Modeling and Simulations of the Migration of Pelagic Fish

Modeling and Simulations of the Migration of Pelagic Fish Alethea Barbaroy{ Baldvin Einarssonyz{ Björn Birniryz{ Sven Sigurðssonz Héðinn Valdimarsson§ Ólafur Karvel Pálsson§ Sveinn Sveinbjörnsson§ Þorsteinn Sigurðsson§ June 16, 2008 Abstract We model the spawning migration of the Icelandic capelin stock using an interacting particle model with added environmental fields. Without an artificial forcing terms or a homing instinct, we qualitatively reproduce several observed spawning migrations using available temperature data and approximated currents. The simulations include orders of magnitude more particles than many similar models, affecting the global behaviour of the system. Without environmental fields, we discuss how various parameters scale with respect to one another. In particular, we present scaling behaviour between the size of the time step, radii of the sensory zones and the number of particles in the system. We then discuss how environmental data are incorporated into the model. y Department of Mathematics, University of California, Santa Barbara, USA. z Department of Mathematics, University of Iceland, Reykjavík, Iceland. z Department of Computer Science, University of Iceland, Reykjavík, Iceland. § Marine Research Institute of Iceland. { Center for Complex and Nonlinear Science, University of California, Santa Barbara, USA. Keywords: Collective motion, Fish migration, Icelandic capelin (Mallotus villosus), Interacting particle model. Corresponding author: Baldvin Einarssson, [email protected] 1 Introduction take can be observed, as can the approximate tim- ing, the surrounding environment and other char- acteristic factors. There are year-to-year variations Many species of marine animals travel great dis- which could be influenced by a number of variables tances between feeding and spawning grounds, in- including temperature and other environmental fac- cluding pelagic species of fish like the capelin (Mal- tors such as food density, salinity, bottom topography lotus villosus) and herring (Clupea harengus) (Vil- and oceanic currents. Internal variables, such as the hjálmsson, 1994, 2002; Vilhjálmsson and Carscadden, physiological state, stage of maturity and egg content 2002; Misund et al., 1998). Fish migrations are yet could also play a role. not completely understood and probably never will be in full detail. However, the general route the fish The capelin is an example of a species of pelagic 1 fish which covers several hundred kilometers in the In this paper, we use an interacting particle model course of its migration. The stock upon which will based on the work of Hubbard, Babak, Sigurðsson, be focused in this paper is the stock inhabiting the Magnússon, and Einarsson to simulate the spawn- Iceland Sea, (Mallotus villosus), hereafter referred to ing migration of the Icelandic capelin and predict as the Icelandic capelin. A brief account will be given their migration route given data about temperature of pertinent details of this species here, but the in- and current (Hubbard et al., 2004; Magnússon et al., terested reader is referred to (Vilhjálmsson, 1994), 2004b). The most notable difference is the absence of which provides extensive details of the stock and the any artificial forcing terms attracting the particles to capelin’s life cycle. a specific spawning location. The Icelandic capelin spends the first 1-2 years of The ouline of the paper is as follows: in Section 2 its life in the waters to the north of Iceland, along the mathematical model is presented with no environ- the edge of the continental shelf. When it approaches mental factors added. Section 3 analyzes that model maturity, usually either during the spring of its sec- and presents various scaling behaviour between pa- ond or third year, it undertakes an extensive migra- rameters. The environmental factors are incorporated tion, herein referred to as the feeding migration, to into the model in Section 4. The full model is then the plankton-rich waters of the Iceland Sea as far applied to the Icelandic capelin stock and simulations north as to the island of Jan Mayen. There, zooplank- from three different years are presented and then dis- ton are plentiful as they feed on the vernal phyto- cussed in Section 5. Details of the implementation are plankton bloom in the region. The maturing capelin also given in that section. eat these zooplankton and grow extensively. In Octo- ber and November, fully-grown capelin return to the waters northwest and north of Iceland. In January, 2 Mathematical Model this portion of the stock undertakes a spawning migration around Iceland to the southern and western The model presented here is an interacting parti- coasts. The spawning migration generally approaches cle model. In the classical interacting particle mod- along the continental shelf edge to the northeast and els available in the literature such as (Vicsek et al., east of Iceland. However, in some years a portion 1995), the models are discrete in time and spatially of the capelin stock has been observed to migrate represent individuals rather than keeping track of the against the coastal current and take a westerly di- density of a population. Particles look to their neigh- rection to the spawning grounds. The capelin spawn bors to determine their directional heading at each in February-March and then die, leaving the eggs to time step, averaging the neighbors’ directional head- hatch and the larvae to drift with the coastal currents ings to determine their own. This allows the parti- to the continental shelf waters north of Iceland and cles to move together as a group. The model intro- begin the cycle again. duced in (Vicsek et al., 1995; Czirók et al., 1997, 1999; A separate capelin stock resides in the Barents Sea Czirók and Vicsek, 2000) is hereafter called the CV (Mallotus villosus Müller), north of the coast of Nor- model. This type of model originated in physics and way and Russia. This stock has been widely studied was adapted by the biological community to model and exhibits similar migration patterns between feed- group dynamics of social animals. It has been applied ing and spawning grounds (Gjøsæter, 1998; Gjøsæter to herds of mammals, swarms of locusts, and schools et al., 1998). Much effort has been put into modeling of fish (Vicsek et al., 1995; Czirók et al., 1997; Buhl that stock, see (Sigurðsson et al., 2002; Magnússon et al., 2006; Couzin et al., 2002). A European project et al., 2004a; Fiksen et al., 1995; Huse et al., 1999; called STARFLAG1 uses similar models on swarms Huse, 2001; Huse et al., 2004). However, the Barents of starling and explores various interdisciplinary con- Sea migration route differs from the Icelandic one, nections. since they migrate within an ocean containing no is- lands or other obstacles. 1http://angel.elte.hu/starling 2 Figure 1: Feeding (top) and spawning (bottom) migration routes of the Central North Atlantic stock of capelin. From (Vilhjálmsson, 1994). Partridge (Partridge, 1982) pointed out that fish see for example (Chuang et al., 2007; Couzin et al., tend to adjust their speeds according to their neigh- 2002; D’Orsogna et al., 2006; Hemelrijk and Kunz, bors. This feature was first introduced into the CV 2005; Kunz and Hemelrijk, 2003; Kunz et al., 2006; model in (Hubbard et al., 2004) and is represented in Topaz et al., 2006). equation (1). With this model, Birnir (Birnir, 2007), We follow (Aoki, 1982; Huth and Wissel, 1992) analyzed the continuous time limit of the model and and employ three sensory zones around each particle found several solutions and symmetries. determining its reaction to the particles around it. Biologists have found that fish interact differently Unlike similar models, including (Aoki, 1982; Huth with each neighbor, dependent on the distance to the and Wissel, 1992; Couzin et al., 2002; Kunz and neighbor. Fish use both their vision and their lateral Hemelrijk, 2003; Hemelrijk and Kunz, 2005; Kunz lines, sense organs running down the sides of many et al., 2006; Hemelrijk and Hildenbrandt, 2008; Vis- species of fish, to align themselves with neighbors cido et al., 2004, 2005), we do not employ a blind and organize themselves into schools (Partridge and region behind a particle. It is ambiguous whether or Pitcher, 1980; Partridge, 1982). Fish tend to aggre- not this blind region is biologically relevant in the gate and avoid collisions when traveling and distance case of fish since the lateral line should allow a fish to nearest neighbors seems to play a role in the orga- to sense the region behind it as it swims. In addition, nization of a school (Grünbaum et al., 2005). the presense of such a region doesn’t seem to affect the outcome of the simulations (Huth and Wissel, To simulate the internal dynamics of a group of 1994). interacting animals, many interacting particle models therefore incorporate different sensory regions into In (Barbaro et al., 2007), solutions to the model their simulations. The distance between two parti- presented in (Birnir, 2007) were verified numerically. cles determines how they react to each other and With sensory zones added, the model exhibits rich also, in some models, the strength of this interac- behaviour and swarming solutions induced by noise tion. These models include both individual and con- were found. The paper explored the interdependence tinuum (density) models, and the shape and size of of noise and the size and weights of the sensory zones the zones tends to differ depending on the model, in eliciting certain behaviours from the model. 3 The particles update their speeds as follows: 1 X vk(t + ¢t) = vj(t); (1) jOkj A j2O k k O and their positions: k R k qk(t + ¢t) = qk(t) :::µ ¶ r r cos(Ák(t + ¢t)) (2) +¢t ¢ vk(t + ¢t) : sin(Ák(t + ¢t)) r o Here, Ák(t) is the directional angle of particle k. At r a each time step, a particle chooses its next directional heading based on the previous positions and loca- tions of its neighboring particles.

Load more