A Stochastic Dierential Equation Model for Predator-Avoidance Fish Schooling Aditya Dewanto Hartonoa Linh Thi Hoai Nguyenc T on Vi e .t Ta.ab

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A Stochastic Diﬀerential Equation Model for

Predator-Avoidance Fish Schooling

Aditya Dewanto Hartonoa, Linh Thi Hoai Nguyenc, Tˆon Viˆe

.t Ta

.a,b

aMathematical Modeling Laboratory, Department of Agro-environmental Sciences, Kyushu

University

bCenter for Promotion of International Education and Research, Kyushu University

cInstitute of Mathematics for Industry, Kyushu University

Abstract

This paper presents a system of stochastic diﬀerential equations (SDEs) as mathe-

matical model to describe the spatial-temporal dynamics of predator-prey system in

an artiﬁcial aquatic environment with schooling behavior imposed upon the associ-

ated prey. The proposed model follows the particle-like approach where interactions

among the associated units are manifested through combination of attractive and

repulsive forces analogous to the ones occurred in molecular physics. Two hunting

tactics of the predator are proposed and integrated into the general model, namely

the center-attacking and the nearest-attacking strategy. Emphasis is placed upon

demonstrating the capacity of the proposed model in: (i) discovering the predator-

avoidance patterns of the schooling prey, and (ii) showing the beneﬁt of constituting

large prey school in better escaping the predator’s attack. Based on numerical simu-

lations upon the proposed model, four predator-avoidance patterns of the schooling

prey are discovered, namely Split and Reunion, Split and Separate into Two Groups,

Scattered, and Maintain Formation and Distance. The proposed model also suc-

cessfully demonstrates the beneﬁt of constituting large group of schooling prey in

mitigating predation risk. Such ﬁndings are in agreement with real-life observations

October 11, 2022

arXiv:2210.03989v1 [math.DS] 8 Oct 2022

of the natural aquatic ecosystem, hence conﬁrming the validity and exactitude of the

proposed model.

Keywords: Predator-Prey System, Swarm Behavior, Particle-Based Model,

Stochastic Diﬀerential Equations, Predator-Avoiding Patterns, Fish Schooling

2020 MSC: 92-10, 60H10, 68W10

1. Introduction

Swarm dynamics, one of the most commonly observed phenomena in the real world,

have attracted the interest of many researchers from diverse ﬁelds including biology,

mathematics, and computer engineering [1,2,3,4,5,6]. As one manifestation of

collective behavior in nature, swarm dynamics oﬀer many advantages to the associ-

ated individuals: higher foraging success [7,8,9], improve protection upon predators

[8,10,11,12,13,14], increase mating chance [8,10], and more eﬃcient energy

consumption due to hydrodynamic or aerodynamic advantages [8,9,10]. Based

on observational and empirical investigations of the interaction between congregat-

ing animals and their neighbors, researchers have constructed artiﬁcial life [15], for

example, mathematical models of schooling prey in the presence of a predator.

As far as mathematical modeling of predator-prey interaction with schooling be-

havior upon the associated prey is concerned, two diﬀerent approaches of modeling

are routinely considered. The ﬁrst approach examines the predator-prey system from

the standpoint of tracking population densities of the associated entities. In this

framework, mathematical models are typically constructed by integrating functional

responses into the general description (see, for example, [16,17,18,19,20,21,22,

23]). The second approach of modeling examines the predator-prey system as self-

propelled units in an artiﬁcial ecosystem. Here, the associated entities are treated as

agents (or particles) that move around and interact with each other perpetually in the

prescribed domain. Hereinafter, the discussion concerning mathematical modeling

of the predator-prey system follows the latter approach.

Let us review some existing studies. Oboshi et al. [24] proposed a mathematical

model to describe predator-prey interaction in an aquatic ecosystem. Therein, they

designed an artiﬁcial habitat where schooling prey ﬁsh coexist with solitary predator.

They used a local rule that each prey ﬁsh in the school chooses one way of action

among four possibilities according to the closest mate to construct a model of diﬀer-

ence equations for anti-predator schooling behavior. Their model is able to return

single predator-avoidance maneuver that ﬁts real observation at the Port of Nagoya

Public Aquarium. Such an evasive maneuver is also observed in [12,25,26,27,28].

Later on, the same authors [29] extended their assessment by incorporating three

diﬀerent conditions for prey-predator encounters in the artiﬁcial habitat, namely

predator faces center, side, and vicinity of the schooling prey ﬁsh. By assimilat-

ing such conﬁgurations into their model, they managed to obtain two additional

anti-predator patterns.

Nishimura [30] investigated mechanisms for a predator to select an individual

prey from a group. Therein, the author introduced a priority function as pertinent

parameter determining the selectivity of predator upon a particular prey in the group.

The study is later extended [31] by including three diﬀerent priority functions, each

corresponds to the selection of nearest, peripheral, and split individuals among the

schooling prey. It is shown that, from the predator’s viewpoint, the selection of

peripheral individual is the best of the three available options: such a strategy yields

the highest average number of successfully captured prey.

Zheng et al. [32] considered the evasive maneuver of schooling prey ﬁsh as complex

collective traits resulting from the combination of schooling, cooperative, and selﬁsh

behavior. Three mathematical models of prey ﬁsh behavior are constructed, each

corresponds to a fundamental trait. A combination of such models yields one anti-

predator pattern which is similar to the one obtained by Oboshi et al. [24].

Lee et al. [33] implemented molecular dynamics (MD) to model the dynamical

behavior of schooling prey in response to predator’s attack. They retrieved one

predator-avoidance pattern of the schooling prey, namely the crescent-shaped ma-

neuver. In a separate study [34], the transitional regimes associated with such a

maneuver are elaborated further.

Demˇsar and Lebar Bajec [35] studied three hunting tactics of the predator (attack

center, nearest, and isolated prey) in association with two distinct behavior of the

prey (schooling and solitary) using a mathematical model based on fuzzy logic. They

found that compared to individualistic behavior, schooling is the optimal defense

mechanism to avoid predation. Later on, Demˇsar et al. [14] extended the assessment

by including composite hunting tactics of the predator. Two composite hunting

tactics were proposed: (i) the predator may choose one of three simple hunting

strategies (attack nearest, center, and peripheral prey) in successive attacks based

on probability, and (ii) the predator was set to initially disperse the schooling prey

and then pursue an isolated individual. They concluded that predator’s confusion

plays an important role upon the evolution of the composite tactics. Moreover, they

conveyed that when confusion is taken into account, the latter composite tactic is

the better predation mode that provides favorable outcome for the predator.

Despite the vibrant research activity in the ﬁeld of mathematical modeling of

predator-prey system, up to our knowledge, there is still no mathematical model

that provides not only one but all real predator-avoidance patterns mentioned above.

Moreover, there is also no mathematical model which shows the beneﬁt of making a

large school of ﬁsh in surviving predator’s attack. In this paper, we aim to construct

a mathematical model based on stochastic diﬀerential equations (SDEs) for predator-

avoidance ﬁsh schooling which can show not only simulated patterns adapting to the

real patterns but also the beneﬁt of making a large school of ﬁsh in surviving preda-

tor’s attack. The SDE model is chosen as it is well-suited for capturing complex

movements and interaction rules of individuals in diverse environment. Further-

more, an SDE model is amenable for undertaking numerical simulations in computer

platform.

In [36,37], we introduced a system of stochastic diﬀerential equations to describe

the process of ﬁsh schooling in a free, unbounded environment based on a set of

behavioral rules of ﬁsh schooling proposed by Camazine et al. [38]. The rules are

stated as follows:

(a) The school has no leaders and each ﬁsh follows the same behavioral rules.

(b) Each ﬁsh uses some form of a weighted average of the position and orientation

of its nearest neighbors to decide where to move.

the imperfect information-gathering ability of a ﬁsh, and the imperfect execu-

tion of the ﬁsh’s actions.

However, these rules do not consider obstacle-avoiding mechanisms and the foraging

process of a school of ﬁsh. In accordance with this, in [6,39], we proposed behav-

ioral rules for both phenomena, respectively. Furthermore, using the model in [36],

we constructed two mathematical models for obstacle-avoiding mechanisms and the

foraging behavior of ﬁsh schools. As a result, four obstacle-avoiding patterns are

discovered. In addition, the probability of foraging success is observed to escalate up

to an optimal value of school size, then gradually decreases as the school size gets

larger.

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AStochasticDierentialEquationModelforPredator-AvoidanceFishSchoolingAdityaDewantoHartonoa,LinhThiHoaiNguyenc,T^onVi^e.tTa.a,baMathematicalModelingLaboratory,DepartmentofAgro-environmentalSciences,KyushuUniversitybCenterforPromotionofInternationalEducationandResearch,KyushuUniversitycInstituteofMath...

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A Stochastic Dierential Equation Model for Predator-Avoidance Fish Schooling Aditya Dewanto Hartonoa Linh Thi Hoai Nguyenc T on Vi e .t Ta.ab

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