Sigmoid functional responses and population stability

Sigmoid functional responses are known to stabilize the differential Lotka-Volterra predator-prey model. However, we have found that they have no such effect in a comparable discrete generation model. The difficulty in stabilizing this model results from the one-generation time delay between changes in predator population density and the level of prey mortality. By contrast, sigmoid functional responses can stabilize the system if the predator population remains relatively constant, as is more likely of generalist predators.     

The effect of long time delays in predator-prey systems

Past studies have indicated that a time delay longer than the natural period of a system will generally cause instability; however here it is shown that including long maturational time delays in a general predator-prey model need not have this effect. In each of the three cases studied (a predator delay, a prey delay, and both), local stability can persevere despite the presence of arbitrarily long time delays. This perseverence depends upon an interaction between delayed and undelayed features of the model. Delayed processes always act to destabilize the model. For example, prey self-regulation, usually a source of stability, becomes destabilizing if subject to a long delay. However, the effect of such a delay is offset by undelayed regulatory processes, such as a stabilizing functional reponse. In addition, the adverse effects of delayed predator recruitment can be reduced by the nonreproductive component of the numerical reponse, a feature not usually involved in determining stability. Finally, it is shown that long time delays are not necessarily more disruptive than short delays; it cannot be assumed that lengthening a time delay progessively reduces stability.     

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.     

A comparison of two predator-prey models with Holling's type I functional response

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.     

Stabilizing effects of spatial heterogeneity in predator-prey systems

A general predator is assumed to divide its hunting time between two sub-habitats with different prey species, spending a larger fraction (φ) of search time in an area as the relative prey abundance there increases. This always causes switching in the model, and changes a functional response from one that imposes a risk on the average prey that decreases with prey density in the direction of one that imposes an increasing risk. I discuss the conditions for a response that is density dependent, and those predatory attributes that make such a response more likely. Transit time between subhabitats always increases the density dependent effect, and is necessary for “system stability” in a Lotka-Volterra model with two prey species. Experiments have confirmed the model's basic assumption. General predators do not fit easily into classical predator-prey models of simple “closed” communities, and then the degree of density dependence of the functional response becomes a useful measure of a predator's short-term stabilizing effect on a prey species. The model demonstrates how spatial heterogeneity can be stabilizing.     

Enrichment and foodchain stability: the impact of different forms of predator-prey interaction

We propose a simple model of an ecological foodchain of arbitrary length. The model is very general in nature and describes a whole class of foodchains. Using the methods of qualitative analysis the model's stability can be analysed without restricting the predator-prey interaction to any specific functional form. The model can therefore be used to study the effect of different functional forms on the stability of the foodchain. We demonstrate that the stability of steady states may strongly depend on the exact functional form of the interaction function used. It is shown that a class of interaction functions exists, which are similar to the widely used Holling functions but bestow radically different stability properties upon the model. An example is shown in which enrichment has a stabilizing effect on the foodchain. By contrast enrichment destabilizes steady states if Holling functions are used.     

The effect of spatial heterogeneity on the persistence of predator-prey interactions

The effect of spatially discontinuous environments on predator-prey systems is examined by using a computer simulation model. It is shown that increasing prey dispersal and decreasing predator dispersal do not necessarily have a stabilizing influence on the interaction, as had been concluded by previous workers. The stability of predator-prey interaction depends on the interaction of the dispersal process with normal reproduction and feeding of the predator and prey species.     

Predator functional response changed by induced defenses in prey

Functional responses play a central role in the nature and stability of predator-prey population dynamics. Here we investigate how induced defenses affect predator functional responses. In experimental communities, prey (Paramecium) expressed two previously undocumented inducible defenses--a speed reduction and a width increase--in response to nonlethal exposure to predatory Stenostomum. Nonlethal exposure also changed the shape of the predator's functional response from Type II to Type III, consistent with changes in the density dependence of attack rates. Handling times were also affected by prey defenses, increasing at least sixfold. These changes show that induced changes in prey have a real defensive function. At low prey densities, induction led to lower attack success; at high prey densities, attack rates were actually higher for induced prey. However, induction increased handling times sufficiently that consumption rates of defended prey were lower than those of undefended prey. Modification of attack rate and handling time has important potential consequences for population dynamics; Type III functional responses can increase the stability of population dynamics and persistence because predation on small populations is low, allowing a relict population to survive. Simulations of a predator-prey population dynamic model revealed the stabilizing potential of the Type III response.     

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.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

Linking saturation,stability and sustainability in food webs with observed equilibrium structure

Stability of a dynamic equilibrium in a predator-prey system depends both on the type of functional response and on the point of equilibrium on the response curve. Saturation effects from Holling type II responses are known to destabilise prey populations, while a type III (sigmoid) response curve has been shown to provide stability at lower levels of saturation. These effects have also been shown in multi-trophic model systems. However, stability analyses of observed equilibria in real complex ecosystems have as yet not assumed non-linear functional responses. Here, we evaluate the implications of saturation in observed balanced material-flow structures, for system stability and sustainability. We first make the effects of the non-linear functional responses on the interaction strengths in a food web transparent by expressing the elements of Jacobian ‘community’ matrices for type II and III systems as simple functions of their linear (type I) counterparts. We then determine the stability of the systems and distinguish two critical saturation levels: (1) a level where the system is just as stable as a type I system and (2) a level above which the system cannot be stable unless it is subsidised, separating a stable materially sustainable regime from an unsustainable one. We explain the stabilising and destabilising effects in terms of the feedbacks in the systems. The results shed light on the robustness of observed patterns of interaction strengths in complex food webs and suggest the implausibility of saturation playing a significant role in the equilibrium dynamics of sustainable ecosystems.     

Predator-prey models with delay and prey harvesting

It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate is increased. Computer simulations are carried to explain the mathematical conclusions. Received: 1 March 2000 / Revised version: 7 September 2000 /?Published online: 21 August 2001     

Age structure in predator-prey systems. II. Functional response and stability and the paradox of enrichment

We employ the general model of predator-prey systems incorporating age structure in the predator, developed in the previous paper, to study the role of functional response in stability and the paradox of enrichment. The destabilizing effect of age structure leads to both qualitatively and quantitatively new results, including a lower bound to prey density for a stable equilibrium, a feature not present in models without age structure.     

Intuition, functional responses and the formulation of predator-prey models when there is a large disparity in the spatial domains of the interacting species

1. The disparity of the spatial domains used by predators and prey is a common feature of many terrestrial avian and mammalian predatory interactions, as predators are typically more mobile and have larger home ranges than their prey. 2. Incorporating these realistic behavioural features requires formulating spatial predator-prey models having local prey mortality due to predation and its spatial aggregation, in order to generate a numerical response at timescales longer than the local prey consumption. Coupling the population dynamics occurring at different spatial scales is far from intuitive, and involves making important behavioural and demographic assumptions. Previous spatial predator-prey models resorted to intuition to derive local functional responses from non-spatial equivalents, and often involve unrealistic biological assumptions that restrict their validity. 3. We propose a hierarchical framework for deriving generic models of spatial predator-prey interactions that explicitly considers the behavioural and demographic processes occurring at different spatial and temporal scales. 4. The proposed framework highlights the circumstances wherein static spatial patterns emerge and can be a stabilizing mechanism of consumer-resource interactions.     

A mechanistic model for interference and Allee effect in the predator population

We present the results of simulations in an individual-based model describing spatial movement and predator-prey interaction within a closed rectangular habitat. Movement of each individual animal is determined by local conditions only, so any collective behavior emerges owing to self-organization. It is shown that the pursuit of prey by predators entails predator interference, manifesting itself at the population level as the dependency of the trophic function (individual ration) on predator abundance. The stabilizing effect of predator interference on the dynamics of a predator-prey system is discussed. Inclusion of prey evasion induces apparent cooperation of predators and further alters the functional response, giving rise to a strong Allee effect, with extinction of the predator population upon dropping below critical numbers. Thus, we propose a simple mechanistic interpretation of important but still poorly understood behavioral phenomena that underlie the functioning of natural trophic systems.     

Global stability of predator-prey interactions

Summary A Lyapunov function is given that extends functions used by Volterra, Goh, and Hsu to a wide class of predator-prey models, including Leslie type models, and a biological interpretation of this function is given. It yields a simple stability criterion, which is used to examine the effect on stability of intraspecific competition among both prey and predators, of a refuge for the prey, and of Holling type II and type III functional responses. Although local stability analysis of these specific models has been done previously, the Lyapunov function facilitates study of global stability and domains of attraction and provides a unified theory which depends on the general nature of the interactions and not on the specific functions used to model them.     

Should "handled" prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory

Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered.     

Age-dependent predation is not a simple process. I. Continuous time models

The stability of models of age-dependent predation in continuous time with predators exhibiting a functional response are analyzed. A number of new features of biological importance emerge that are not present in simpler models. These include limits to the length of juvenile periods (both upper and lower) for stability, and the possibility that increases or decreases in any of the model parameters can be stabilizing or destabilizing. Hence, increased delays are not necessarily destabilizing. The variance in the length of the juvenile period is shown to be an important factor determining stability. Additionally, the relative stability of predation only on juveniles or only on adults is compared.     

Can spatial variation alone lead to selection for dispersal?

The stability of models of age-dependent predation in continuous time with predators exhibiting a functional response are analyzed. A number of new features of biological importance emerge that are not present in simpler models. These include limits to the length of juvenile periods (both upper and lower) for stability, and the possibility that increases or decreases in any of the model parameters can be stabilizing or destabilizing. Hence, increased delays are not necessarily destabilizing. The variance in the length of the juvenile period is shown to be an important factor determining stability. Additionally, the relative stability of predation only on juveniles or only on adults is compared.