Elsevier

Progress in Oceanography

Volume 134, May 2015, Pages 232-243
Progress in Oceanography

Analysis of functional response in presence of schooling phenomena: An IBM approach

https://doi.org/10.1016/j.pocean.2015.02.002Get rights and content

Highlights

  • Functional responses emerge from an IBM in which schooling occurs.

  • Collective behaviour of prey causes higher functional responses.

  • Holling type II functional response emerges when prey aggregation does not occur.

  • Holling type III functional response emerges if prey aggregate.

Abstract

The aim of this paper is to analyse the emergence of functional response of a predator–prey system starting from diverse simulations of an Individual-Based Model of schooling fish. Individual characteristics can, indeed, play an important role in establishing group dynamics. The central question we address is whether or not aggregation influences predator–prey relationships.

To answer this question, we analyse the consequences of schooling when estimating functional response in four configurations: (1) no schooling of either prey nor predators; (2) schooling of prey only; (3) schooling of predators only; and (4) schooling of both prey and predators. Aggregation is modelled using the rules of attraction, alignment and repulsion.

We find important differences between the various configurations, highlighting that functional response is largely affected by collective behaviour. In particular, we show: (1) an increased predation efficiency when prey school and (2) different functional response shapes: Holling type II emerges if prey do not school, while Holling type III emerges when prey aggregate.

Introduction

The scientific community today is called upon to solve many compelling challenges dealing with crucial issues such as climate changes, overexploitation of resources and the necessity of sustainable economic development. In addition, there is a need for intelligent management of living resources. This goal requires a deep knowledge of the interactions within different species as well as between species and the environment. In particular, predation is one of the most important factors influencing the ecological structure and the development of communities, as already stressed by Bax, 1998, Geritz and Gyllenberg, 2013, who showed that predation is a key process in ecosystem functioning which must not be neglected in longer-term management. This is especially true in marine ecosystems due to both the complexity of the food net and to intensive fishing activities, which could cause marked cascading effects (Scheffer et al., 2005). Predator–prey dynamics are usually represented by a functional response, which is the amount of prey eaten per predator and per unit of time. This function is a proxy of the flux of matter from one trophic level to another as it determines the transfer of biomass in the food chain (Poggiale, 1998). Typically, a predator–prey model focuses on the interactions between two isolated species (Geritz and Gyllenberg, 2013), taking into account some aspects that are considered nodal to explain the dynamics. These interactions depend on the nature of the studied species. Crucial among these characteristics are collective behaviours, especially in the context of marine ecosystems. In fact, in these ecosystems, schooling and swarming are dominant features (over 50% of bony fish species school (Shaw, 1978, Major, 1978)).

Over the last three decades, considerable attention has been paid to this phenomenon in the literature. Aggregates displaying collective behaviours are present in many different systems, from non-living ones (such as nanoparticles clusters) to living ones (schooling fish, swarming ants or flocking birds). Important common features can be identified in all these cases (Giardina, 2008):

  • collective behaviour emerges in the absence of centralised control;

  • the mechanism of group formation is very general and transcends the detailed nature of its components;

  • some collective properties, known as emergent properties, arise from the set of individuals.

From a modelling point of view, the challenge is to build a model that begins with the description of individual interactions and goes on to reproduce the group formation and predict its dynamics. Many examples of clustering modelling can be found in the literature. A statistical physics approach was introduced by Vicsek et al. and became known as the “Vicsek Model” (VM) (Vicsek et al., 1995, Vicsek and Zafiris, 2012). This model is based on the assumption that the movements of living organisms are the result of self-propulsion, of interactions with neighbours and of randomness (Vicsek et al., 1999). In the literature, other models presenting additional rules governing interaction can be found. Hubbard et al. (2004) add an environmental gradient and Czirok et al. (1999) provide an extra spatial dimension (3D model). Other authors have built models characterised by the rules of attraction and repulsion. Couzin et al. (2002), for example, formulate a model for three-dimensional schooling fish (or flocking birds) in which repulsion, alignment and attraction interactions take place (the so-called “A/R/A model”). They test how behavioural differences among organisms influence aggregation processes. Moreover, they demonstrate sharp transitions between four collective behaviours (swarm, torus, dynamic parallel group, highly parallel group) by changing model parameters. Attraction–repulsion rules are also present in the work of Inada and Kawachi, 2002, Grégoire et al., 2003. The first study analyses order and flexibility in the motion of fish schools by changing the number of interacting neighbours and the randomness of motion. Simulations show that school order is strongly affected by randomness and by the number of interacting fish. High interconnection among fish leads to patterns of escape in the presence of a predator. Grégoire et al. (2003) combine Lennard-Jones potential with alignment and study phase transitions in such a system of self-propelled particles.

Improvements in the description of collective behaviours has been made possible by the huge development of Individual-Based Models (IBM). These models are interesting because of the novel way in which they approach the topic: they describe the dynamics at the individual level, by setting the rules of movement and the characteristics of every individual agent, while their outputs provide a representation of the whole system. One can actually see collective properties emerging from individual behaviours. Moreover, by simply testing different values of the parameters, it is possible to estimate how a change at individual scale can impact the whole system.

Consequently, IBMs are widely used to study aggregation phenomena of animals. Nevertheless, the majority of individual-based schooling models presented in the literature rarely focus on functional response to determine whether schooling, or more generally aggregation, has consequences for predator–prey dynamics. Attention is often paid to either aggregate behaviour under attack, such as Inada and Kawachi (2002) and in Lee et al. (2006), or to evolutionary topics. In this regard, Wood and Ackland (2007) consider the evolution of various aggregating features to examine which flock configurations may be selected in optimising foraging, or in defending against predation. They find that two types of flocks emerge when predators are present. The first is a slow-moving, milling group, characterised by a low orientation radius and a high turning angle. The second is a fast moving, dynamic group, with a large orientation zone.

Concerning functional response, we can cite (Tyutyunov et al., 2008, Cosner et al., 1999, Poggiale, 1998). The first paper analyses the way in which different assumptions about individual movements lead to various kinds of functional response. Aggregation is not explicitly expressed in this IBM, but predator prey-taxis and evasion of predators by prey individuals is considered, in addition to random displacement. The taxis stimulus of each species is the odour of the other species. The distribution of the odour of several individuals is obtained by superimposing all individual odours. Consequently, denser zones exist that mainly attract (or repulse) individuals. Depending on the intensity of taxis and on predator density, the predator population exhibits varying degrees of interference. Hence, functional response results as being prey-dependent if no directional movements are considered, and predator-dependent if predators actively hunt the prey. Moreover, for particular values of predator density and taxis, ratio-dependent responses appear. The latter two papers do not deal with IBM. Cosner et al. (1999) examine how existing predator–prey models (from “traditional” to ratio/dependent models) can be derived in a unified way from mass action principles. Indeed, the authors start from a generalisation of the functional response and analyse how the total encounter rate between predator and prey is influenced by their spatial heterogeneity. However, this theoretical work does not explain how individual behaviours lead to different kinds of clustering. Poggiale (1998) studies spatial heterogeneity effects on functional response when different time scales occur. By using aggregation methods, this work links functional response to individual behaviour in a multi-patch environment.

Published works have thus either focused primarily on possible aggregate responses to predator attacks or they have explored the theoretical formulation of functional response.

The objectives of this paper are: (1) to test the emergence of functional response and its qualitative properties in the presence of schooling phenomenon, with no prior hypothesis concerning defence or attack strategies and (2) to compare these properties with the emerging functional response in the absence of schooling.

For this purpose, we first consider interactions between two species, then we define a set of aggregation rules according to the A/R/A model and finally we analyse the consequences of schooling in predator–prey dynamics. We make the assumption that predators are attracted by prey situated within a visual-range distance.

We analyse four cases: (i) a simple predator–prey model in which no schooling behaviour is present; (ii) the presence of schooling prey only; (iii) predators only school; and (iv) both prey and predators school.

In the first part of this paper we explain the model rules (Section ‘Material and methods’). In the second part we provide various model studies and results (Section ‘Results’). Finally, we discuss the results (Section ‘Discussion’) and conclude (Section ‘Conclusions’).

Section snippets

State variables and rules

The state variables we are dealing with are agents, virtually representing fish, moving in a two dimensional space, a disk of radius L (see Table 1 for numerical values) and split in two types: prey and predators. The position of each agent is defined in polar coordinates by a radius and an angle (Eq. (1)):rip(t)=(rip(t),θip(t))i=1,,NrjPr(t)=(rjPr(t),θjPr(t))j=1,,Pwhere p stands for prey and Pr for Predators. If we note IR,I=[0;L], we have both rpI and rPrI,θp and θPr[0;2π] and finally t

Results

Hereafter we analyse the functional responses obtained from simulations. We display simulation data for all four cases and data trend estimated by non parametric regression over 50 simulations (bandwidth selected by cross validation (Simonoff, 1998)). Moreover, we present the mean of the 50 Holling type II and III functional responses.

In Fig. 2(a) the simplest configuration is presented, where no schooling phenomenon takes place. For schooling predators and no schooling prey, illustrated in

Discussion

Consumer–resource interactions are basic in ecology. In many modelling studies, attention is primarily focused on collective behaviour facing a single predator, or on different schooling escape strategies, or again on the theoretical formulation of functional response in population dynamics. However, in this paper we focus on the emergence of functional response using an individual-based approach. We realise simulations of predator–prey interactions to verify if the schooling process could

Conclusions

In this paper, our primary aim has been to show the consequences of school formations on predator–prey interactions and to show that the occurrence of inhomogeneous interacting groups influences predation efficiency. We propose an IBM describing predator–prey dynamics in the presence of schooling phenomena. Our rules have been built with the aim of constructing as simple a model as possible, one which can be adapted to different species simply by changing a few parameter values, or which can be

Acknowledgements

The authors would like to thank Mathias Gauduchon for interesting discussions and the anonymous referees for their detailed and useful comments, which helped us to improve the manuscript. We acknowledge the support of the French National Research Agency, under the Grant ANR-09-CEP-003.

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