Prey-dependent Functional Response Research Articles

Ratio-Dependence in Predator-Prey Systems as an Edge and Basic Minimal Model of Predator Interference

The functional response (trophic function or individual ration) quantifies the average amount of prey consumed per unit of time by a single predator. Since the seminal Lotka-Volterra model, it is a key element of the predation theory. Holling has enhanced the theory by classifying prey-dependent functional responses into three types that long remained a generally accepted basis of modeling predator-prey interactions. However, contradictions between the observed dynamics of natural ecosystems and the properties of predator-prey models with Holling-type trophic functions, such as the paradox of enrichment, the paradox of biological control, and the paradoxical enrichment response mediated by trophic cascades, required further improvement of the theory. This led to the idea of the inclusion of predator interference into the trophic function. Various functional responses depending on both prey and predator densities have been suggested and compared in their performance to fit observed data. At the end of the 1980s, Arditi and Ginzburg stimulated a lively debate having a strong impact on predation theory. They proposed the concept of a spectrum of predator-dependent trophic functions, with two opposite edges being the prey-dependent and the ratio-dependent cases, and they suggested revising the theory by using the ratio-dependent edge of the spectrum as a null model of predator interference. Ratio-dependence offers the simplest way of accounting for mutual interference in predator-prey models, resolving the abovementioned contradictions between theory and natural observations. Depending on the practical needs and the availability of observations, the more detailed models can be built on this theoretical basis.

Deterministic and Stochastic Holling-Tanner Prey-Predator Models

A modified version of the so called Holling-Tanner prey-predator models with prey dependent functional response is introduced. We improved some new results on Holling-Tanner model from Lotka-Volterra model on real ecological systems and studied the stability of this model in the deterministic and stochastic environments. The study was focused on three types of stability, namely, stable node, spiral node, and center. The numerical schemes are employed to get the approximated solutions of the differential equations. We have used Euler scheme to solve the deterministic prey-predator model and we used Euler-Maruyama scheme to solve stochastic prey-predator model.

Emergent patterns in diffusive Turing-like systems with fractional-order operator

Patterns obtained in abiotically homogeneous habitats are of specific interest due to the fact that they require an explanation based on the individual behavior of chemical or biological species. They are often referred to as ‘emergent patterns,’ which arise due to nonlinear interactions of species in spatial scales that are much more larger than the individuals characteristic scale. In this work, we examine the spatial pattern formation of diffusive fractional predator–prey models with different functional response. In the first model, we investigate the dynamics of the Riesz fractional predation of Holling type-II functional response with the prey Allee effects, while the second model describes prey-dependent functional response of Ivlev-case and fractional reaction–diffusion. In order to give good guidelines on the correct choice of parameters for numerical simulation experiment of full fractional-order reaction–diffusion systems, we discuss the dynamics of each system in the biologically meaningful region $$u\ge 0$$ and $$v\ge 0$$ and give conditions for the existence of Hopf bifurcation, and Turing instability with either homogeneous (zero-flux) boundary conditions which imply no external input or Dirichlet boundary conditions. A novel alternating direction implicit based on backward Euler scheme with either the homogeneous Neumann (zero-flux) or Dirichlet boundary is applied for the numerical solution. The performance of this method is compared with that of the shifted Grunwald formula in terms of accuracy and computational time. Numerical experiments which justify our theoretical findings exhibits some fractional-order controlled patterns of stripes, spots and chaotic spiral-like structures that are mostly found in animal coats.

Trypanosome-host interaction revealed through the zebrafish looking glass

As a first approximation of immune-mediated within-host parasite dynamics we can consider the immune response as a predator, with the parasite as its prey. In the ecological literature of predator–prey interactions there are a number of different functional responses used to describe how a predator reproduces in response to consuming prey. Until recently most of the models of the immune system that have taken a predator–prey approach have used simple mass action dynamics to capture the interaction between the immune response and the parasite. More recently Fenton and Perkins (2010) employed three of the most commonly used prey-dependent functional response terms from the ecological literature.In this paper we make use of a technique from computing science, process algebra, to develop mathematical models. The novelty of the process algebra approach is to allow stochastic models of the population (parasite and immune cells) to be developed from rules of individual cell behaviour. By using this approach in which individual cellular behaviour is captured we have derived a ratio-dependent response similar to that seen in the previous models of immune-mediated parasite dynamics, confirming that, whilst this type of term is controversial in ecological predator–prey models, it is appropriate for models of the immune system.

Studying Controversies: Unification, Contradiction, Integration

My aim here is to show that approximate truth as a paraconsistent notion (neutral to the realism/anti-realism debate) can be successfully incorporated into the analysis of scientific unification, thus advancing towards a more realistic representation of theory development that takes into account the controversies that often loom alongside the progress of research programmes. I support my analysis with a case study of the recent debate in ecology centred around the existence of the paradox of enrichment and the controversy between ecological models of predation that employ prey-dependent and ratio-dependent functional responses. These models were initially proposed as equally good representations of the basic aspects of predator–prey dynamics. However, both models generated inconsistent observational consequences and, therefore, introduced a contradiction within predator–prey theory. I argue that by accepting these models as approximately true representations of predator–prey dynamics we can convey how the available observational data have been successfully systematized in a consistent way under them. This first step in resolving the controversy relied on building a series of contrastive arguments based on both models’ derivations about population dynamics and the available empirical data. The heightening of this contrast between the models, in turn, was also essential in defining a limiting function which can be used to integrate both models and reach a new unified expression of predator–prey dynamics.

Analysis on the existence of the steady-states for an ecological–mathematical model with predator–prey-dependent functional response

We establish a reaction–diffusion ecological–mathematical model with predator–prey dependent functional response and the Neumann boundary conditions. First, we study the bifurcation solution emitting from the unique positive constant solution by considering the local bifurcation. Finally, we mainly analyze the coexistence of the prey and predator with the help of the fixed point index theory, where we use a different approach in calculating one of the fixed point indices.

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra- and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.

Analysis on dynamics of a population model with predator–prey-dependent functional response

We consider a reaction–diffusion population model with predator–prey-dependent functional response. Firstly, we discuss the conditions which ensure the model has a unique positive constant solution. Secondly, we investigate the dynamical properties of the model, including the large time behaviors of the nonconstant solutions and the local and the global asymptotic stability of the positive constant solution.

Evaluation of alternative prey-, predator-, and ratio-dependent functional response models in a zooplankton microcosm

There is strenuous debate among ecologists regarding the inclusion of predator density into the originally prey-dependent functional response. We provided comprehensive empirical comparisons of alternative functional response models for the predatory ostracod Heterocypris incongruens (Ramdohr, 1808) and the rotifer Brachionus calyciflorus (Pallas, 1766) as its prey in small freshwater microcosms. Prey killed was measured at factorial combinations of four predator densities and five prey densities, and was recorded at 3 min intervals over 60 min experiments. To support the potential effect of predator interference on per capita kill rate, we recorded ostracod activity and aggression. Kill rate increased following a saturating function with increasing prey density and decreased with increasing predator density. Model evaluation using an information–theoretic approach indicated that the Arditi–Ginzburg type II ratio-dependent model performed best, followed by the Arditi–Akcakaya and Beddington–DeAngelis type II predator-dependent models, suggesting that predator interference was important in predicting kill rates. Interference among predators increased and their activity decreased with increasing predator density, providing confirmation that interference was responsible for the predator-dependent effect. By combining a microcosm experiment and behavioral observations, our results suggest that predator interference at realistic population densities influences ostracod kill rates and this form of interference was best accommodated by predator-dependent models.

Interaction modification effects on ecological networks are affected by ratio dependence and network topology

Interaction modification (IM), where one species modifies the strength of the density-mediated direct interaction between two other species, is an important ecological process, but little is known about the collective effect of multiple IM on overall community dynamics. We use stochastic bioenergetic modelling of ecological networks with different network topologies, functional responses and parameter values, to investigate the effects of IM connectance and IM strength on ecosystem properties including the evenness of species abundances and variability of system biomass. We found that the maximum system biomass that could potentially be attained by the model systems increased with IM connectance and strength when the models had nonrandom topology and prey-dependent functional responses as opposed to random topology and ratio-dependent responses. The maximum potential species evenness increased with IM strength but decreased with increasing IM connectance, when all modifications were negative. These findings underscore the importance of accounting for multiple IM across the community for understanding complex community dynamics.

Stage-structured ratio-dependent predator–prey models revisited: When should the maturation lag result in systems’ destabilization?

Time delay is often introduced into models of theoretical ecology to take into account stage structuring of real populations. The well-known characteristic benchmark of time delay models is eventual destabilization of systems’ equilibria for a sufficiently large maturation/gestation time period. Here we argue, however, that some delay formulations might lack a sound biological rationale and, more importantly, the use of different delay formulations in models might result in rather different outcomes in terms of stability loss. To illustrate this idea we consider a family of predator–prey models with a ratio-dependent predator functional response with a maturation time lag of predators. In such models the functional response depends on the ratio between the predators and the available prey as opposed to a prey-dependent functional response. To describe the effects of delay, we use two different formulations from the literature: one based on the work by Beretta and Kuang (1996), which we call the conventional approach, with delay being included only into the per-capita numerical response of predator. The other formulation is the Wangersky–Cunningham (1957) approach, where delay is introduced in the overall predator numerical response. Unlike the previous studies, we focus here on deriving the explicit conditions of stability of the interior equilibrium (assuring species coexistence) in the presence of delay in terms of model parameters. We investigate three scenarios of prey growth rate parametrization: (i) the prey growth is given by the logistic function, (ii) the prey growth is subject to a strong Allee effect and (iii) there is a weak Allee effect in prey. In the latter two cases the per capita growth rate is an increasing function at low prey density. We show that the use of the two above delay formulations eventually result in completely different outcomes: with the conventional approach, the interior predator–prey equilibrium will be eventually destabilized for a supercritical time lag, whereas, implementation of the Wangersky–Cunningham approach predicts an absolute stability of the equilibrium within a large range of parameters, i.e. the system cannot be destabilized by means of delay. We find that for the models parameters, where delay-induced destabilization in the system with an Allee effect is possible (the interior equilibrium is conditionally stable), the stability loss eventually results in population collapse and extinction of both species.

Using process algebra to develop predator–prey models of within-host parasite dynamics

As a first approximation of immune-mediated within-host parasite dynamics we can consider the immune response as a predator, with the parasite as its prey. In the ecological literature of predator–prey interactions there are a number of different functional responses used to describe how a predator reproduces in response to consuming prey. Until recently most of the models of the immune system that have taken a predator–prey approach have used simple mass action dynamics to capture the interaction between the immune response and the parasite. More recently Fenton and Perkins (2010) employed three of the most commonly used prey-dependent functional response terms from the ecological literature.In this paper we make use of a technique from computing science, process algebra, to develop mathematical models. The novelty of the process algebra approach is to allow stochastic models of the population (parasite and immune cells) to be developed from rules of individual cell behaviour. By using this approach in which individual cellular behaviour is captured we have derived a ratio-dependent response similar to that seen in the previous models of immune-mediated parasite dynamics, confirming that, whilst this type of term is controversial in ecological predator–prey models, it is appropriate for models of the immune system.

An impulsively controlled pest management model with n predator species and a common prey

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.

Prey–predator dynamics in rotifers: density‐dependent consequences of spatial heterogeneity due to surface attachment

Classical models of prey-predator interactions assume that per capita prey consumption is dependent on prey density alone and that prey consumption (functional response) and consumer proliferation (numerical response) operate on the same timescales and without time lags. Several modifications have been proposed for resolving this timescale discrepancy, including variants where the functional response depends on both prey and predator densities. A microcosm system with the rotifer Brachionus 'Nevada' feeding on the prasinophyte Tetraselmis sp. showed significant (P < 0.0005) increases in steady-state biomasses of both prey and predators with increasing carrying capacity (represented by total phosphorus of the growth medium), which is inconsistent with predictions based on the traditional prey-only-dependent functional response. We provide data indicating that surfaces where the predator can attach provide a high-quality habitat for rotifers, which can result in a predator-dependent functional response. We also show that partitioning between the attached and free-swimming habitats was fast compared to the timescale of the numerical response. When attached to surfaces, rotifers maximized net energy gain by avoiding the high cost of swimming and by increased food capture due to reduced viscous drag. A mathematical model with prey-dependent functional response and wall-attached and free-swimming fractions of the population describes our data adequately. We discuss the implications of this finding for extrapolating microcosm experiments to systems with other surface-to-volume ratios, and to what extent our findings may apply to other popular model organisms for prey-predator interaction.

Bifurcation analysis of a ratio-dependent prey–predator model with the Allee effect

There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee effect. We both consider the cases of a strong Allee effect (the population growth rate is negative at low species density) and the case of a weak Allee effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee effect results in a paradox of biological control.

Climate, season, and social status modulate the functional response of an efficient stalking predator: the Eurasian lynx

1. Predation plays a major role in shaping the structure and dynamics of ecological communities, and the functional response of a predator is of crucial importance to the dynamics of any predator-prey system by linking the trophic levels. For large mammals, there is a dearth of field studies documenting functional responses, and observations at low prey density are particularly scarce. Furthermore, there is a lack of understanding about how variables such as season, social status and climate modulate the functional response curves. 2. We analysed kill rate data collected over a 10-year period based on radio-marked lynx (Lynx lynx) mainly preying on roe deer (Capreolus capreolus) along a steep prey density gradient in south-eastern Norway. 3. The asymptotic kill rate was reached at a very low prey density for both solitary individuals and family groups (i.e. females with their dependent kittens), indicative of an efficient predator. This highlights the importance of understanding the interplay between predator and prey at low prey densities. 4. A purely prey-dependent functional response was a poor descriptor of the data, as the curve was strongly modulated by season and differences between lynx of different social status. In addition, there was a clear effect of abiotic climatic factors (indexed by the North Atlantic Oscillation) on observed kill rates in the more snow-rich portion of our study area. 5. Our analysis suggests that simple functional response curves might be poor descriptors of predator consumption rates in complex natural system, and that auxiliary factors are likely to induce complexity into any predator-prey systems that would not be captured by simple deterministic approaches.

Locating the equilibrium points of a predator–prey model by means of affine state feedback

Abstract A predator–prey model with prey-dependent functional response is considered. The set of all points in the positive quadrant of the state plane that can be made equilibrium points by means of an affine state-feedback control law is determined, and the values of the control parameters ensuring the desired equilibria are provided. It is shown how the asymptotic stability of the equilibrium points depends on simple geometric conditions. The problem of stabilizing unstable equilibrium points is also briefly discussed.

Experimental measurements of functional response: What is the relevance for food web theory?

The functional response describes how the rate of consumption changes in response to a change in the density of the resource. Since HOLLING’s (1959) seminal study, ecologists have measured and categorized the functional response for a large number of specific consumer-resource relationships (JESCHKE et al. 2002). Functional response curves are ecologically important because they allow inferences about rates of resource depletion, predator preferences for certain prey types, and competitive hierarchies among consumers. Theoretical ecologists, however, have a more general interest in the functional response. They use functional responses as crucial link functions between consumers and their resources in generic multi-species food web models that predict community dynamics over multiple generations. Such models are particularly appropriate for planktonic communities, characterized by short generation times, and make predictions about the stability, biomass distributions, and the importance of indirect effects in these aquatic food webs. The dynamics of such model food webs, however, are extremely sensitive to the type and mathematical formulation of the functional response (WILLIAMS & MARTINEZ 2004, FUSSMANN & BLASIUS 2005). The cumulative evidence gathered from field observations and experiments can help ecologists choose functional response models representative of natural communities. I investigated whether the functional response is predatoror prey-dependent (or, more generally, consumeror resourcedependent) and how this question is best approached in experiments with aquatic microorganisms. Prey-dependence is the traditional assumption in food web models, under which the per-capita resource uptake (g) by the consumer is only a function of the resource concentration (x): g = f(x). Over the past 2 decades this view has been challenged under the banner of the predator-dependence hypothesis, which claims that per-capita food uptake should depend on both consumer concentration (y) and prey concentration: g = f(x, y). Ratio-dependence is a special form of predator-dependence; the functional response is a function of the ratio of resource and consumer concentrations: g = f(x/y). This problem is essentially unresolved, and the arguments that proponents of either view have put forward to strengthen their cases are not discussed here (see ABRAMS & GINZBURG 2000). However, all debaters agree on 2 points: (1) The question is of great importance for food web theory because models predict dramatically different dynamics depending on whether predatoror prey-dependent functional responses are chosen. (2) Targeted experimentation with real communities can bring us closer to a solution to this debate that has been characterized by an exchange of largely theoretical viewpoints. In recent years, aquatic and terrestrial ecologists have conducted experiments explicitly designed to distinguish between consumerand resource-dependence (SKALSKI & GILLIAM 2001, FUSSMANN et al. 2005, SCHENK et al. 2005, TSCHANZ et al. 2007). Although this trend is generally welcomed, some experiments have also met with criticism (JENSEN et al. 2007). It appears that a current point of debate is how such experiments ought to be designed and conducted so they provide discriminatory power and are not biased to back either hypothesis by design (FUSSMANN et al. 2007, JENSEN et al. 2007).

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.

The interaction of spatial scale and predator-prey functional response

Predator-prey models with a prey-dependent functional response have the property that the prey equilibrium value is determined only by predator characteristics. However, in observed natural systems (for instance, snail-periphyton interactions in streams) the equilibrium periphyton biomass has been shown experimentally to be influenced by both snail numbers and levels of available limiting nutrient in the water. Hypothesizing that the observed patchiness in periphyton in streams may be part of the explanation for the departure of behavior of the equilibrium biomasses from predictions of the prey-dependent response of the snail-periphyton system, we developed and analyzed a spatially-explicit model of periphyton in which snails were modeled as individuals in their movement and feeding, and periphyton was modeled as patches or spatial cells. Three different assumptions on snail movement were used: (1) random movement between spatial cells, (2) tracking by snails of local abundances of periphyton, and (3) delayed departure of snails from cells to reduce costs associated with movement. Of these assumptions, only the third strategy, based on an herbivore strategy of staying in one patch until local periphyton biomass concentration falls below a certain threshold amount, produced results in which both periphyton and snail biomass increased with nutrient input. Thus, if data are averaged spatially over the whole system, we expect that a ratio-dependent functional response may be observed if the herbivore behaves according to the third assumption. Both random movement and delayed cell departure had the result that spatial heterogeneity of periphyton increased with nutrient input.