause anti-synchronized and cryptic population dynamics

Population dynamics and evolutionary dynamics can occur on similar time scales, and a coupling of these two processes can lead to novel population dynamics. Recent theoretical studies of coevolving predator-prey systems have concentrated more on the stability of such systems than on the characteristics of cycles when they are unstable. Here I explore the characteristics of the cycles that arise due to coevolution in a system in which prey can increase their ability to escape from predators by becoming either significantly larger or significantly smaller in trait value (i.e., a bidirectional trait axis). This is a reasonable model of body size evolution in some systems. The results show that antiphase population cycles and cryptic cycles (large population fluctuation in one species but almost no change in another species) can occur in the coevolutionary system but not systems where only a single species evolves. Previously, those dynamical patterns have only been theoretically shown to occur in single species evolutionary models and the coevolutionary model which do not involve a bi-directional axis of adaptation. These unusual dynamics may be observed in predator-prey interactions when the density dependence in the prey species is strong.     

Testing for predator dependence in predator-prey dynamics: a non-parametric approach

The functional response is a key element in all predator-prey interactions. Although functional responses are traditionally modelled as being a function of prey density only, evidence is accumulating that predator density also has an important effect. However, much of the evidence comes from artificial experimental arenas under conditions not necessarily representative of the natural system, and neglecting the temporal dynamics of the organism (in particular the effects of prey depletion on the estimated functional response). Here we present a method that removes these limitations by reconstructing the functional response non-parametrically from predator-prey time-series data. This method is applied to data on a protozoan predator-prey interaction, and we obtain significant evidence of predator dependence in the functional response. A crucial element in this analysis is to include time-lags in the prey and predator reproduction rates, and we show that these delays improve the fit of the model significantly. Finally, we compare the non-parametrically reconstructed functional response to parametric forms, and suggest that a modified version of the Hassell-Varley predator interference model provides a simple and flexible function for theoretical investigation and applied modelling.     

一类食饵具有常数投放率系统的恢复率

在现实的生态系统中,特别当捕食者的密度大或捕食能力强时,人们常采用食饵补充的方式来控制捕食者和食饵种群数量的稳定,以达到生态平衡．本文在Chisholm,R．A．和Filotas,E．研究的捕食-食饵模型基础上,定义了食饵具有常数投放率的捕食系统,讨论了此系统的恢复率与食饵最大容量之间的关系以及投放率对恢复率和预警长度的影响．最后通过算例分析了递减的恢复率作为系统转移的指示器与投放率与食饵最大容量的关系,说明食饵的投放率可增强预警效果．     

Effects of warming on predator–prey interactions – a resource‐based approach and a theoretical synthesis

We theoretically explore consequences of warming for predator–prey dynamics, broadening previous approaches in three ways: we include beyond‐optimal temperatures, predators may have a type III functional response, and prey carrying capacity depends on explicitly modelled resources. Several robust patterns arise. The relationship between prey carrying capacity and temperature can range from near‐independence to monotonically declining/increasing to hump‐shaped. Predators persist in a U‐shaped region in resource supply (=enrichment)‐temperature space. Type II responses yield stable persistence in a U‐shaped band inside this region, giving way to limit cycles with enrichment at all temperatures. In contrast, type III responses convey stability at intermediate temperatures and confine cycles to low and high temperatures. Warming‐induced state shifts can be predicted from system trajectories crossing stability and persistence boundaries in enrichment‐temperature space. Results of earlier studies with more restricted assumptions map onto this graph as special cases. Our approach thus provides a unifying framework for understanding warming effects on trophic dynamics.     

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.     

Are all prey created equal? A review and synthesis of differential predation on prey in substandard condition

Our understanding of predator-prey interactions in fishes has been influenced largely by research assuming that the condition of the participants is normal. However, fish populations today often reside in anthropogenically altered environments and are subjected to many kinds of stressors, which may reduce their ecological performance by adversely affecting their morphology, physiology, or behaviour. One consequence is that either the predator or prey, or both, may be in a substandard condition at the time of an interaction. We reviewed the literature on predator-prey interactions in fishes where substandard prey were used as experimental groups. Although most of this research indicates that such prey are significantly more vulnerable to predation, prey condition has rarely been considered in ecological theory regarding predator-prey interactions. The causal mechanisms for increased vulnerability of substandard prey to predation include a failure to detect predators, lapses in decision-making, poor fast-start performance, inability to shoal effectively, and increased prey conspicuousness. Despite some problems associated with empirical predator-prey studies using substandard prey, their results can have theoretical and applied uses, such as in ecological modelling or justification of corrective measures to be implemented in the wild. There is a need for more corroborative field experimentation, a better understanding of the causal mechanisms behind differential predation, and increased incorporation of prey condition into the research of predator-prey modellers and theoreticians. If the concept of prey condition is considered in predator-prey interactions, our understanding of how such interactions influence the structure and dynamics of fish communities is likely to change, which should prove beneficial to aquatic ecosystems.     

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.     

Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.     

Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems

Despite growing awareness of the significance of body-size and predator-prey body-mass ratios for the stability of ecological networks, our understanding of their distribution within ecosystems is incomplete. Here, we study the relationships between predator and prey size, body-mass ratios and predator trophic levels using body-mass estimates of 1313 predators (invertebrates, ectotherm and endotherm vertebrates) from 35 food-webs (marine, stream, lake and terrestrial). Across all ecosystem and predator types, except for streams (which appear to have a different size structure in their predator-prey interactions), we find that (1) geometric mean prey mass increases with predator mass with a power-law exponent greater than unity and (2) predator size increases with trophic level. Consistent with our theoretical derivations, we show that the quantitative nature of these relationships implies systematic decreases in predator-prey body-mass ratios with the trophic level of the predator. Thus, predators are, on an average, more similar in size to their prey at the top of food-webs than that closer to the base. These findings contradict the traditional Eltonian paradigm and have implications for our understanding of body-mass constraints on food-web topology, community dynamics and stability.     

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.     

Predator prey interactions with time delays

Summary A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which stable periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.