Roger Arditi

Coupling in predator-prey dynamics: Ratio-Dependence

In continuous-time predator-prey models, the per capita rate of consumption (the functional response or “trophic function”) is usually interpreted as a behavioral phenomenon. The classical assumptions are that predators encounter prey at random and that the trophic function depends on prey abundance only. We argue that this approach is not always appropriate. The trophic function must be considered on the slow time scale of population dynamics at which the models operate—not on the fast behavioral time scale. We propose that, in cases where these two time scales differ, it is reasonable to assume that the trophic function depends on the ratio of prey to predator abundances. Several field and laboratory observations support this hypothesis. We compare the consequences of the two types of dependence with respect to the dynamical properties of the models and the responses of population equilibria to variations in primary production. In traditional prey-dependent models, only the predator population responds to primary production, while both levels respond in ratio-dependent models. This result is generalized to food chains. We suggest that the ratio-dependent form of the trophic function is a simple way of accounting for many types of heterogeneity that occur in large scale natural systems, while the prey-dependent form may be more appropriate for homogeneous systems like chemostats.

Empirical Evidence of the role of Heterogeneity in Ratio‐dependent Consumption

Classical models describing the number of prey consumed by predators rest on an analogy with the law of mass action and, consequently, the functional response of predators depends only on the density of prey. An alternative model is that the functional response depends on the ratio of prey and predator densities. We hypothesize that the applicability of one or the other model depends on the degree of heterogeneity of predators and prey in space, the prey-dependent model being appropriate in homogeneous situations while the ratio-dependent model is appropriate in heterogeneous situations. We have de- signed experiments to test this hypothesis, using cladocerans filter-feeding on algae. The design is such that the two types of dependence can be discriminated by observation of equilibrium patterns. Daphnia magna and Simocephalus vetulus, the two cladoceran species tested, differ in their spatial distributions. Daphnia has homogeneous distribution whereas Simocephalus has heterogeneous distribution. Experimental results support the hypothesis: D. magna follows the prey-dependent model and S. vetulus follows the ratio-dependent model. By artificially modifying the environment of the two species, we forced D. magna to a heterogeneous distribution and S. vetulus to a homogeneous distribution. As a con- sequence, each species changed its dependence, further confirming our hypothesis.

Variation in Plankton Densities Among Lakes: A Case for Ratio-Dependent Predation Models

McCauley et al. (1988) have compared the predictions of simple traditional predator-prey models with the patterns of abundance exhibited by Daphnia and its algal food supply. On one hand, the data gathered from several natural lakes of different nutrient status show a concurrent increase of phytoplankton and Daphnia biomasses. On the other hand, all models considered by McCauley et al. (1988) predict that any increase of the parameters quantifying algal productivity would lead to an increase in Daphnia biomass only. The algal biomass should not respond to a productivity increase. To resolve this contradiction, the authors suggest that enrichment affects not only algal production but also Daphnia parameters. The four models considered by McCauley et al. (1988) can be written in a single generalized form:

Ratio‐Dependent Predation: An Abstraction That Works

Recent papers opposing ratio dependence focus on four main criticisms: (1) the empirical evidence we present is insufficient or biased, (2) ratio-dependent models exhibit pathological behavior, (3) ratio dependence lacks a logical or mechanistic base, and (4) more general models incorporate both prey and ratio dependence and there is no need for either of the two simplifications. We review these arguments in the light of empirical evidence from field and experimental studies. We argue that (1) empirical evidence shows that most natural systems are closer to ratio dependence than to prey dependence, (2) pathological dynamics in a mathematical sense is not only realistic, but the lack of such dynamics in prey-dependent models actually makes them pathological in a biological sense, (3) the mechanistic base of ratio dependence is (direct and indirect) interference and resource sharing, and (4) although more general models (with extra parameters) can never fit natural patterns worse than either prey- or ratio-dependent models, there are theoretical, practical, and pedagogical reasons for attempting to find simpler models that can capture the essential dynamics of natural systems.

Logistic Theory of Food Web Dynamics

Classical food-web theory arises from Lotka-Volterra models. As an alternative, we develop a model from the logistic concept of demand and supply. We first extend the logistic to an arbitrary species in a trophic chain or stack by developing a simple equation for any population X i : 1/X i dX i /dt=a i -b i X i -X i /c i X i-1 -d i X i+1 /X i , which includes the effects of intra-specific competition for fixed resources (the term b i X i ), intra-specific competition for renewable resources in the lower trophic level (the term X i /c i X i-1 ), and consumers in the upper trophic level (the term d i X i+1 /X i ). This equation emerges from the basic logistic concept of demand and supply, as captured the consumer/resource ratios, and fulfills all the requirements for a pausible food-chain equation. We then generalize the equation to any polation in a food web of arbitrary complexicity. 1/X i -dX i /dt=a 1 -b i X i -X i /Σ j c ij X j F j r(i) -Σ k d ik X kFk c(i) /X i , where F j r(i) is the fraction of population X j that is a resource for i, and F k c(i) is the fraction of population X k that consumes i. This equations meets all the requirements for a general food web model. Some properties of the model are discussed.

Food web structure at equilibrium and far from it: is it the same?

Theoretical studies, based on stability analysis of equilibria, predict that food webs should be link-poor: naturalists, however, report link-rich webs. We argue that this discrepancy is due to the fact that theoreticians and naturalists do not always study the same situations: the former usually examine systems at equilibrium, whereas the latter observe trophic interactions in systems that may be far from equilibrium. We present a dynamic food web model that can reconcile these approaches by being link-poor at equilibrium and link-rich far from it. The number of interactions within the food web diminishes as the system approaches equilibrium.

Lakemaker: A general object-oriented software tool for modelling the eutrophication process in lakes

A proposition for an integrated modelling tool and simulation environment is presented. The aim of this tool is to help modelling the eutrophication processes in shallow lakes. The system architecture is divided into three main modules plus one that handles the communication between them and the user. The modules are: the Domain Base, which involves a structural description of all components and processes that make components interact; the Model Base, which contains the equations related to the components (dynamic models) and to the processes (static models); and the Database into which input and output data are stored and can be accessed. The program was implemented on a personal computer with MS-DOS® operating system.

Modelling fluctuations and optimal harvesting in perch populations

Tyutyunov, Yu., Arditi, R., Bfittiker, B., Dombrovsky, Yu. and Staub, E., 1993. Modelling fluctuations and optimal harvesting in perch populations. Ecol. Modelling, 69: 19-42. A mathematical model of perch (Perca fluviatilis L.) was elaborated, based on the data for ten Swiss lakes. The aim is management of a commercially exploited population. This required elucidation of dynamics of perch stocks and catches. Statistical data analysis demonstrated complicated population dynamics, including cycles of different periods and chaos. The model describes age and sex structure, nonlinear reproduction (including the effects of cannibalism and competition), individual growth, and harvesting, which depends on fishing effort and mesh size of nets. Bifurcation analysis of the model showed the presence of chaotic dynamics and, hence, high sensitivity to initial conditions. Sensitivity to model parameters was also investigated. Simulation analysis of the fishing process was done on the basis of data from Lake Constance. A multiobjective problem of harvest maximization combined with variance minimization was investigated. The isopleths of the annual harvest and of its dispersion were calculated. Pareto-effective strategies, accounting for trade-off between average harvest and its stability were evaluated. A recommendation was given for optimal mesh size and fishing effort. User-friendly software was developed for IBM PC compatible computers, which can be used to investigate a broad class of age-structured populations.

Optimal foraging in nonpatchy habitats. I. Bounded one-dimensional resource

Abstract We present a model of optimal foraging in habitats where the food has an arbitrary density distribution (continuous or not). The classical models of foraging strategies assume that the food is distributed in patches and that the animal divides its time between the two distinct behaviors of patch exploitation and interpatch travel. This assumption is hard to accept in instances where the food distribution is continuous in space, and where travel and feeding cannot be sharply distinguished. In this paper, the habitat is assumed to be one-dimensional and bounded, and the animal is assumed to have a limited foraging time available. The problem is treated mathematically in the context of the calculus of variations. The optimal solution is to divide the habitat in two subsets according to the food density. In the richer subset, the animal equalizes the density distribution; in the poorer subset, it travels as fast as possible.

Scale invariance is a reasonable approximation in predation models : reply to Ruxton and Gurney

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