Simple models for studying complex spatiotemporal patterns of animal behavior

Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Simple models for studying complex spatiotemporal patterns of animal behavior Yuri V. Tyutyunov a,b,n, Lyudmila I. Titova b a b

Institute of Arid Zones, Southern Scientific Centre of the Russian Academy of Sciences, Chekhov Street, 41, 344006 Rostov-on-Don, Russia Vorovich Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Stachki street, 200/1, 344090 Rostov-on-Don, Russia

art ic l e i nf o

Keywords: Simulation Animal movements Spatial behavior Schooling, swarming Attraction–repulsion Movement stimuli Self-organization Taxis-diffusion-reaction Individual based model

a b s t r a c t Minimal mathematical models able to explain complex patterns of animal behavior are essential parts of simulation systems describing large-scale spatiotemporal dynamics of trophic communities, particularly those with wide-ranging species, such as occur in pelagic environments. We present results obtained with three different modelling approaches: (i) an individual-based model of animal spatial behavior; (ii) a continuous taxis-diffusion-reaction system of partial-difference equations; (iii) a ‘hybrid’ approach combining the individual-based algorithm of organism movements with explicit description of decay and diffusion of the movement stimuli. Though the models are based on extremely simple rules, they all allow description of spatial movements of animals in a predator–prey system within a closed habitat, reproducing some typical patterns of the pursuit-evasion behavior observed in natural populations. In all three models, at each spatial position the animal movements are determined by local conditions only, so the pattern of collective behavior emerges due to self-organization. The movement velocities of animals are proportional to the density gradients of specific cues emitted by individuals of the antagonistic species (pheromones, exometabolites or mechanical waves of the media, e.g., sound). These cues play a role of taxis stimuli: prey attract predators, while predators repel prey. Depending on the nature and the properties of the movement stimulus we propose using either a simplified individual-based model, a continuous taxis pursuit-evasion system, or a little more detailed ‘hybrid’ approach that combines simulation of the individual movements with the continuous model describing diffusion and decay of the stimuli in an explicit way. These can be used to improve movement models for many species, including large marine predators. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction Mathematical models of spatiotemporal dynamics of trophic communities are effective and indispensable tools allowing researchers to analyze a variety of biological, behavioral, climatic and hydrological processes that determine the complex dynamics of marine ecosystems. Solving the problems of harvest control, of natural resource management, of monitoring species invasion, of protecting the endangered endemic animals requires deep understanding of the underlying processes that cause the emergence of the spatial patterns in the communities of interacting species. Climatic changes, in particular, the observed trends of n Corresponding author at: Institute of Arid Zones, Southern Scientific Centre of the Russian Academy of Sciences, Chekhov Street, 41, 344006 Rostov-on-Don, Russia. E-mail addresses: [email protected] (Y.V. Tyutyunov), [email protected] (L.I. Titova).

global warming (Huang et al., 2015), increase the significance of the theoretical modelling studies that help scientists to explain and forecast the response of migrating species to spatial variation in ocean productivity (Hobday et al., 2013, 2015; Dell et al., 2015; Young et al., 2015). Due to strong observed and projected trends in environmental factors, statistical models cannot give a reliable prognosis of the ecosystem development. Models that reflect mechanisms of ecological and behavioral processes will be much more useful in predicting the impact of climate change on harvested and/or vulnerable species (Plagányi et al., 2011). Stressing the importance of developing new approaches to modelling spatial movements of large marine predators, Boschetti and Vanderklift (2015) noted that a better understanding of the animal foraging behavior improves density estimates and provides more accurate information for decision-making. Along with the detailed simulation systems describing large marine ecosystems in great detail (Dragon et al., 2015; Lehodey et al., 2015; Senina et al., 2015), simple, so-called ‘minimal’ or

http://dx.doi.org/10.1016/j.dsr2.2016.08.010 0967-0645/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

Y.V. Tyutyunov, L.I. Titova / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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‘conceptual’ theoretical models capable of reproducing complex dynamics observed in the natural communities play a highly important role in applied ecological studies (Medvinsky et al., 2001; Petrovskii and Li, 2005; Tyutyunov et al., 2007). In seeking the minimal description of the ecological problem, researchers must test alternative theories that explain a given observation and are commensurate with the complexity of the problem. The simplest and the most general model will be potentially more robust in practical applications (Ginzburg and Colyvan, 2004). A clear and justified mathematical description should be used for every process that governs the dynamics of the ecosystem being studied: species reproduction, mortality, passive and active migrations, feeding, growth, etc. Ideally, this will ensure that assembly of the simulation system of any complexity consists of simple, sensible, and empirically proven model elements. If any element fails, we must not complicate the simulation model just to procure a desired result, but rather, we must modify or revise the theory. Eventually, if thorough quantitative validation of the more complicated modelling tools built for ecosystem management is impossible or difficult, qualitative validation of the basic models ensures their reliability. Therefore, building simple minimal models of ecological processes is an extremely useful task for both theory and practice. Such simple minimal models have been developed for a range of purposes over recent decades (Accolla et al., 2015; Couzin, 2002; Edelstein-Keshet et al., 1998; Okubo et al., 1977; Parrish et al., 2002; Pitcher et al., 1993; Tsyganov et al., 2004; Tyutyunov et al., 2007, 2008, 2009, 2013). In this paper we consider only one particular process underlying spatiotemporal dynamics of marine population systems, namely the movement behavior causing animal aggregation. This is important as the spatial behavior of species is linked to their trophic relationships (e.g., Arditi et al., 2001; Tyutyunov et al., 2008; Willis, 2007, 2014). We consider three approaches to modelling the pursuit-evasion behavior of prey and predators: (i) individual-based models (IBM) of animal spatial behavior; (ii) continuous taxis-diffusion-reaction systems using partialdifference equations (PDEs); and (iii) ‘hybrid’ approaches combining the individual-based algorithm approach of organism movements with the explicit description of decay and diffusion of the movement stimuli. All these models are based on the common hypothesis of inertia: a delayed response of animals (both predator and prey species) to changing distribution of antagonistic species. We will demonstrate that though these models are based on extremely simple rules, due to this hypothesis they all produce quite complex realistic heterogeneous regimes under minimal modeling assumptions.

2. Statement of the problem Spatial heterogeneity of population systems is highly dynamic. Formation of animal aggregations (i.e., swarming, schooling, etc.) occurs much faster than other key processes of population dynamics (Ritz et al., 2011), particularly as spatial aggregations occur at smaller spatial scales relative to the scale of environmental heterogeneity (Levin, 1992). Therefore, adequate models should demonstrate spatially inhomogeneous solutions induced only by the animals’ spatial behavior. Neither the patchiness of environmental factors, nor population kinetics, i.e., birth/death demographic processes should play a crucial role in small-scale pattern formation. This is particularly true for highly developed pelagic species capable of active movements in space. Building adequate models for such species requires understanding of the mechanisms of the spatial movements of an animal being alone or within an aggregation. New approaches should be developed because most of the classical spatial population dynamics results

were obtained during the second half of the 20th century on the basis of theoretical diffusion and taxis models and these models were primarily intended to describe microbiological and planktonic communities (Murray, 2003; Okubo and Levin, 2001). Direct transfer of these results to the modelling of larger species is fraught. Thus, practical needs for developing applied models of spatiotemporal dynamics of large ecosystems in changing environments raise new theoretical and methodological multidisciplinary problems at the intersection of ecology, ethology, mathematical modelling and cybernetics. One of the tasks is explaining the effects of self-organization observed in the populations of social animals such as schooling, swarming, flocking and synchronized behavioral responses to external signals and stimuli (e.g., Ritz et al., 2011). Studying collective movements in fish populations, Pitcher and Parrish (1993) classified behavioral patterns and described typical examples of spatial configurations (flash expansion, ball, avoidance, herd, cruise, split, joinder, vacuole, hourglass, etc.), caused, in particular, by mutual attraction–repulsion interactions of prey and predators (see also Lee et al. (2006), Parrish et al. (2002) and Ritz (1991)). Such pursuit-evasion movements occur at much faster temporal scales than birth–death processes, and even faster than consumption of prey by predators because the predation efficiency (their ability to capture prey after an attack) may be quite low for some species, especially if prey exhibit active escape behavior to reduce the risk of predation (Lee et al., 2006; Ritz, 1991). The estimation of the proportion of successful attacks made for various species indicate that only few attacks performed by the predators pursuing their prey are successful (MacKenzie and Kiørboe, 1995; Hammerschlag et al., 2006; Roth and Lima, 2007).

3. The models We first present in detail each of the three approaches that we suggest to use for modelling pursuit-evasion behavior in the predator-prey systems. 3.1. Individual-based model An IBM is the most natural and intuitive way of modelling population dynamics (e.g., Huston et al., 1988). The individualbased approach describing a multi-agent system consisting of a finite number of members is especially effective for studying the behavior of animals at low abundance. Such models are highly useful for theoretical studies of various spatial phenomena in population dynamics. In particular, IBMs demonstrate that spatial heterogeneity can be a key factor altering the dynamics described by the classical (not resolved spatially) population models, which are, either explicitly or implicitly, based on the mass-action hypothesis borrowed from the chemical kinetics models (Tyutyunov et al., 2008; Accolla et al., 2015). IBM simulations are quite often used to unravel the mechanisms of the emergence of non-linear effects due to the spatial behavior of interacting animals, including such phenomena as patchiness of population distribution, density-dependent predation (e.g., predator interference) and the Allee effect (Cosner et al., 1999; Viscido et al., 2004; Ramanantoanina et al., 2011; Romanczuk and SchimanskyGeier, 2012; Tyutyunov et al., 2008, 2013; Harris and Blackwell, 2013; Accolla et al., 2015; Reuter et al., 2016; MacPherson and Gras, 2016). Additionally, taking into account the energy balance of interacting organisms opens new perspectives to building the realistic simulation models of spatiotemporal dynamics of biological systems (Maury and Poggiale, 2013; Politikos et al., 2015). However, most of the individual-based models for collective behavior were subject to criticism (Katz et al., 2011; Romanczuk

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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and Schimansky-Geier, 2012) for the explicit inclusion of some sort of velocity-alignment mechanisms to coordinate the movements of an individual with the velocities of other individuals (Vicsek et al., 1995; Couzin et al., 2002; Sugawara et al., 2007). Though such models are capable of reproducing realistic-looking spatial configurations of animal groups, experimental studies of collective behavior in fish (Katz et al., 2011; Herbert-Reada et al., 2011) do not confirm the existence of an explicit velocity-alignment pairwise interaction of moving individual with its neighbors. In particular, Katz et al. (2011) reported that they found no evidence for explicit matching of body orientation in the experiments with swimming golden shiners (Notemigonus crysoleucas). In more realistic models for animal spatial behavior, coordination and alignment of moving organisms emerge as the result of the selforganization of individuals responding independently to the local conditions (or to the nearest neighbors). Such a response manifests itself as attracting or repulsing forces affecting the motion of the self-propelled individuals (Sugawara et al., 2007; Katz et al., 2011; Herbert-Reada et al., 2011; Romanczuk and SchimanskyGeier, 2012; Reuter et al., 2016). The hypothesis of deterministic behavior is often used in order to take account the animal ability to sense the environment. Thus each individual is supposed to be constantly aware of the exact position, the direction and the velocity of all the other conspecific individuals as well as of the current distribution and the movement characteristics of the food objects and the natural enemies. For the sake of plausibility, the awareness of such individual is considered limited by a certain interaction radius (Vicsek et al., 1995; Couzin et al., 2002) or sensory-range distance (Romanczuk and Schimansky-Geier, 2012; Accolla et al., 2015). Additionally, various sophisticated hypotheses can be included into the IBM to improve its biological realism, such as a blind perception zone behind each organism, cognitive memory effects, interactions with physical environment, and swimming biomechanics (Huth and Wissel, 1994; Viscido et al., 2004; Gautrais et al., 2008; Lett and Mirabet, 2008). Considering the minimal approaches to the modelling of the animal movements, we now describe a simple individual-based predator–prey model in which the spatial behavior of each animal is determined by the heterogeneity of distribution of movement stimulus, i.e., either some material substance or a signal emitted by the individuals of the antagonistic species. We first outline the basic modelling assumptions and then present the results of

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simulations. More details can be found in Tyutyunov et al. (2008, 2013). In the model, space is continuous and time is discrete ðt ¼ 0; 1; 2; …Þ. Populations of prey and predators, consisting of n and p individuals respectively, inhabit a closed rectangular area
 Ω ¼ ½0; Lx   0; Ly with reflective boundaries ∂Ω. The states of each prey i ði ¼ 1; 2; …; nÞ and predator j ðj ¼ 1; 2; …; pÞ at time t P are given by their coordinates xN i;t ; x j;t A Ω and their velocities by N P vi;t , vj;t (the upper indexes N and P stand for prey and predator respectively). It is assumed that every individual emits a specific cue, either some biological or chemical substance (e.g., exometabolite, scent, pheromone) or mechanical signal (e.g., sound, water vibrations), that spreads and decays much faster than the animal moves from one position to another. The distribution of individual cues can be considered as normal, centered at the individual location xi;t ; with root-mean-square deviations σ N for the prey and σ P for the predator respectively. The total cue of the whole population (SN ðx; t Þ for the prey and SP ðx; t Þ for the predator) in every position of the modelling domain is the sum of the cues emitted by all individuals. The movements of the j-th predator are described by the P equation xPj;t þ 1 ¼ xPj;t þ ξj;t þvPj;t ; where xPj;t is the vector specifying the position of theindividual at the discrete moments t ¼ 0; 1; 2; ::;  vector vPj;t ¼ κ P ∇SN xPj;t þ ηP vPj;t  1 describes the directed motion of the predator upward following the gradient of the prey's cue SN at P the point xPj;t ; and ξj;t is the random vector with the exponentially    P   distributed magnitude ξj;t   Exp λP representing the undirected movements of predators. For the i-th prey the formulae are analogous differing only by N N N the swapping the indices N and P: xN i;t þ 1 ¼ x i;t þ ξi;t þv i;t ;   N N vN i;t ¼ κ N ∇SP x i;t þ ηN vi;t  1 : The taxis coefficients κ N and κ P specify the intensity of the directed movements; coefficients ηN ; ηP A ½0; 1 define their inertia. The direction of the motion is determined by the sign of kappa: κ N r 0 when the predator's cue is a repellent for a prey; κ P Z 0 if the prey's cue is an attractant for a predator. In this way, by changing the parameter values, we can effectively model various strategies of the predator and prey motions, obtaining a variety of spatial patterns observed in the natural predator–prey systems (Tyutyunov et al., 2008). As an example, Fig. 1 (right panel) represents a snapshot of the simulation performed with the following parameter values: Lx ¼ 10, Ly ¼ 6,

Fig. 1. Spatial patterns, generated by the pursuit-evasion behavior of trophically-linked species in the natural system (left panel) and in the individual-based model (right panel). Smaller blue dots and bigger red points represent prey and predator individuals respectively; gradations of grey color show the distribution of the cues emitted by the predators, acting as repellent for prey individuals (darker patches correspond to higher concentration of the repellent). The concentration field of the cues emitted by prey (attractive to predators, not shown) is correlated with the prey distribution. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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N ¼ 5000, σ N ¼ 0:5, λN ¼ 0:1, κ N ¼ 0:03, ηN ¼ 0:05, P ¼ 25, σ P ¼ 0:35, λP ¼ 0:15, κ P ¼ 0:003, ηP ¼ 0:05. The results are qualitatively similar to the behavioral patterns observed in the aquatic ecosystems (see Fig. 1), including ‘flash expansion’, ‘avoidance’, ‘herd’, ‘split’, ‘joinder’, ‘vacuole’, and ‘hourglass’ patterns described in earlier studies (Parrish et al., 2002; Pitcher and Parrish, 1993). The coordination and alignment effects observed in natural systems emerge in the model solely from the movement of each individual animal in response to its local conditions. Although an IBM is the most natural and perhaps the only effective way of modeling the spatial dynamics for low-abundance populations, this approach cannot be used for populations with high abundance, because tracking individual animal movements in such cases incurs a high computational burden. Fortunately, the continuous taxis-diffusion-reaction models based on partial differential equations (PDEs) provide perfectly adequate descriptions of the spatiotemporal dynamics of high abundance populations. Essentially, the individual-based (Lagrangian models) models underlie most of the continuous (Eulerian) models that describe the population spatial dynamics. There is a very large body of work on various methods that allow the continuous models to be derived by a limit transition from the individual-based models with the number of individuals tending to infinity; the latter allows assuming the hypothesis of continuity of the population distribution. Inasmuch as this particular technical problem lies beyond the scope of the paper, interested readers can refer to a number of existing papers (Patlak, 1953; Keller and Segel, 1971; Grünbaum, 1994; Czárán, 1998; Flierl et al., 1999; Okubo and Levin, 2001; Hillen and Hadeler, 2005; Hillen and Painter, 2009). Many commonly used continuous models are based on the classical Patlak–Keller–Segel equation of the population density flow, which arises from the assumption that the individuals are moving permanently, changing their direction with frequency that depends on the stimulus concentration (Patlak, 1953; Keller and Segel, 1971; Okubo and Levin, 2001). Our own experience of deriving the density flow equations from an IBM shows that the Patlak–Keller–Segel model can also be obtained assuming that individuals stop their motion and then spontaneously leave their positions with frequency (probability) that depends on the local stimulus (Tyutyunov et al., 2010). This result proves the universality of the Patlak–Keller–Segel model and its applicability for the description of animal's directional movements under different plausible assumptions. In the next section we present an example of a simple continuous pursuit-evasion model based on the Patlak–Keller–Segel equation of the population density flow. 3.2. Continuous model In considering the application of the continuous models to the description of the pursuit-evasion process common to pelagic species, we shall first note that although conventional continuous taxis-diffusion-reaction models perfectly describe spatial dynamics of planktonic and microbial communities (Berezovskaya et al., 1999; Berezovskaya and Karev, 1999; Tsyganov and Biktashev, 2004; Tsyganov et al., 2003, 2004, 2007), these models cannot explain the pursuit-evasion phenomena of larger marine species, due to the stability of the homogeneous equilibrium in such models in the absence of nonlinear terms of local population kinetics (e.g., the birth/death processes) (Tyutyunov et al., 2007). In order to resolve this problem, we use an alternative model capable of reproducing the spatial clustering and heterogeneous wave regimes induced solely by attraction–repulsion movements of prey and predators within a closed spatial domain Ω at relatively fast time-scales. In fact, the model dynamics represent a kind of reciprocal movement of prey and predators that changes their local density while total population sizes remain constant.

This can be achieved by hypothesizing very similar mechanisms as those in the IBM presented in Section 3.1. The model is given by the following system of partial differential equations: 8 ∂N=∂t ¼ divðN∇SN Þ þDN ΔN; > > > > < ∂P=∂t ¼  divðP∇SP Þ þ DP ΔP; ð1Þ ∂S =∂t ¼ κ N P  θN SN þDSN ΔSN ; > > > N > : ∂SP =∂t ¼ κ P N  θP SP þ DS ΔSP : P Here N and P are the densities of prey and predator populations respectively; t is the time; ∇ denotes the gradient; Δ is the Laplacian; SN and SP are the stimuli of the prey and the predators movements, i.e., the potentials of the taxis velocities of prey and predators respectively: vN ¼  ∇SN , vP ¼ ∇SP . According to (1) we can interpret SN and SP as substances like exometabolites (pheromones, scents, etc.) emitted by prey and predators with the rates κ N and κ P respectively. Thus, directed predator movements are stimulated by the attractive substance SP being continuously emitted by prey, and vice versa, prey movements are stimulated by the repellent substance SN emitted by predators. Parameters DN , DP , DSN and DSP are the diffusion coefficients of the model variables; θN and θP are the decay rates of the prey and predator taxis stimuli respectively. Note that applying the gradient operator to the balance equations of the stimuli SN and SP we obtain the equivalent alternative formulation of the model (1) that is different in kinematic terms of taxis acceleration and velocity: 8 ∂N=∂t ¼  divðNvN Þ þDN ΔN; > > > > < ∂P=∂t ¼  divðPvP Þ þ DP ΔP; ð2Þ ∂vN =∂t ¼  κ N ∇P  θN vN þ DSN ΔvN ; > > > > : ∂vP =∂t ¼ κ P ∇N  θP vP þ DS ΔvP : P According to the equations describing the dynamics of the species velocities in the system (2) accelerations of the population densities (i.e., derivatives of velocities vN and vP with respect to time) have the terms proportional to the gradients of the antagonistic population densities ∇P and ∇N . For that reason and as the higher values of κ N and κ P correspond to the higher intensity of animals’ directed movements, we call parameters κ N and κ P the taxis coefficients. Note also that in the kinematic formulation (2) of the system (1) the decay terms θN vN and θP vP can be interpreted as the friction forces, while diffusions DSN and DSP turn into the viscosity coefficients. The system of PDEs (1) [as well as the equivalent system (2)] with zero-flux boundary conditions ∂N=∂nj∂Ω ¼ ∂P=∂nj∂Ω ¼ ∂SN =∂nj∂Ω ∂SP =∂nj∂Ω ¼ 0;

ð3Þ

where n is the external normal to the boundary ∂Ω, entirely describes the dynamics of the model within a closed spatial domain Ω. With the boundary conditions specified in (3), the system (1) is conservative with respect to the spatially averaged population densities, i.e., Z Z 1 1 ð4Þ N0 ¼   Ndx ¼ const; P 0 ¼   Pdx ¼ const:

Ω

Ω

Ω

Ω

Rescaling the population densities N ¼ N=N 0 , P ¼ P=P 0 , and considering κ P N 0 and κ N P 0 as the new taxis coefficients, the proposed model (1, 3) can be nondimensionalized so that the average population densities (4) became N 0 ¼ P 0  1. The nondimensionalized model (1, 3) has a spatially homogeneous stationary regime N ¼ 1; P ¼ 1; SN ¼ κ N =θN ; SP ¼ κ P =θP :

ð5Þ

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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However, besides the homogeneous solution (5), various patchy solutions emerge in the model if the parameters satisfy particular condition. This condition can be determined by standard techniques of the linear analysis allowing one to predict the response of a model to small heterogeneous perturbations of its homogeneous equilibrium (Murray, 2003). Linear analysis of model (1, 3) (see details in Tyutyunov et al. (2007)) shows that the homogeneous stationary regime (5) becomes unstable when the product of the prey and predator taxis coefficients exceeds the critical bifurcation value:

κN κP 4

  D þ k; DN ; DP ; DSN ; DSP ; θ N ; θ P

: k ðDN þ DP þ DSN þ DSP Þ þ 2ðθN þ θ P ÞðDN þ DP þDSN þ DSP Þk þðθ N þ θ P Þ k 6

2

4

2

ð6Þ   Here the numerator D þ k; DN ; DP ; DSN ; DSP ; θN ; θP is a positive polynomial of the wavenumber k, the diffusion and the decay coefficients, which is not given here because of its large expression. The instability condition (6) suggests that with any admissible values of the model parameters there exists a critical value of the product κ N κ P that corresponds to the oscillating instability of the homogeneous stationary regime (5). The following emergence of the spatially heterogeneous wave dynamics can be interpreted as pursuit-evasion in predator–prey system. Numerical experiments showed that the further increase of the taxis coefficients complicates the dynamics causing the cascade of bifurcations including the doubling of the period of oscillations and emergence of chaos. According to inequality (6) the spatial self-organization in the model (1, 3) requires the ability of both prey and predator to perform active directed movements. Increasing the taxis coefficient of one species can compensate for lowering the taxis activity of the antagonistic species. The result is independent of the geometry of the domain Ω: formula (6) is valid for onedimensional segment, two-dimensional rectangle and circular domain (Tyutyunov et al., 2007). Fig. 2 shows an example of the wave dynamics in the model (1, 3) that can be interpreted as a pursuit-evasion process in the rectangular domain Ω ¼ Lx  Ly . This dynamic regime stabilized in the numerical experiments with the following parameter values: Lx ¼ 4; Ly ¼ 3; N 0 ¼ 30; P 0 ¼ 10; κ N ¼ 0:003; κ P ¼ 0:005; DN ¼ 0:1; DP ¼ 0:15; DSN ¼ DSP ¼ 0:07; θN ¼ θP ¼ 10  5 :

3.3. Hybrid model Besides deterministic stimuli, various random factors may influence the direction and velocity of animal displacements. They can be more easily taken into account within the framework of individual-based (Lagrangian) models. However, the IBM presented in Section 3.1, dependent on the hypothesis of fast dispersal and decay of the movement stimuli, has a substantial restriction for potential application even to species systems with low abundance. While this hypothesis greatly simplifies the algorithm used to calculate the stimulus field and its gradients, it does not allow application to the situations where persistent chemical marks are left by the animals, which can be used by their antagonistic species as navigational signals (e.g., by predators tracking their prey). For such slowly decaying chemical cues, a more general ‘hybrid’ approach combining the algorithm of individual movements with explicit (i.e., continuous) modelling of the stimuli decay and

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diffusion is appropriate. This ‘hybrid’ model is formulated as follows: 8 p   X > > > δ x  xPj  θN SN þDSN ΔSN ; > ∂SN =∂t ¼ γ P > > j¼1 > > > > n > X   < ∂SP =∂t ¼ γ N δ x  xNi  θP SP þ DSP ΔSP ; ð7Þ > i¼1 > > > > N N N > > > x€ i ¼ κ N ∇SP  χ N vi þ ξi ; ði ¼ 1; 2; …; nÞ; > > > : x€ P ¼ κ P ∇SN  χ vP þ ξP ; ð j ¼ 1; 2; …; pÞ: j

P j

j

Here SN and SP are the movement stimuli emitted by the individuals of predator and prey population respectively; γ P and γ N are the emission rates; xNi ; xPj A Ω are the current positions of the prey and predator individuals within a closed domain Ω; δðxÞ is the Dirac delta-function, such that ( ZZ 1; x ¼ 0; δ ðx Þ ¼ and δðxÞdσ ¼ 1: ð8Þ 0; x a 0; Ω The other parameters are as defined in earlier sections. According to Eqs. (7)–(8), the sources of the cue-stimuli in the model are the constantly moving prey and predator individuals. Thus, here we simultaneously solve a reaction-diffusion system of the PDEs describing decay and dispersal of two movement stimuli, and kinematic equations of organism motions. These kinematic equations result from Newton's second law and describe the acceleration of each prey and predator individual (i.e., the second € that is caused by the following individual forces: the derivative x) self-propelling force κ ∇S, proportional to the gradients of pursuitevasion stimuli; the friction force χ v, assumed to be proportional to _ and the individual velocities of each moving organism (recall v ¼ x); ξ with the exponentially distributed magnitude random force    ξj  Exp λ , that accounts for the elements of uncertainty and randomness that may also influence animal motion. In order to reproduce the pursuit-evasion dynamics, the values of the attraction/repulsion coefficients should have correct signs: κ P 4 0, κ N o 0. Additionally, if necessary, the forces affecting the attraction–repulsion reaction in response to conspecific individuals in the prey and/ or predator population can be easily added into Eq. (7). Finally, we integrate numerically the second-order differential kinematic equations of the system (7) and simultaneously solve the PDEs using the standard method of lines (Schiesser, 1991) discretizing the spatial domain Ω with a regular grid of m  k nodes. Fig. 3 shows example output of the predator–prey ‘hybrid’ model that describes spatial movements of animals and distributions of stimuli emitted by each individual within a rectangular domain Ω ¼ Lx  Ly . This result was obtained with the following values of the model parameters: Lx ¼ 40; Ly ¼ 30; n ¼ 50; p ¼ 5; γ N ¼ 100:5; γ P ¼ 100:5; κ N ¼  30:0; κ P ¼ 33:0; λN ¼ 0:01; λP ¼ 0:015; χ N ¼ 2:5; χ P ¼ 2:5; DSN ¼ 0:5; DSP ¼ 0:5; θN ¼ θP ¼ 0:5; m ¼ k ¼ 50: Note that due to the relative persistence of the movement stimuli in this numerical experiment, the responses of animals to the displacements of their antagonists in the model are delayed. The examples A and B in Fig. 3, suggest that as the prey and predator individuals are well separated in space, their displacements at the micro-scale might seem uncorrelated: quite expectedly, prey leave places heavily marked by predators (gradations of red in Fig. 3B), while predators spend more time searching within the patches with higher level of stimulus emitted by prey individuals (gradations of grey in Fig. 3A). However, examining the individual trajectories in Fig. 3C reveals good agreement between animal trails reflecting quite complex chase and escape behavior. Simple IBMs cannot reproduce such behavior patterns.

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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Prey density, N

Predator density, P

t = 28

t = 36

t = 44

t = 52

Fig. 2. Wave dynamics demonstrated by continuous pursuit-evasion model (1, 3).

4. Discussion We have reviewed three simple (minimal) theoretical models that could effectively be used for representing the pursuit-evasion pattern observed in natural predator–prey systems. Each of the

models has its advantages and restrictions, but depending on the particular species being studied, these models can be used as the elementary bricks of a complex simulation system designed to explore applied and theoretical ecosystem management problems. In particular, in the simulation systems developed for marine

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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Fig. 3. Distribution of 50 prey (blue points) and 5 predators (yellow points) overlapped with the cues emitted by prey (A) and predators (B) in the ‘hybrid’ model. Plot (C) displays the movement trajectories of prey (black lines) and predators (red lines) during the simulation period t A ½0; 10. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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pelagic environments these models can be used to describe seasonal migrations, schooling and collective foraging movements common for fish and invertebrate species. For modelling the spatial behavior of animals within small populations, an IBM based on the hypothesis about the fast-decay of stimuli of pursuit-evasion movements in a predator–prey system could be developed. Though the potential of the IBM approach for analytical studies is quite limited, such models give nearly unlimited freedom in planning and performing numerical simulations. In particular, we have demonstrated one IBM of a predator–prey system that reproduces the type of pursuit-evasion behavioral patterns observed in nature (Fig. 1). This style of IBM has also been used for studying the influence of animal spatial movements on the functional response of predator populations (Tyutyunov et al., 2008, 2013). A similar theoretical study with an IBM was performed by Accolla et al. (2015) who tested the emergence of functional response in relation to the fish schooling phenomenon. It would be interesting to generalize the results of these two studies, unifying the approaches to model animal movements and simplifying the rules of attraction and repulsion in the predator–prey interactions. Spatial dynamics of large populations with high local densities can be described with taxis-diffusion-reaction PDE systems. This combination of powerful numerical simulations and pure analytical techniques is widely used, and is useful for studying dispersal when the heterogeneity of the environment and/or local kinetics of the populations are important. As we have shown, the proposed method of taxis modelling [see Eqs. (1)–(3)] produces complex spatiotemporal dynamics that can be interpreted as pursuitevasion waves even in the absence of local kinetics, i.e., without description of the birth/death processes in both prey and predator populations. Therefore, this technique of modeling of animal movements significantly simplifies the model and potentially reduces the number of parameters that must be estimated to deal with a specific applied problem. Continuous models are unsuitable for small populations or subpopulations that are sparsely distributed in space. In turn, the use of IBMs as developed here for predator–prey dynamics is inappropriate in the case of persistent signals where the fast-decay hypothesis does not hold. Thus, depending on the nature of animal movement stimuli, either a simplified IBM, or a more detailed ‘hybrid’ approach that combines simulation of individual movements with a continuous model explicitly describing diffusion and decay of the stimuli can be used. Comparison of the outputs of the IBM model with the outputs of the ‘hybrid’ one reveals interesting patterns (Fig. 4). A simulation obtained with the ‘hybrid’ model under an assumption of fast dispersal and decay of the stimuli emitted by both prey and predator individuals (recall this assumption corresponds to the hypothesis stated in the simplified IBM) is shown in Fig. 4B. The parameters of this ‘hybrid’ model version were: Lx ¼ 40; Ly ¼ 30; n ¼ 1000; p ¼ 25; γ N ¼ 100:5; γ P ¼ 100:5; κ N ¼ 30:0; κ P ¼ 3:0; λN ¼ 0:02; λP ¼ 0:013; χ N ¼ 2:5; χ P ¼ 2:5; DSN ¼ 10:5; DS P ¼ θN ¼ θP ¼ 10:5; m ¼ k ¼ 50: As might be expected by using 10:5; high values for the diffusion (DSN , DSP ) and decay ( θN , θP ) of the stimuli, the qualitative spatiotemporal dynamics of the ‘hybrid’ model are similar to the output of the simplified IBM model (Fig. 4A), in which the center of the stimulus patch of each individual always coincides with the individual's position. Although the IBM, as specified in Section 3.1 runs more than 100 times faster, the ‘hybrid’ model is more flexible, with a wider spectrum of dynamic regimes and possibilities for theoretical and practical applications. With regard to marine ecosystems the ‘hybrid’ model with appropriate initial and boundary conditions can effectively be applied to analyze migration strategies of pelagic species, which use some persistent

Please cite this article as: Tyutyunov, Y.V., Titova, L.I., Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II (2016), http://dx.doi.org/10.1016/j.dsr2.2016.08.010i

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8

Fig. 4. Example visualization of the individual-based (A) and ‘hybrid’ (B) models. Bigger red and smaller blue points show predators and prey respectively.

chemical or biological cues for their orientation or for pursuit of prey, or emit pheromones to mark their habitats (Hay, 2009). Examples of chemically-driven spatial behavior can be found in both pelagic and benthic communities of fish and invertebrate species (e.g., Ritz et al., 2011). The model (7) can be particularly useful in the studies of trophodynamics and patchiness in two-dimensional benthic communities (Erlandsson and Kostylev, 1995; Tyutyunov et al., 2009, 2010; Azovsky et al., 2012). All three models can be also generalized for three-dimensional space, which is necessary to include vertical diving behavior of large pelagic predators (Young et al., 2015; Williams et al., 2015). Despite of the simplicity of the basic assumptions, all three models presented here provide realistic description of pursuitevasion behavior, while ignoring demographic processes such as births the deaths in the populations of interacting species. These example models are built based on a common hypothesis: directional movements of prey and predators in the system are determined by the gradients of cues (e.g. exometabolit, pheromone, scent, sound) emitted by the individuals of the predator and prey species. These cues act as movement stimuli such that prey attract the predator, while the predator repels prey. In all three models, the patterns of collective behavior emerge due to the ability of animals to move directionally. Inasmuch as the animal movements are determined only by local conditions, the observed collective behavior emerges essentially due to self-organization. The obtained results and the basic simplicity of the proposed models allow selecting them as the good candidates to serve as elementary units within detailed simulation systems. We suggest that the complex ecosystem models being developed for marine systems, including pelagic fish species (Dragon et al., 2015; Lehodey et al., 2015; Senina et al., 2015) and sea turtles (Abecassis et al., 2013), could consider including these model building blocks. Simplified description of animal movements should potentially promote robustness of the models with respect to the effects of spatial behavior of predators and prey at the population and community levels (Maury and Poggiale, 2013).

Acknowledgments We thank two anonymous reviewers and Dr. Alistair Hobday for highly constructive comments that helped us to improve the paper. We also thank Dr. Inna Senina for detailed discussion of the

theoretical approach. The work was funded by the research project 01-16-04 of the Institute of Arid Zones, SSC RAS “Development of GIS-based methods of modelling marine and terrestrial ecosystems”, and by the State Assignment for Research, project 2014/174 of the Southern Federal University “Development of mathematical tools in support of decision taking and ensuring the stable development of the region”.

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