A model of the formation of fish schools and migrations of fish Simon Hubbard, Petro Babak, Sven Th.Sigurdsson,Kjartan G. Magnússon, ecological modeling,elsevier,2003 Presentation:nafiseh babaee

Outline: Introduction basic model presented by Vicsek et al.(1995a) Model description Simulation results Conclusions suggestions

Introduction

Introduction: feeding and spawning migrations of pelagic fish species timing and route of the migrations are influenced by :  Environmental conditions  internal variables

Environmental conditions: boundaries between warm &cold water masses certain isotherms Oceani ccurrents density of food

Internal variables: physiological state of each fish state of maturity of each fish

Fig. 1. Feeding and spawning migrations of the Central North Atlantic stock of capelin.

features have been observed: Schools tend not to cross fronts between cold and warmer waters The geographical path is fairly constant The spawning stock enters the spawning grounds along different routes …

The objective with this work: to construct and explore a realistic simulation model of fish schools to describe and predict migrations and spatio-temporal distributions of fish in specified oceanic regions To be used to explore and test assumptions and hypotheses about factors,which may influence the migrations of fish

Vicsek et al.(1995a) model

motion of self-driven interacting particles in a plane All particles have the same fixed speed Each particle has a direction of motion given by an angle θ Direction angle is modified by averaging the direction of motion of all neighbours+uniform, zero mean noise. So direction in migration is arbitary & random Periodic boundary conditions are imposed

Current model: variable speeds, “directional” noise instead of uniform noise boundaries and boundary conditions and force fields due to temperature and food density.

Model description

Dynamical equations Collection of particles moving in a plan: Particle i has a speed of vi and a direction of motion θi.

Dynamical equations The term ξti is a random perturbation of the direction angle. Unit vector in direction of velocity vector has two terms.first:

Dynamical equations The other unit vector is: T(x, y) is gradient function for example temperature/food density The speed is calculated by:

Dynamical equations The weighted average of two unit vectors is: And the velocity is therefor:

Dynamical equations

Formulation of noise Uniform noise in Vicsek et al model but directional noise in this model. θ0i (t) shows the direction of attracting point which typically is in the spawning region.

Formulation of noise Probability density function of ξ, the random perturbation in direction angle. The desired direction,θ0 the actual direction of motion of the particle,θi and the range of possible perturbations

Formulation of noise The probability distribution of ζ, the random component of speed is given by:

Boundaries and boundary conditions

Simulation results

Simulation of one migration cycle between feeding and spawning grounds.

Simulation results Simulation of one migration cycle between feeding and spawning grounds.

Simulation results Simulation of one migration cycle between feeding and spawning grounds.

Simulation results A hypothetical food density surface with the associated gradient field

Simulation results

spawning migration feeding migration Five sample paths for five different particles.

Simulation results spawning migrationfeeding migration Two snapshots of the migration cycle with the temperature boundary in a more westerly location

Conclusions

suggestions

Refrence: Simon Hubbard, Petro Babak, Sven Th. Sigurdsson, Kjartan G. Magnússon,2003,” A model of the formation of fish schools and migrations of fish”,ecological modeling,Elsevier.