# Models of kleptoparasitism on networks: the effect of population structure on food stealing behaviour

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## Abstract

The behaviour of populations consisting of animals that interact with each other for their survival and reproduction is usually investigated assuming homogeneity amongst the animals. However, real populations are non-homogeneous. We focus on an established model of kleptoparasitism and investigate whether and how much population heterogeneities can affect the behaviour of kleptoparasitic populations. We consider a situation where animals can either discover food items by themselves or attempt to steal the food already discovered by other animals through aggressive interactions. Representing the likely interactions between animals by a network, we develop pairwise and individual-based models to describe heterogeneities in both the population structure and other individual characteristics, including searching and fighting abilities. For each of the models developed we derive analytic solutions at the steady state. The high accuracy of the solutions is shown in various examples of populations with different degrees of heterogeneity. We observe that highly heterogeneous structures can significantly affect the food intake rate and therefore the fitness of animals. In particular, the more highly connected animals engage in more conflicts, and have a reduced food consumption rate compared to poorly connected animals. Further, for equivalent average level of connectedness, the average consumption rate of a population with heterogeneous structure can be higher.

### Keywords

Kleptoparasitism Food stealing Structured populations Networks Individual-based models Pairwise models### Mathematics Subject Classification

05C57 91A43 92B05 91A40 91B72 92D50## 1 Introduction

In many biological situations animals attempt to steal food already discovered by others for their survival. This is a very common form of feeding, usually referred to as *kleptoparasitism*. Different forms of kleptoparasitic behaviour are observed in many species in the animal kingdom, for example species of spiders (e.g., Coyle et al. 1991), birds (e.g., Brockmann and Barnard 1979), snails (e.g., Iyengar 2002), lizards (e.g., Cooper and Pérez-Mellado 2003), fish (e.g., Hamilton and Dill 2003), primates (e.g., Janson 1985), carnivores (e.g., Carbone et al. 2005) and insects (e.g., Erlandsson 1988). This behaviour of animals has been well documented in a review paper (Iyengar 2008).

There are a number of theoretical models focused on the kleptoparasitic behaviour of animals using different mathematical methods, in particular evolutionary game theory (e.g., Barnard and Sibly 1981; Broom and Ruxton 1998). The model of Broom and Ruxton (1998) is a model of kleptoparasitism which employs game theory to consider the ecological conditions under which attacking to steal the food from other animals when the opportunity arises is the best strategy that foraging animals should adopt in order to maximise their food intake rate and consequently their fitness. Food in this model comes as single indivisible items, which must be consumed completely by an animal. Thus, food can never be shared and challenging animals attempt to steal the whole item from the owner, or not.

A series of publications has appeared developing the original model of Broom and Ruxton (1998) in a number of ways (e.g., Broom and Ruxton 2003; Broom and Rychtář 2007; Luther et al. 2007; Yates and Broom 2007; Broom et al. 2008; Broom and Rychtář 2009, 2011; Hadjichrysanthou and Broom 2012). Crowe et al. (2009) provides a brief review on the main theoretical work on kleproparasitism prior to the investigation of a stochastic model of kleptoparasitism in finite populations. A comparison between some main models of kleptoparasitism is discussed in Vahl (2006) and an alternative model is presented. There is also a series of related mechanistic, but not game-theoretic, models which investigate interference competition, where foraging animals engage in aggressive interactions in order, for example, to defend their territory, resulting in negative effects on their foraging efficiency (e.g., Beddington 1975; Ruxton et al. 1992; van der Meer and Ens 1997; Vahl 2006; Smallegange and van der Meer 2009; van der Meer and Smallegange 2009).

The game-theoretical model of Broom and Ruxton (1998) and the subsequent work assumed that the population of foraging animals is infinitely large, homogeneous and well-mixed, where every animal has the same foraging and fighting abilities and is equally likely to meet and interact with any other animal. However, in natural situations, animals differ in these abilities and usually forage in small groups forming complex relationships and social structure (e.g., Krause et al. 2007; Croft et al. 2008). A number of stochastic models have been developed to consider the dynamics of kleptoparasitic populations of finite size (see Yates and Broom 2007; Crowe et al. 2009). However, the effect of the population structure and other heterogeneities on the behaviour of such populations remains a research question.

In this paper, we explore the role of the underlying connectivity and other heterogeneities between animals in the dynamics of kleptoparasitic populations. Interactions between animals are represented by a static network; each animal is assumed to occupy a node, the links between nodes represent interactive relationships. We first revisit the original model of Broom and Ruxton (1998), which provides a baseline approximation to the key features of the system that we will consider, in particular the handling ratio, the proportion of animals handling food items at any one time. We then extend the model to networks. We develop and analyse an individual-based model which incorporates an arbitrarily complex population structure and heterogeneities in the model parameters and obtain a more precise estimate of these features, albeit with the downside that a large number of parameters and distinct estimates are required. To reduce the number of parameters and equations we also describe the system by a pairwise approximation model. We consider the effect of the population structure and compare the numerical solution of the two models and analytical solutions derived with the results of stochastic simulations on theoretical and empirical networks.

## 2 Models of kleptoparasitism in homogeneous well-mixed populations

### 2.1 The model of Broom and Ruxton (1998)

*P*the population density, by

*S*the density of searchers and by

*H*the density of handlers. When a foraging animal encounters an animal in the handling state it attacks it to steal the prey. There is a constant density of food items

*f*available and searchers cover an area \(\nu _{f}\) per unit of time whilst searching for food, so animals find food at rate \(\nu _{f}f\). If a handler animal is not attacked, it consumes its food item in a time drawn randomly from an exponential distribution with mean \(t_h\). Attacked animals always defend their food and a fight takes place. Searchers encounter handlers and engage in a fight at rate \(\nu _{h}H\). A fight lasts for a time drawn randomly from an exponential distribution with mean \(t_a/2\). In the model of Broom and Ruxton (1998) it was assumed that animals involved in an aggressive interaction are equally likely to win the fight and obtain the food. Here, we allow different competitive abilities between the attacker and the attacked animals, i.e. the probability of the attacking animal winning and obtaining the food, \(\alpha \), varies between 0 and 1, as happens in natural situations. This was an extension to the original model introduced by Broom et al. (2004). We denote by

*A*and

*R*the density of attacking searchers and defenders engaged in a fight, respectively. The loser of the fight returns to the searching state while the winner starts handling the food item. The model notation is summarised in Table 1.

Notation of the basic game-theoretical model of kleptoparasitism

Population’s densities | Meaning |
---|---|

| Density of the population |

| Density of searchers, handlers, attackers and defenders |

Model parameters | Meaning |
---|---|

\(\nu _{f}f\) | Rate at which foragers find undiscovered food |

\(\nu _{h}H\) | Rate at which foragers encounter handlers |

\(t_{h}\) | Expected time for a handler to consume a food item if it is not attacked |

\(t_{a}/2\) | Expected duration of a fight |

\(\alpha \) | The probability that the attacker wins the fight |

### 2.2 The searchers-handlers relationship at the steady state and the handling ratio

*handling ratio*\(\frac{H}{P}\), the proportion of handlers at the steady state, is given by

## 3 Models of kleptoparasitism on networks

### 3.1 An individual-based model

We extend the model (1)–(4) at the individual-level to allow heterogeneity in the parameters and the underlying connectivity between animals. We assume that the population can be represented by a network of *N* nodes. Each node is occupied by a single animal. A link between two nodes represents a connection between the two animals occupying the nodes. The network can be described by the adjacency matrix \(A_M=[a_{ij}]\), where \(a_{ij} = 1\) if node *i* is connected to node *j* and \(a_{ij} = 0\) otherwise, \( i, j \in \{1, 2,\ldots , N\}\). All the networks that we consider in this study are static networks. It is also natural to assume that the networks are undirected, i.e. an animal that can attack another animal can also be attacked by the same animal.

*i*, \(d_i\), is given by:

*i*is either in the searching state with probability \({\langle S_i\rangle }\), the handling state with probability \({\langle H_i\rangle }\), or it is fighting with a connected animal in some node

*j*. Animal

*i*searches for food at speed \({v_f}_i\) and discovers a food item at rate \({v_f}_if_i\), where \(f_i\) is the food availability for the animal in node

*i*. A searcher \(S_i\) also searches for handlers at rate \(v_{h,i}\). If a searcher encounters a handler, they engage in a fight over the food. Let \({\langle A_{i}R_{j}\rangle }\) denote the probability that animals in nodes

*i*and

*j*are engaged in a fight following the attack of \(S_i\) on \(H_j\). Clearly, \(\sum _{j}{\langle A_{i}R_{j}\rangle }\) is equal to the overall probability of an animal in node

*i*being an attacker, \(\langle A_{i}\rangle \), and \(\sum _{i}{\langle A_{i}R_{j}\rangle }\) is equal to the overall probability of an animal in node

*j*being a defender, \(\langle R_{j}\rangle \). The expected fight duration between a searcher

*i*and a handler

*j*is \(t_{a,ij}/2\). The fight time \(t_{a,ij}/2\) might differ from \(t_{a,ji}/2\); for example the length of the contest might depend upon the persistence of the attacking animal, which can be different in the two cases. At the end of the fight between animals in nodes

*i*and

*j*, the attacking searcher wins with probability \(\alpha _{ij}\) and takes the food. A handler consumes the food at rate \(t_{h,i}^{-1}\), thus taking an expected time of \(t_{h,i}\) to consume the item, and then returns back to the searching state.

*i*encounters a handler

*j*during the time it invests in searching for handlers. Hence, we could simply assume that the adjacency matrix is a weighted matrix \(A_M'\) whose element \(a'_{i,j}\) is equal to \(v_{h,i}a_{ij}\), \(\forall i,j\in \{1,\ldots , N\}\). However, to be consistent with the notation of previous models developed we separate the two terms. In addition, as \({v_f}_i\) and \(f_i\) always appear as a product, we have replaced \({v_f}_if_i\) by \(v_ff_i\).

The system (9)–(11) consists of \(N(2+d)\) equations, where *d* is the average degree of a node. This system assumes statistical independence between animals that are not fighting each other. This assumption can be relaxed by extending the system further to include the dynamics of pairs of animals at the searching, handling and fighting state (see for example Sharkey 2008 for relevant epidemic models on networks). However, such complex extensions are more computationally expensive and as the model (9)–(11) is very accurate (see Sect. 4), we focus on this to study the effect of the population structure in kleptoparasitic populations as described by Broom and Ruxton (1998).

#### 3.1.1 Steady states of the individual-based model and the handling ratio

In Luther and Broom (2004) the authors proved rapid convergence to a unique solution for the considerably simpler original models of Ruxton and Moody (1997) and Broom and Ruxton (1998). Despite the complexity of the system (9)–(11) we believe that this has a unique steady-state solution too. Although the complexity of the system has meant that we are unable to prove this, the output of stochastic simulations supports this statement.

(19) is a system of *N* simultaneous equations with *N* unknowns, the individual handling ratios, which are the probabilities of each animal handling a food item at any time point at the steady state.

A potential limitation of the solution of Eq. (19), which is based on the assumption (17), is that this is independent of the probability of an attacking searcher *i* winning the fight with handler *j*, \(\alpha _{ij}\), \(\forall i,j\in \{1,\ldots , N\}\). However, as discussed in Sect. 4, the value of the probability \(\alpha _{ij}\), \(\forall i,j\in \{1,\ldots , N\}\), does not affect the system in homogeneous populations placed on complete or regular networks and its effect on the average handling ratio in populations placed on heterogeneous networks, if any, is negligible.

#### 3.1.2 A special case: a homogeneous population on a regular undirected network

*d*is the degree of each of the nodes. The biologically relevant solution of the above equation is

*d*. Hence, decreasing the number of neighbouring animals in a homogeneous population placed on a regular network has the same effect as decreasing the rate at which foragers encounter handlers in a homogeneous well-mixed population, i.e. as decreasing \(\nu _{h}\).

### 3.2 A pairwise model

Although the individual-based model developed in the previous section can describe more realistic systems, it is characterised by a large number of variables and parameters. In this section, to reduce complexity, we consider an alternative model for the description of the dynamics of a structured kleptoparasitic population. This is a pairwise model developed using the pair approximation method (Matsuda et al. 1992; van Baalen and Rand 1998; Keeling 1999; Eames and Keeling 2002; House and Keeling 2011). The method assumes regularity of the network and identical parameters for every individual.

Assume that animals of a finite homogeneous population occupy the nodes of a regular network of degree *d*, i.e. every animal has exactly *d* neighbours. Let [*X*] be the number of animals in state *X*, [*XY*] the number of \(X-Y\) pairs between an animal in state *X* and an animal in state *Y*, and [*XYZ*] the number of triples of type \(X-Y-Z\). *X*, *Y* and *Z* represent any of the foraging states; the searching state *S*, the handling state *H*, the attacking state *A* and the resisting state *R*. Two connected animals in the fighting state might fight each other or they might fight with some other animal. We distinguish these different types of pairs of animals by denoting by \([AR_J]\) the number of pairs of animals which are fighting each other, and by [*AR*], [*AA*] and [*RR*] the number of pairs of animals which are involved in a fight, either by attacking or resisting, but which are not fighting each other (so that both, one or neither could be an *A*). \(X-X\) pairs are counted twice (once in each direction, and thus [*XX*] is always even), whereas \(X-Y\) pairs are counted once in each direction, and \([XY]=[YX]\).

*d*is the degree of each node in the network.

Hence, the system (24)–(38), together with the closure approximation (39), is a system of 15 differential equations, which is significantly lower than the number of equations in the individual-based model (9)–(11), which has two equations for every individual and for every link.

*A*there is one animal

*R*,

## 4 Numerical examples: effect of network structure on the handling ratio

We compare the solution of the models developed in the previous section with the results of stochastic simulations on a number of theoretical and empirical networks; a random network, a random regular network, a scale-free network and the Guppy network. The stochastic simulation is described in “Appendix B”. The random network we consider is an Erdős–Rényi type network (Erdős and Rényi 1959) generated as described in Lindquist et al. (2011). The random regular network was generated in the same way as the random network with the restriction that every vertex has the same number of connections. The scale-free network is a network that has a power-law (or scale-free) degree distribution. This was generated following the algorithm of preferential attachment (Barabási and Albert 1999; Albert and Barabási 2002). The random, random regular and scale-free networks developed are undirected and unweighted. Unless otherwise stated, the number of nodes of each of these networks that are considered in the current study is 100 and the average degree is 4. The Guppy network is a real network dataset that represents the contacts of a well-known freshwater fish species, the guppy (*Poecilia reticulata*), also known as the millionfish (Croft et al. 2011). This network is complex and highly structured, and exhibits small-world network properties (Watts and Strogatz 1998). It is a weighted network consisting of 99 nodes and 726 undirected links, where each node represents a guppy and each link a social interaction between two guppies. The average connectivity of each node is 14.7 and the clustering coefficient (Watts and Strogatz 1998) of the network is 0.77. To be consistent with the other example networks considered in this work, the Guppy network was converted to an unweighted network. In Croft et al. (2011) there is a detailed consideration of some local and global parameters of the Guppy network, such as the average geodesic (the smallest number of edges by which one node can be reached from the other), and other social characteristics. To illustrate the effect of the population structure in each of the examples, the solution of the models on the complete network (with \(N=100\)), where every animal is connected to everyone else, is also presented and compared with the other solutions. Clearly, in large homogeneous well-mixed populations all models become equivalent and exact.

### 4.1 Case 1: identical parameter values among individuals

*H*] in (60). Since (23) is a solution of the system (19) in the special case of homogeneous populations on regular undirected networks, the solutions of (19) and (23) are equivalent on well-mixed and random regular networks.

In Fig. 1, we illustrate the variation in the density of handlers over time on the different networks when \(v_{h,i}= v_h/d_{i}\). This example can describe the case where the search for handlers effectively requires a time consuming action such as a ‘visit’ to a neighbour’s site, where \(v_h\) is some basic rate at which animals can perform the search. It is observed that the density of handling animals (and similarly animals in other states) is not changed significantly by the change in the population structure, especially when animals are placed on a regular (or at least not a highly-structured) network. This is mainly because the number of connections is the same (or almost the same) for every animal, and therefore every animal has the same chance of engaging in an aggressive interaction.

A more pronounced structural effect is observed when the structure is highly heterogeneous, for example a structure which has the features of a scale-free network (see Fig. 1d). The existence of animals whose degree greatly exceeds the average reduces significantly the overall number of fights taking place over food, resulting in important changes of the population food intake rate. In particular, the increased number of connections of an animal increases its chance to be involved in a fight over food with another animal. This has a negative effect on its food consumption rate (see for example Fig. 2a). On the other hand, lowly connected animals can search, find and consume food with a very small risk of being engaged in an aggressive interaction. Since in networks with a power-law degree distribution the number of highly connected nodes is small, the average handling ratio, and thus the food intake rate of the population, increases compared to that of the respective well-mixed population of the same size, or the infinite homogeneous well-mixed population of the model of Broom and Ruxton (1998).

It should be noted that the rate at which foragers find undiscovered food, \(\nu _{f}f\), and the expected time for a handler to consume a food item, \(t_{h}\), seem to have the most important influence on the average intake rate of kleptoparasitic populations irrespective of their structure. However, the duration of a fight between two animals, \(t_{a}/2\), is the parameter whose increase yields higher differences between the average intake rate in structured populations and that in homogeneous well-mixed populations (see Fig. 3). The choice of the value of the probability of an attacking searcher *i* winning the fight with a defender *j*, \(\alpha _{ij}\), is generally insignificant (see for example Fig. 3b). In particular, in homogeneous well-mixed populations, after any contest there is one winner and one loser both connected to the same number of animals and thus the population distribution at the steady state is the same irrespective of \(\alpha _{ij}\), \( i, j \in \{1, 2,\ldots , N\}\). The solution of the pairwise model at the steady state found in “Appendix A”, which we have not conclusively proved but we believe to be the unique solution, is also independent of the value of \(\alpha \). Moreover, even in populations with extreme structural heterogeneity different values of \(\alpha \) make little or no difference; although the value of \(\alpha \) can have an effect on the handling ratio of each animal individually, the change of the average handling ratio, if any, is negligible. This is verified both by the results of stochastic simulations and the numerical solution of the individual-based model (9)–(11).

An example of the case where \(v_{h,i}=v_h\), \(\forall i\in \{1,\ldots , N\}\), is illustrated in Fig. 4. This example can describe a case in which foraging animals can search for animals handling a food item from their position by scanning the entire space, for example by rotating at rate 360 degrees per \(v_h^{-1}\) seconds, and thus in the same amount of time they can see an arbitrary number of animals. However, their location or geographical position might affect the likelihood that they observe others. The animals can interact only if they can see each other, and a higher degree of a node in this case means a higher number of visible animals.

### 4.2 Case 2: inter-individual variability in parameter values

In order to illustrate the effectiveness of the individual-based model in highly heterogeneous environments, we also consider two different examples where the individual model parameters are not the same for every individual. In both examples, the individual parameters \(v_ff_i\), \(t_{h,i}^{-1}\), \(t_{a,ij}^{-1}\), \(i,j \in \{1,\ldots , N\}\), are drawn independently from an exponential distribution with mean equal to 1. \(\alpha _{ij}\) is drawn from the uniform distribution on the unit interval [0, 1]. In the first example, illustrated in Fig. 5, \(v_{h,i}=1/d_i\). In the second example, illustrated in Fig. 6, \(v_{h,i}\) is also drawn from an exponential distribution with mean equal to 1. The parameter values for each individual are the same on the different structured populations, and we compare only the results on the complete network, the random regular network, the random network and the scale-free network that have the same population size.

## 5 Discussion

We have extended the original model of kleptoparasitism proposed by Broom and Ruxton (1998) onto networks to study the effect of the population structure and other heterogeneities on the behaviour of kleptoparasitic populations. To describe the dynamics of a population placed on a network, we first developed an individual-based model which enabled us to incorporate individual-level mechanisms in the original model, for example to introduce heterogeneity in the consumption rates of animals, their ability to find food and other animals handling food, their fighting abilities, as well as in the fighting times over food between pairs of animals. Despite the accurate prediction of this model in describing the output of stochastic simulations, it consists of a large number of equations, which is \(O({ dN})\). Assuming identical behavioural parameters and regular population structures, we also developed a pairwise model that has a significantly lower number of equations. For each model, we derived either approximate or exact solutions of the handling ratio, which is the rate at which an animal consumes food at the steady state. In highly heterogeneous structures the individual-based model performed much better than the pairwise model. However, in populations placed on regular or random networks, the solution of the pairwise model agreed very well with the results of stochastic simulations.

We have shown that the population structure may not greatly affect the dynamics of the population, mainly due to the fact that animals can discover and consume items of food independently of the population structure. The only effect of the structure is due to the change of the rate at which each animal is involved in aggressive interactions. Hence, in populations placed on a regular network, the decrease of the network degree has the same effect as that of the decrease of the rate at which foraging animals encounter handlers in a homogeneous well-mixed population. A more important influence of the population structure was observed on degree-heterogeneous structures, where the chance of encountering an animal and being engaged in a fight, either as a searcher or a handler, is not the same for every animal in the population, due to the different degree of connectivity. In particular, a significant effect was observed in scale-free networks, where the variance in the degree distribution is high. Clearly, in such networks, highly connected animals are more likely to fight over food with a neighbouring animal, affecting negatively their food consumption rate, whereas poorly connected animals handling a food item have a higher chance of consuming the food before being challenged by other animals. As the highly connected animals, and therefore the number of aggressive interactions among animals, on such structures are few, the average intake rate of the population increases compared to the respective rate in a well-mixed population.

This study is the first step in developing tools for the modelling of food-stealing interactions in more realistic settings, building upon models developed in idealised ‘well-mixed’ populations, to create models that can accommodate heterogeneous spatial and social relationships. In this first development of classic models of kleptoparasitism on networks animals do not have the option to optimise their foraging strategy and maximise their food intake rate in different ecological conditions. The influence of the population structure on the evolution of kleptoparasitic populations when animals can choose from a range of foraging strategies (see for example Broom et al. 2004) constitutes an interesting subject for subsequent studies. The general methodology could also apply more widely to related scenarios of animal interaction, such as that of predator interference (e.g. van der Meer and Ens 1997; Vahl 2006; van der Meer and Smallegange 2009).

## Notes

### Funding

M. Broom and J. Rychtář were supported by funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 690817. J. Rychtář was also supported by Simons Foundation Grant 245400.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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