Abstract

Methods of migrations of school of fish often involve directional movements. The paper suggests aggregation of cosine of angels formed by the members of the school and finding the most preferred direction (mean) and the spread (Standard deviation). The percentage of fish likely to be found in a band of (Mean ± k.SD) is given for k= 1, 2, and 3. The suggested method helps to identify trajectory of a school of fish and can be verified using automated methods. Better tracking of movements of fish may help ecologists to develop effective conservation and management strategies to protect fish populations and their habitats.

Keywords:Oriented swimming movements; School of fish; Cosine of angels; Mean; Standard Deviation

Introduction

Migratory behaviour of fish mainly for feeding and reproduction is a regular phenomenon. Major types of migration are “anadromous”, where adult fish migrate from sea into fresh water to spawn, like salmon, striped bass, and the sea lamprey, etc. and “catadromous” where adult fish migrate from fresh water into salt water to spawn like eels, etc. Marine forage fish like anchovies, sardines, shad, and menhaden etc. often undertake large migrations between their spawning, feeding and nursery grounds. Methods of migrations by Denatant (swimming with the water current) or Contranatant movement (swimming against the current) are:

By drifting: fishes are carried passively by water currents and may result in directional movements. Random locomotory movements: random in direction, lead to a uniform distribution or to an aggregation. Oriented swimming movements: in a particular direction:
(a) Towards or away from the source of stimulation.
(b) At some angle to an imaginary line running between them and the source of stimulation.

The paper suggests aggregation of cosine of angels formed by the members of the school and find the most preferred direction and also the spread.

Literature Survey

Orientation refers to coordinated movement in a given direction. Coordination (in motion) has been defined by Herbert-Read (2016) [1], as the synchronisation of individuals’ movements in time and space facilitating movements in groups through non-independent interactions with one another. The mechanism of collective motion or orientation during fish migration was reviewed by Able (1980) [2]; Vicsek and Zafeiris (2012) [3]. Aoki (1982) [4], undertook simulation study on the schooling mechanism in fish and determined simple interaction rules between neighbouring individuals generated coordinated motion, similar to animal groups. Similar interaction rules were found by Reynolds (1987)[5], for flocks, herds and schools. Couzin et al. (2002)[6], investigated interaction rules in three dimensions with changes in size of the zones of alignment. Hemelrijk and Kunz, (2005)[7]; Kunz and Hemelrijk, (2003) [8], followed different models for fish schools. The studies assumed that group movements of animals involve interactions among each other with effective social forces like attraction, repulsion, and alignment. While attraction encourages individuals to aggregate in groups, repulsion prevents collisions, and alignment induces individuals to follow the same direction as their neighbors. Following similar assumptions, Herbert-Read (2016) [1], investigated relative positions of an individual (N_1) with a focal individual (N_2) based on the bearing angle (θ) and the distance to that neighbour (d) to correlate speed or turning angle (α) of N_2 with θ and d to see how the individuals are interacting. If change of speed or direction of a focal individual are adopted by neighbouring individuals (maybe after some time gap), leader–follower relationships can be concluded.

However, instead of the two criteria θ and d, one van use appropriate trigonometric function say tanθ or cosθ. In addition, changes in interaction rules affect the general properties of moving groups.

Suggested method:
Let θ_1, θ_2, θ_3,……..,θ_k are the angels formed by each of the k-members of the school of fish with the imaginary line running between them. Without loss of generality, let us assume that the angles. 〖Cosθ〗_i can be computed for vectors of unit length.

Mean and SD of 〖Cosθ〗_ifor a school of fish can be obtained by the method suggested by Rao (1973) [9].

Mean or most preferred direction is estimated by θ ̅=〖Cot〗^ (-1) (Σ_(i=1)^k▒〖Cosθ〗_i )/(Σ_(i=1)^k▒〖Sinθ〗_i ) and the dispersion by

√ (1-r^2 ) where r^2=(〖(Σ▒〖Cosθ〗_i )/k)〗^2+ 〖((Σ▒〖Sinθ〗_i )/k)〗^2

Convert X_cand X_0to π_cand π_0where π_ic=√ (X_ic/ (‖X_c ‖)) and π_i0=√(X_i0/(‖X_0 ‖)) so that
‖π_c ‖^2=‖π_0 ‖^2=1.

Thus, sample mean and sample dispersion of 〖Cosθ〗_i can be computed respectively by
Cos (θ ̅) = Cos (〖Cot〗^ (-1) (Σ▒〖Cosθ〗_i)/ (Σ▒〖Sinθ〗_i )) (1)
and SD = √ (1-([〖 (Σ▒〖Cosθ〗_i)/k)〗^2+ 〖((Σ▒〖Sinθ〗_i )/k) 〗^2 )]

Equation (1) gives the most preferred direction of movement of the school. Equation (2) indicates the spread. Assuming normal distribution of 〖Cosθ〗_i, following can said:
68.2% of fish move within a band of Cos (θ ̅) ± SD
95.4% of fish move within a band of Cos (θ ̅) ± 2SD
99.7% of fish move within a band of Cos (θ ̅) ± 3SD

Equation (1) and (2) help to identify trajectory of a school of fish and get verified using automated methods. Better tracking of movements of fish is important to understanding their ecology and behaviour, based on which ecologists may develop effective conservation and management strategies to protect fish populations and their habitats.

Future studies may be undertaken for empirical verification of the suggested method along with its robustness and comparison with method of fish tracking using deep learning like trackingby- detection, deep features combined with correlation filtering methods, Siamese networks, etc.. One can look forward to the fish tracking method combined with Transformer, aiming to provide a reference for accelerating the promotion of smart fishery and precision farming.

References

1. Herbert-Read JE (2016) Understanding how animal groups achieve coordinated movement. J Exp Biol 219(Pt 19): 2971-2983.
2. Able KP (1980) Mechanisms of orientation, navigation and homing. Animal migration, orientation and navigation pp: 283-373.
3. Vicsek T, Zafeiris A (2012) Collective motion. Phys Rep 517: 71-140.
4. Aoki I (1982) A simulation study on the schooling mechanism in fish. Bull Jpn Soc Sci Fish 48(8): 1081-1088.
5. Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. Comput Graph 21(4): 25-34.
6. Couzin ID, Krause J, James R, Ruxto GD, Franks NR (2002) Collective memory and spatial sorting in animal groups. J Theor Biol 218(1): 1-11.
7. Hemelrijk CK, Kunz H (2005) Density distribution and size sorting in fish Schools: an individual-based model. Behav Ecol 16(1): 178-187.
8. Kunz H, Hemelrijk CK (2003) Artificial fish schools: collective effects of school size, body size, and body form. Art Life 9(3): 237-253.
9. Rao CR (1973) Linear Statistical Inference and its Application. 2^nd Edition, Wiley Eastern Private Limited, New Delhi