A Simple Game-Theoretic Model for Upstream Fish Migration

Theory Biosci. DOI 10.1007/s12064-017-0244-3

ORIGINAL PAPER

A simple game-theoretic model for upstream ﬁsh migration

Hidekazu Yoshioka1

Received: 28 December 2016 / Accepted: 25 April 2017 Ó The Author(s) 2017. This article is an open access publication

Abstract A simple game-theoretic model for upstream fish modulators of biochemical processes, and transport vectors migration, which is a key element in life history of (Winemiller and Jepsen 1998; Flecker et al. 2010). In diadromous fishes, is proposed. Foundation of the model is addition, many of them are economically and culturally a minimization problem on the cost of migration with the valuable fishery resources. Both their abundance and swimming speed and school size as the variables to be diversity have been highly affected by loss and degradation simultaneously optimized. Finding the optimizer ultimately of habitats and migration routes (Guse et al. 2015; Logez reduces to solving a self-consistency equation without et al. 2013; Radinger and Wolter 2015). Physical barriers, explicit solutions. Mathematical analytical results lead to such as dams and weirs, are the major factors that fragment the sufficient condition that the self-consistency equation habitats and migration routes of fish (Jager et al. 2015;Yu has a unique solution, which turns out to be identified with and Xu 2016). Analyzing fish migration has, therefore, the condition where the unique optimizer exists. Behavior been a key topic in current biological and ecological of the optimizer is analyzed both mathematically and research areas (Becker et al. 2015; Wang et al. 2012; White numerically to show its biophysical and ecological conse- et al. 2011). quences. The analytical results demonstrate reasonable Since many of the fishes migrate by forming schools agreement between the present mathematical model and (Partridge and Pitcher 1980; Partridge 1982), analyzing the theoretical and experimental results of upstream migration dynamics of fish schools is a crucial topic. migration of fish schools reported in the past research. Migration of individuals and groups of organisms has been studied from both theoretical and practical viewpoints. For Keywords Fish migration Á Fish school Á Game theory Á migration of isolated individual fishes, swimming path and Swimming speed Á School size velocity have been analyzed with the help of optimization theory where some hydrodynamic cost is minimized. The physiological energy minimization principle has been a Introduction core to determine the preferred swimming speed of individual fishes given a flow speed and water quality (Clair- Migratory fishes, such as salmonids and sturgeons, are eaux et al. 2006; Tucker 1975). Such models have recently central aquatic species of environment and ecosystem of been a basis of a generalized biological theory for the surface water systems, since they shape and create linkages physiological energy minimization principle (Pa- among food webs as consumers, ecosystem engineers, padopoulos 2009, 2013). Some authors assumed that a migrating animal selects a path that minimizes the total resistance force (Bejan and Marden 2006; Lindberg et al. & Hidekazu Yoshioka 2015; McElroy et al. 2012). Sto¨cker and Weihs (2001) [email protected] analyzed a conceptual optimization model based on Weihs (1974) for energy-saving conceptual burst swimming of 1 Faculty of Life and Environmental Science, Shimane University, Nishikawatsu-cho 1060, Matsue, fishes. For migration of fish schools, increasing hydrody- Shimane 690-8504, Japan namic efficiency for each individual has been considered as 123 Theory Biosci. one of the most significant advantages to form schools research results on behavior of fish schools subject to water (Mayer 2010). Increasing foraging efficiency and vigilance current under experimental settings by Onitsuka and his co- against predators have also been considered to be advan- workers, which can be effectively utilized for development tageous (Mayer 2010). On the other hand, increasing the and validation of mathematical models (Onitsuka et al. foraging efficiency would not necessarily increase the net 2009, 2012a, b). Development of a conceptual model for available food for each individual in a school (Beecham the dynamics of fish schools based on these results can and Farnsworth 1999). Day et al. (2011) experimentally significantly advance comprehending and analyzing fish observed that smaller fish schools of Poecilia reticulate migration. This is the main motivation of this paper. (guppy) demonstrated higher foraging efficiency than the Based on the above research background, this paper larger ones. presents a minimal mathematical model for upstream Macroscopic models based on conceptualization for migration of fish schools in water current based on a simple effectively analyzing animal group’s dynamics have been game theory: an optimization theory. The model focuses on proposed to comprehend group dynamics from phe- prolonged swimming behavior of fishes rather than burst nomenological viewpoints. In these models, one of the most ones, the latter being relevant for much shorter timescale. important parameters for describing migration of animal Game-theoretic models have extensively been utilized for groups is the size: total number of individuals in the group. A ecological and biophysical modelling of animals (Kabalak common assumption shared by most of the models is that et al. 2015; Mariani et al. 2016; Metz et al. 1992; Pereira there is a size to compromise between advantages and dis- and Martinez 2010; Riechert and Hammerstein 1983; advantages of forming groups. Gueron and Levin (1995) Tanaka et al. 2011); however, application of such models mathematically analyzed stability of the equilibrium of to biophysics of fish migration is still rare. The variables to group size with the different fusion and fission rates. be optimized in the present model contain the swimming Anderson (1981) proposed a stochastic differential equation speed and school size that should be chosen by the fish model for temporal evolution of the size of fish schools. Niwa school, so that the net cost of migration is minimized. The (1996, 1998, 2005) established modelling frameworks for present model is an extended counterpart of the model for analyzing group-size distributions based on probabilistic upstream migration of isolated individuals (Yoshioka and descriptions of the merging and splitting phenomena. Shirai 2015; Yoshioka et al. 2015, 2016a). Assuming that On the other hand, microscopic, individual-based mod- certain simplifications reduce finding the optimizer to els with repulsion and attraction forces acting among the solving a self-consistency equation. Unfortunately, general individuals have also been utilized to study group exact expressions of its solution have not been found. To dynamics of animals. Behavior of individual-based deter- overcome this issue, this paper proposes an algorithm to ministic and stochastic models for migrating fish schools numerically solve the self-consistency equation. An ana- has been studied both from mathematical and numerical lytical estimate of the error between the true solutions and viewpoints (Ebeling and Schweitzer 2001; Uchitane et al. approximate solutions is also presented. Behavior of the 2015; Ngyuen et al. 2016; Ta et al. 2014). Sumpter et al. optimizer is then mathematically and numerically studied (2008) numerically investigated information transfer within to show its ecological and biophysical consequences. moving animal groups using a 1-D time-discrete model of Consistency between the present model and the experi- migration. They implied that each group has an optimal mental results of swimming behavior of migratory fishes size where the group density is close to a phase transition validates its performance. point from disordered state to ordered state. Lee (2010) The rest of this paper is organized as follows. Section 2 studied a noisy self-propelled particles system to analyze presents the mathematical model for upstream migration of stability of collective motion of particles against period and isolated individuals. An extended mathematical model to intensity of the noise. Katz et al. (2011) evaluated repulsive deal with the migration of fish schools, which is the main and attractive forces acting among individuals of a fish focus of this paper, is also presented in this section. Sec- school based on a trajectory tracking technique. tion 3 performs mathematical and numerical analyses of As reviewed above, both individual-based microscopic the models. Section 4 concludes this paper and provides and conceptual macroscopic models have been proposed future perspectives of this research. and extensively studied so far; however, the conventional theoretical research lacks analyzing migration of fish schools subject to water current, which are of great eco- Mathematical models logical and environmental interest. Important such exam- ples include assessing passage efficiency of river reaches This section presents two mathematical models for and hydraulic structures (Baek et al. 2015; Kerr et al. 2015; migration of isolated individual fishes and fish schools, the Vowles et al. 2015). There exist a series of extensive latter being an extension of the former. In this paper, an 123 Theory Biosci. isolated fish is regarded as a fish school whose total number Model for upstream migration of an isolated of individuals is exactly one. The models assume that the individual fishes are migrating toward upstream in a water current defined along the 1-D space R with the flow velocity of Ordinary differential equation V [ 0. A fish school is thus conceptualized as a particle moving along R. The swimming velocity of the fish school The model for upstream migration of isolated individuals is denoted as u, which gives its ground velocity as starts from the ODE Vg ¼ u À V. The school is, therefore, migrating toward dX t ¼ V À u; ð1Þ upstream when Vg [ 0. The total number of individuals in dt the fish school is denoted as N, which is regarded as a where t is the time and X is the 1-D position of the indi- continuous variable. The variable N is hereafter referred to t vidual fish along the flow at the time t. The ODE (1)is as the size, which is considered as another control variable equipped with the initial condition X 0, which is to be optimized. Although the continuous approximation of 0 ¼ assumed without the loss of generality. The ODE (1)is the school size N, which is a discrete variable in reality, then solved as maybe problematic, it is well known that discrete optimization problems are far more technically and mathe- Xt ¼ðV À uÞt: ð2Þ matically difficult than the continuous ones. Therefore, to avoid the above-mentioned difficulty, this paper focuses on the analysis with the school size N as a continuous vari- Objective function able. This approximation becomes more accurate as N increases (Niwa 2003). Although the conventional defini- The range U of the variable u is denoted as U ¼½0; þ1Þ. tion of the fish school (Partridge and Pitcher 1980; Par- Under this setting, it is possible to identify a swimming velocity as a swimming speed because of the non-nega- tridge 1982) applies to N 3, this paper assumes N [ 0 for the sake of simplicity of analysis. It should be noted that tivity of the former. Assuming positive rheotaxis of the fish (Hinch et al. 2005; Keefer and Caudill 2014), the objective biologically relevant solutions should satisfy N 1. The function / u to be minimized through its migration pro- variable u (the couple of variables ðu; NÞ) is optimized in ð Þ cess is formulated as the model for isolated individual (fish school). We assume Z u [ V, since the focus is upstream migration. s /ðuÞ¼ f ðuÞdt; ð3Þ It should be noted that water current along a real open 0 channel such as a river is not spatially homogenous. Such a where s is the time at which Xs ¼ L, and thus, situation is more realistic than that considered in this paper. À1 We mention that the present model can serve as the basis of s ¼ Lðu À VÞ . The objective function / measures the analyzing fish migration in the real open channels if it is total physiological energy consumed during the migration effectively incorporated in that for the recently developed process. The cost function f is assumed to be given by the mathematical models based on the dynamic programming power law-type function: principle (Yoshioka 2016; Yoshioka et al. 2016b). This 1 f ðuÞ¼ unþ1; ð4Þ research topic is beyond the scope of this paper and will, n þ 1 therefore, be addressed elsewhere. with a parameter n 1. The cost function f in (4) shows As shown in what follows, the present model has a that the fish is more high speed-averse for larger n, because number of key non-dimensional parameters, as summa- it more rapidly grows for larger n. The cost function f in rized in Table 1. (4) is convex and increasing with respect to u; which is in

Table 1 Summary of the key non-dimensional parameters Parameter Range or relationship Meaning n Oð100Þ (Yoshioka et al. 2016a) Growth rate of the hydrodynamic cost m Oð10À1Þ (Mayer 2010) Discount rate of the hydrodynamic cost k Larger than or equal to Oð10À1Þ Growth rate of the non-hydrodynamic cost y kðnþ1Þ Non-linearity of the self-consistency equation y ¼ mþk [ 1 hik z m 1 m y mþk Inverse of the power function of the ﬂow speed V z ¼ knÀm z~, where z~¼ bkðnþ1Þ yÀ1 V

123 Theory Biosci. good accordance with the conventional experimental where the coefficient a 1 represents the discount of the results on swimming behavior of fishes (Brodersen et al. hydrodynamic cost, which is motivated by the theoretically 2008; Cucco et al. 2012; Mori et al. 2015; Roche et al. and experimentally validated fact that increasing the size N 2013; Svendsen et al. 2010; Yoshioka et al. 2016a). The can effectively reduce the hydrodynamic cost per individ- objective function is rewritten with (2)as ual (Marras et al. 2015; Mayer 2010). Using a conceptual model with probabilistic consideration, Mayer (2010) L unþ1 /ðuÞ¼ : ð5Þ derived that the discount of the energy for each individual n 1 u V þ À À1 is proportional to N 3, which leads to Assuming that the migration process is temporally m homogeneous, we can simply set L ¼ 1 without the loss of aðNÞ¼N ; ð9Þ generality. 1 with m ¼ 3 in the present model. The conventional research suggests that schooling has Optimization problem many aspects of benefits, such as improvements of navi- gational performance (Berdahl et al. 2014; Torney et al. According to the classical optimization theory (Bonnans 2015), hearing perception (Larsson 2009, 2012), and for- et al. 2006), the optimal swimming speed uÃ that minimizes aging efficiency (Di-Poi et al. 2014; Wang et al. 2016). the objective function has to solve the equation derived These benefits are modelled in a lumped manner as from the first-order condition of optimality: Z s La f 0 uÃ uÃ V f uÃ : 6 /2ðu; NÞ¼À adt ¼ /ðuÞ¼À ; ð10Þ ð Þð À Þ¼ ð Þ ð Þ 0 u À V This Eq. (6) admits two solutions; one of them is uÃ ¼ 0 using a positive constant a. It seems to be reasonable to and the other is formulate an objective function / by summing up /1 and /2 n þ 1 as / ¼ / þ / . However, this formulation leads to the uÃ ¼ V [ V; ð7Þ 1 2 n trivial optimizer N ¼þ1, which is not reasonable from a biological viewpoint. We, therefore, conjecture that this / the latter being inferred as a biophysically relevant optimal lacks the fact that schooling is not always definitely benefi- swimming speed to represent upstream migration. Actu- cial. Schooling would negatively affect collective perfor- ally, it is straightforward to check that the candidate (7) mances of fishes, such as passage efficiency (Lemasson et al. minimizes /. 2014) and information transmission among individuals (Castellano et al. 2009; Shang and Bouffanais 2014). Other Model for upstream migration of a fish school disadvantages of forming a school such as competitions among the individuals in the school would also exist. In Ordinary differential equation addition, various environmental disturbances would also affect the fish migration. We thus assume that the net non- This section presents a mathematical model for upstream hydrodynamic cost of forming a school /3 is formulated as migration of a fish school. This paper assumes that longi- Z tudinal movement of the fish school is also described by the s 1 / ðu; N; gÞ¼ gNq À gpþ1 dt ODE (1) where the variable u serves as the representative 3 cðp þ 1Þ 0 ð11Þ swimming velocity of the school. L 1 ¼ gNq À gpþ1 ; u À V cðp þ 1Þ Objective function where p [ 0, q [ 0, and c [ 0 are parameters, and g 0is The value function for upstream migration of a fish school the intensity of the negative effect of forming the school. is different from that of an isolated individual due to the The first term of (11) represents the cost of forming a appearance of an additional control variable and coeffi- school due to the above-mentioned negative influencing cients. Assume that all the individuals in the school share factors that increase as N increases, while the second term an ideal objective function /; which is derived with a represents the mitigation effect by forming a school. game-theoretic argument as explained below. The cost of The objective function / for the fish school is conse- swimming, namely, the hydrodynamic cost during the time quently set as interval ½0; s is / ¼ / þ / þ / Z 1 2 3 s f u L unþ1 L unþ1 1 ð Þ a gNq gpþ1 : 12 /1ðu; NÞ¼ dt ¼ ; ð8Þ ¼ m À þ À ð Þ 0 aðNÞ n þ 1 ðu À VÞaðNÞ u À V ðn þ 1ÞN cðp þ 1Þ

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not 1 (Mayer 2010), the order of the parameter k can be The objective function is minimized by the fish school 3 À1 itself by choosing appropriate ðu; NÞ, while nature maxi- inferred to be larger than or equal to Oð10 Þ. mizes it by choosing g. Since L can be set as 1 without the loss of generality, the optimization problem reduces to the Optimization problem optimizer ðu; N; gÞ¼ðuÃ; NÃ; gÃÞ that gives Ã Ã Ã By the static optimization problem, as in the case for the U ¼ inf sup /ðu; N; gÞ¼/ðu ; N ; g Þ: ð13Þ Ã Ã u;N g model of isolated individuals, the optimizers u and N have to solve the equations derived from the first-order The optimization problem can be seen as a game with necessary condition of optimality: two non-cooperative players: the fish school and nature. ðuÃÞnþ1 Straightforward calculation shows uÃ n uÃ V b NÃ mþk ; ð Þ ð À Þ¼ Ã m þ ðð Þ À 1Þ ð19Þ 1 ðn þ 1ÞðN Þ gÃðNÞ¼gÃ ¼ðcNqÞp; ð14Þ and and Ã nþ1 mðu Þ kÀ1 U ¼ sup /ðu; N; gÃÞ ¼ bkðNÃÞ : ð20Þ Ã mþ1 u;N ðn þ 1ÞðN Þ nþ1 1 u p q qþ1 ¼ sup À a þ cpNp : ð15Þ With a technical calculation, the equations can be reduced u À V ðn þ 1ÞNm p þ 1 u;N to a self-consistency equation as follows. First, substituting Assuming mathematical consistency between the model (20) into (19) yields hi of isolated individuals and that of fish schools, setting N ¼ 1 m ðuÃÞnðuÃ À VÞ¼ ðuÃÞnþ1 1 þ f1 ÀðNÃÞÀkg : 1 with the objective function / in (12) should reduce to that n þ 1 k in (5). This leads to the constraint of parameters ð21Þ p q Assuming that uÃ [ 0, (21) reduces to a ¼ cp ¼ b; ð16Þ p þ 1 kðn þ 1Þ uÃ ¼ V: ð22Þ with a positive parameter b. Equation (15) is then rewritten kn À m þ mðNÃÞÀk with the positive parameter k ¼ 1 þ q as p Introduce the changes of variables Ã U ¼ sup /ðu; N; g Þ kðn þ 1Þ m 1 u;N y ¼ [ 1; z ¼ ; and nþ1 m þ k kn À m z~ 1 u k k ð23Þ ¼ sup m þ bðN À 1Þ : ð17Þ m y mþk u;N u À V ðn þ 1ÞN z~ ¼ V : bkðn þ 1Þ y À 1 Relevant values of the parameters n, m, and k can be to a Ã Ã certain extent inferred from theoretical and numerical The optimizers u and N are then expressed as y 1 y analysis. On the parameter n, Yoshioka et al. (2016a) uÃ ¼ VwÃ and NÃ ¼ðz~Þk ðwÃÞk; ð24Þ showed Oð100Þ for a variety of migratory fish species. y À 1 Identification of the order of the parameter k is more dif- respectively, with an auxiliary parameter wÃ. The positivity ficult than that for the parameters n and m because of its of uÃ is guaranteed by (18), since y [ 1. The right side of lumped nature. In this paper, it is assumed that the (22) is further reduced to parameters n, m, and k satisfy Ã y Ã kðn þ 1Þ 1 y ðw Þ u ¼ V ¼ V y : kn À m [ 0; ð18Þ kn À m m Ã Àk y À 1 ðwÃÞ þ z 1 þ knÀm ðN Þ which turns out to be a necessary condition for existence of ð25Þ an optimal uÃ with uÃ [ V. A biophysical interpretation of By (25), the equation to be solved for wÃ is consequently the condition (18) is that fishes create fish schools if they found as have to save the hydrodynamic cost of migration, which z occurs if k is large or m is small. This is qualitatively wÃ ¼ FðwÃÞ¼1 À : ð26Þ ðwÃÞy þ z consistent with the model of Sto¨cker and Weihs (2001). No biophysically relevant optimizer that represents upstream Finding the optimizer ðuÃ; NÃÞ is thus ultimately reduced migration is obtained under this condition. The assumption to solving the self-consistency equation (26). A relevant wÃ shows that, provided that m is around Oð10À1Þ even if it is for upstream fish migration should give

123 Theory Biosci. uÃ [ V and NÃ 1: ð27Þ w* 1.0 Introduce the parameter 1 z ¼ yÀyðy À 1ÞyÀ1; ð28Þ for the sake of brevity of descriptions. Straightforward calculation shows that the number of the intersections of the line v ¼ w and the curve v ¼ FðwÞ in the first quadrant of v À w space is zero (z [ z), one (z ¼ z), and two z 0.5 0 (0\z\z). Figure 1 plots the graphs of v ¼ w and v ¼ FðwÞ for z [ z, z ¼ z, and 0\z\z in the square ½0; 1Â½0; 1, which graphically validates the above-mentioned statement. For z ¼ z, in particular, v ¼ w is the tangent line of the curve v ¼ FðwÞ at the point ðw; wÞ, yÀ1 where w ¼ y . The above-mentioned geometrical infor- 0.0 mation shows that there is no positive solution wÃ for large 0.0 0.5 1.0 z, such that z [ z. In addition, for z ¼ z, uÃ is determined 1/ from (24)asuÃ ¼ V, which does not comply with (27). The y analysis in what follows, therefore, assumes 0\z\zwhere Fig. 2 2-D contour diagram of w ¼ wÃ as a bi-variate function of z the two positive solutions w1 and w2, such that and yÀ1 for 0\z\z and 0\yÀ1\1. The white region represents the Ã 0\w \w\w \1; ð29Þ area where w does not exist. The curve that separates the white and 1 2 colored regions in the figure corresponds to the curve z in (32) (color solve (26)(Appendix). It turns out that the larger solution figure online) Ã w ¼ w2 is the biologically relevant solution to give u [ V, Ã which is denoted by w . The condition 0\z\z can be depending on neither b nor V, meaning that the weight b nþ1 expressed as 0\b\CV with a positive constant C for the cost of schooling should not be too large to form a school. Figure 2 shows a 2-D contour diagram of wÃ as a bi- 1.0 variate function of z and yÀ1 for 0\z\zand 0\yÀ1\1. It should be noted that not every wÃ plotted in Fig. 2 com- plies with (27). This is because k is expressed with (23)as 0.8 n þ 1 À y k ¼ [ 0; ð30Þ m 0.6 meaning that a relevant y should satisfy v 1 \yÀ1\1: ð31Þ 0.4 (b ) (c ) (d ) n þ 1 In Fig. 2, the boundary curve that separates the colored and white regions represents

0.2 1À1 yÀ1 (a ) zðqÞ¼qð1 À qÞq ð¼ yÀyðy À 1Þ Þ; ð32Þ with q ¼ yÀ1, which is increasing and convex with respect 0.0 to q. 0.0 0.2 0.4 0.6 0.8 1.0

w Analysis of the optimizers Fig. 1 Plots of the graphs of a the line v ¼ w and the curves v ¼ FðwÞ for b z [ z, c z¼ z, and d 0\z\z in the unit square Qualitative and quantitative analyses of the optimizers uÃ ½0; 1Â½0; 1. This ﬁgure graphically shows the geometrical relation- and NÃ are performed to comprehend their dependence on ships between the line v ¼ w and the curve v ¼ FðwÞ. They have zero, the model parameters. The analysis is also intended to ﬁnd one (multiplicity one), and two positive solutions for z [ z, z z, and ¼ Ã 0\z\z, respectively. For z¼ z in particular, v ¼ w is a tangent line appropriate ranges of the parameters to give realistic u and of v ¼ FðwÞ at the point ðz; zÞ NÃ. 123 Theory Biosci.

Qualitative analysis oz since ob \0. The first inequality in (38) shows positive rheotaxis of the fish school, which has been reported to be Qualitative analysis of the optimal controls uÃ and NÃ is satisfied in experimental fish migration for a series of N first performed before the quantitative analysis in the next against Plecoglossus altivelis (Ayu) (Onitsuka et al. sub-section. The analysis here focuses on the signs of 2012a). The second inequality in (38) shows that increasing partial differentials of the optimal controls with respect to the flow speed potentially increases the optimal number of the model parameters, so that their biophysical and eco- individuals in a fish school, so that the physiological energy logical consequences are clearly indicated. consumed by each individual is effectively reduced. This Proposition 1 The solution w ¼ wÃ to the self-consis- statement has also been experimentally validated against P. tency Eq. (26) is decreasing with respect to z and y, altivelis (Onitsuka et al. 2012a). The two inequalities in owÃ owÃ (39) show that the present mathematical model predicts namely, oz \0 and oy \0: that increasing the weight of the non-hydrodynamic forces Proof of Proposition 1 Partially differentiating both sides a fish school to decide smaller size and slower swimming of (26) with respect to z leads to speed to be optimal. o Ã Ã yÀ1 owÃ Ã yÀ1 owÃ Partial differentials of y with respect to model parame- w yðw Þ oz Ã y yðw Þ oz þ 1 ¼ y Àðw Þ ters further highlight qualitative properties of the opti- oz ðwÃÞ þ z ½ðwÃÞy þ z2 ð33Þ mizers uÃ and NÃ associated with the present mathematical o Ã 0 Ã w 1 model. Rather than dealing with the parameters m and k ¼ F ðw Þ À y 2 ; oz ½ðwÃÞ þ z separately, it is efficient to consider the hybrid parameter mkÀ1 that quantifies the ratio of the discount rate of the which is rewritten as energetic cost and the increasing rate of the schooling cost. Ã ow 1 À1 ¼À \0; ð34Þ The partial derivatives of y with respect to n and mk oz ½ðwÃÞy þ z2½1 À F0ðwÃÞ satisfy 0 Ã oy oy since F ðwÞ\0atw ¼ w by a geometrical consideration. [ 0 and \0; ð40Þ Similarly, using the formula on oðmkÀ1Þ o owÃ respectively. The two inequalities in (40) lead to the ln½ðwÃyÞ ¼ lnðwÃÞþyðwÃÞÀ1 ; ð35Þ oy oy inequalities (26) leads to ouÃ oNÃ \0 and \0: ð41Þ on on owÃ z oðwÃÞy ¼ y 2 Equation (40) then leads to oy ½ðwÃÞ þ z oy ð36Þ z wÃ y owÃ ouÃ oNÃ ð Þ Ã Ã À1 [ 0 and [ 0: ð42Þ ¼ lnðw Þþyðw Þ : o À1 o À1 ½ðwÃÞy þ z2 oy ðmk Þ ðmk Þ Therefore, (36) shows The first inequality in (41) shows that a fish school with a more speed-averse energetic cost (the cost function with owÃ zðwÃÞy lnðwÃÞ ¼ larger n) chooses a smaller swimming speed to be optimal, oy ½ðwÃÞy þ z2½1 À F0ðwÃÞ ð37Þ which associates a smaller school size as indicated in the \0; second inequality. The first inequality in (42) indicates that a higher swimming speed is decided to be optimal for Ã h since lnðw Þ\0. relatively higher efficiency of the physiological energy Proposition 1 has biophysical meanings. By (24), the discount by forming a school. The second inequality in (42) partial derivatives of uÃ and NÃ with respect to V satisfy shows that a larger school size is decided to be optimal as the decreasing rate of the energetic cost becomes higher ouÃ oNÃ than the increasing rate of the schooling cost. o [ 0 and o [ 0; ð38Þ V V Qualitative similarity between the model for isolated oz Ã since oV \0. In addition, the partial derivatives of u and individuals and that for fish schools is also discussed in this NÃ with respect to b satisfy paper. Straightforward calculation shows that the optimal control uÃ for the model of isolated individuals with the ouÃ oNÃ analytical formula (7) is decreasing with respect to n.By o \0 and o \0; ð39Þ b b ouÃ (41), the two models, therefore, predict on \0. In addition,

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ouÃ yz by (7) and (38), they predict oV [ 0. The latter statement 0\h ¼ \1; ð48Þ for isolated individuals is in accordance with the experi- wyÀ1 mental results (Onitsuka et al. 2012b). Actually, it is pos- showing that this numerical algorithm can generate a sible to derive a sharper estimate of the partial derivative decreasing sequence of approximated solutions that rig- ouÃ oV as shown in the proposition below. orously converges to the true solution. It is also shown that a similar algorithm with an initial guess w ¼ w can Proposition 2 The model for fish school leads to 0 also generate a convergent sequence with the limit wÃ, ouÃ which on the other hand is monotonically increasing. For [ 1: ð43Þ oV each couple of the fixe parameter values of y and z, the Proof of Proposition 2 By (7) and (38), the inequality iteration algorithm is stopped at r 1, such that jjw À w \e ¼ 10À10. Decreasing the value of the owÃ o uÃ À V 1 ouÃ uÃ rþ1 r ¼ ¼ À [ 0; ð44Þ threshold e, several orders of the magnitude do not affect oV oV V V oV V the computational results presented below. Details of the namely, iterative method for approximating solutions are described Ã Ã in Appendix. ou u Ã [ ; ð45Þ The non-dimensional quantity WÃ ¼ u ÀV is introduced oV V V here with biophysically more clear meaning (non-dimen- Ã follows. Assuming that u [ V derives (43).h sionalized ground speed) than the quantity wÃ that has Proposition 2 shows that the optimal swimming speed uÃ been employed for the sake of mathematical analysis. Ã uÃÀV increases as the flow speed V does and the rate of its Figure 3a–f shows the contour diagrams of W ¼ V increase is larger than one. The inequality (43) theoreti- complying with the conditions (27) for the integers cally leads to the statement that the difference uÃ À V is 1 n 6. Figure 4a–f shows the contour diagrams of NÃ increasing with respect to V, which has been validated for complying with the conditions (27) for the integers upstream migration of fish schools in experimental flumes 1 n 6. The white region in each panel of Figs. 3 and 4 at least for not too large V (Onitsuka et al. 2012b). The is the area where WÃ is smaller than 0 or NÃ is smaller inequality also applies to the model for upstream migration than 1. Such optimal controls are not biologically rele- of isolated individuals. vant, since at least NÃ should be larger than or equal to 1 and an upstream migration should correspond to a posi- Quantitative analysis tive WÃ. The computational results presented in Figs. 3 and 4 are consistent with the mathematical analytical Quantitative behavior of the optimizers uÃ and NÃ has been results presented in the previous sub-section: WÃ and NÃ performed in the previous sub-section whose results are are decreasing with respect to z in (23), namely, validated in this section through numerical analysis. Their increasing with respect to V. behavior is also quantified here and realistic ranges of the The flow speed V in a river under calm conditions is model parameters are discussed. typically from Oð10À1Þ m/s to Oð100Þ m/s, and would be To carry out the quantitative analysis, the value of the at most Oð101Þ m/s. In addition, the maximum sustained biologically relevant solution w ¼ wÃ to Eq. (26) has to be swimming speed of fish living in rivers is at most accurately approximated; however, it has empirically been Oð100Þ m/s (Onitsuka et al. 2009). Therefore, the biologi- found that this is not possible for generic y. To overcome cally relevant order of the non-dimensional quantity WÃ this difficulty, this paper applies a fixed-point iteration can be at most Oð101Þ, which roughly corresponds to algorithm to generate an approximate sequence of the green-to-blue area in each panel of Fig. 3. It would not be solution w ¼ wÃ. The algorithm is formulated as unrealistic to assume that the range of NÃ includes the Ã wrþ1 ¼ FðwrÞ for r ¼ 0; 1; 2; ...; ð46Þ range 1\N \100. Figure 5a–f shows the area with 0\WÃ\100 and 1\NÃ\100 for the integers 1 n 6. subject to the initial guess w0 ¼ 1. The error between the These figures show that increasing n yields wider area Ã approximate solution wr and the true solution w is ana- satisfying the above-mentioned ranges of 0\WÃ\100 and lytically estimated as 1\NÃ\100. Figure 5 indicates that the lower right end of Ã ðyÀ1Þr the area complying with these conditions is more insensi- 0\wr À w \h for r ¼ 0; 1; 2; ...; ð47Þ tive to the lower left-end, which has been checked to be with true for n up to 10.

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Ã uÃÀV À1 Fig. 3 Contour diagrams of W ¼ V for 0\z\z and 0\y \1 with a n ¼ 1, b n ¼ 2, c n ¼ 3, d n ¼ 4, e n ¼ 5, and f n ¼ 6. The white region represents the area where wÃ does not exist as in Fig. 2 or WÃ\0, which does not verify necessary conditions for relevant solutions

Fig. 4 Contour diagrams of NÃ for 0\z\z and 0\yÀ1\1 with a n ¼ 1, b n ¼ 2, c n ¼ 3, d n ¼ 4, e n ¼ 5, and f n ¼ 6. The white region represents the area where wÃ does not exist as in Fig. 2 or NÃ\1, which does not verify necessary conditions for relevant solutions

Conclusions to solving a self-consistency equation that was not explicitly solved, but its qualitative solution behavior could A minimal mathematical model for upstream migration of be analyzed with a course of elementary calculation. ﬁsh schools in 1-D environment based on a game-theoretic Dependence of the optimal swimming speed uÃ and the formulation was presented and behavior of the associated optimal size NÃ of ﬁsh schools on the model parameters optimizers was analyzed both mathematically and numer- was theoretically analyzed for comprehending their quali- ically. The optimization problem was effectively reduced tative behavior. The analytical results obtained in

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Fig. 5 Diagrams of the area complying with the conditions relevant in this paper. The white region represents the area where 0\WÃ\100 and 1\NÃ\100 with a n ¼ 1, b n ¼ 2, c n ¼ 3, d these conditions are not satisﬁed n ¼ 4, e n ¼ 5, and f n ¼ 6, which are considered to be biologically

‘‘Qualitative analysis’’ showed that the dependence of the when necessary. However, analyzing such an extended optimal swimming speed uÃ and the school size NÃ derived model would require the use of a sophisticated numerical from the present mathematical model on the flow speed V technique and considering an unnecessarily complicated qualitatively agrees well with the experimental results model would not be beneficial from both theoretical and (Onitsuka et al. 2012a, b). This paper thus provided a practical viewpoints. The present mathematical model can theoretical explanation of the experimental results from a be a minimal model for migration of fish schools where the game-theoretic viewpoint, which is a first attempt to the flow of water and both advantages and disadvantages are author’s knowledge. Numerical analysis with the help of a taken into account. Its practical extension includes con- mathematically rigorous technique of fixed-point iteration sidering unsteady and inhomogeneous flow field in the generated a monotonically increasing sequence of ODE for longitudinal movements of fish schools. The approximate solutions to the self-consistency equation. The analytical results presented in this paper can be to some computed solutions to the self-consistency equation with extent valid for analyzing behavior of the optimal controls the appropriate ranges of the model parameters revealed when the rate of change of the flow speed V is not very quantitative behavior of the optimal controls, which vali- large, so that it does not exceeds the maximum swimming dated the theoretical results obtained from the qualitative speed of the individual fishes. Future research will use the analysis. The ranges of the realistic model parameter val- present mathematical models for analyzing upstream fish ues were estimated from the computational results. Cur- migration in existing river systems in Japan as their real rently, we are planning to perform artificially hatching applications with the help of numerical simulation tech- experiments of P. altivelis in a river with local fishery niques. Applicability of the present mathematical mod- cooperatives for the identification of the model parameters elling framework to migration of other animals, such as involved in the objective function. migratory birds (McLaren et al. 2014; Vansteelant et al. This paper could quantify dependence of the optimal 2017), will also be examined in future. controls uÃ and NÃ on the model parameters under sim- plified conditions, so that the most part of the analysis can Acknowledgements The River Fund No. 285311020 in charges of be performed exactly. Theoretically, it is possible to choose The River Foundation, JSPS Research Grant No. 15H06417 and No. 17K15345, and WEC Applied Ecology Research Grant No. 2016-02 more complex parameterization of the coefficients to more support this research. The author thanks to the reviewer for providing appropriately model migration dynamics of fish schools valuable comments and suggestions.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea Combining (53) and (55) then yields tivecommons.org/licenses/by/4.0/), which permits unrestricted use, 0\F0ðwÞ\h for wÃ [ w: ð58Þ distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a The present algorithm is thus a fixed-point type with a link to the Creative Commons license, and indicate if changes were contraction property whose convergence is guaranteed made. under the conditions assumed in this paper (Atkinson and Han 2005, Chapter 5). Appendix: Numerical algorithms Two iteration algorithms for finding the solution w ¼ wÃ, which is a biophysically relevant solution, are This supplementary material presents monotone iteration presented in what follows. The first (second) algorithm algorithms for approximating the largest positive solution generates a monotonically decreasing (increasing) to the self-consistency equation sequence of approximated solutions. The first algorithm is presented as wy w ¼ FðwÞ¼ ; ð49Þ (First algorithm) wy þ z w0 ¼ 1 and wrþ1 ¼ FðwrÞ for r 0: ð59Þ with the parameters y and z, such that y [ 1 and yÀ1 On the other hand, the second algorithm is presented as 0\z\ ðyÀ1Þ . Under the stated conditions, the Eq. (49) yy (Second algorithm) admits two positive solutions w1 and w2, such that w0 ¼ w and wrþ1 ¼ FðwrÞ for r 0: ð60Þ 0\w \w\w wÃ\1; ð50Þ 1 2 The only apparent difference between the two algo- yÀ1 rithms is the initial guess w . The following theorems hold with w ¼ y . Straightforward calculation shows 0 ! for these algorithms. 1 y À 1 ðy À 1ÞyÀ1 w À FðwÞ¼ À z [ 0; ð51Þ Theorem 1 The sequence w generated by the first algo- wy þ z y yy r rithm satisfies yzwyÀ1 wÃ\ ÁÁÁ\w \ ÁÁÁ\w \w ¼ 1; ð61Þ F0ðwÞ¼ [ 0 for w [ 0; ð52Þ r 1 0 y 2 ðw þ zÞ and converges to wÃ: and Theorem 2 The sequence w generated by the second r yÀ2 algorithm satisfies 00 yðy þ 1Þzw y À 1 y F ðwÞ¼ 3 À w for w [ w: Ã ðwy þ zÞ y þ 1 0\w ¼ w0\w1\ ÁÁÁ\wr\ ÁÁÁ\w ; ð62Þ ð53Þ and converges to wÃ: In addition, it is also shown that Proof of Theorem 1 By (51), the inequality w\FðwÞ for w \w\w : ð54Þ 1 2 w1 À w0 ¼ Fðw0ÞÀ1 Combining (49) and (52) yields the estimate ¼ Fð1ÞÀ1 ð63Þ yz ðwÃÞy ðwÃÞy yz \0 0\F0 wÃ ; 55 ð Þ¼ yþ1 Ã y Ã y ¼ yÀ1 ð Þ ðwÃÞ ðw Þ þ zðw Þ þ z ðwÃÞ holds. The inequality Ã Ã which reduces to w0 À w ¼ 1 À w [ 0 ð64Þ 0 Ã 0\F ðw Þ is obvious. Assume yz \ wÃ\w \ ÁÁÁ\w \w ¼ 1; ð65Þ wyÀ1 r 1 0 yÀ1 ð56Þ y ðy À 1Þ for a natural number l ¼ r. Then, for l ¼ r þ 1, the \ wyÀ1 yy inequality ¼ 1: wrþ1 À wr ¼ FðwrÞÀwr

Therefore, (56) shows ¼ FðwrÞÀFðwrÀ1Þ ð66Þ yz \0 0\F0ðwÃÞ\ h\1: ð57Þ wyÀ1 follows from (52) and (65). In addition, the inequality

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Ã Ã Castellano C, Fortunato S, Loreto V (2009) Statistical physics of w À wrþ1 ¼ Fðw ÞÀwrþ1 social dynamics. Rev Modern Phys 81:591–646 Ã ¼ Fðw ÞÀFðwrÞ ð67Þ Claireaux G, Couturier C, Groison AL (2006) Effect of temperature \0 on maximum swimming speed and cost of transport in juvenile European sea bass (Dicentrarchus labrax). J Exp Biol also follows from (52) and (65). Convergence of the 209:3420–3428 Cucco A et al (2012) A metabolic scope based model of fish response sequence wr follows from its monotonicity and bounded- to environmental changes. Ecol Model 237:132–141 Ã ness. The limit of the sequence is unique and is clearly w . Day RL et al (2011) Interactions between shoal size and conformity in Proof of Theorem 1 completes by the application of an guppy social foraging. Anim Behav 62:917–925 inductive argument. Theorem 2 can be proven in an Di-Poi C, Lacasse J, Rogers SM, Aubin-Horth N (2014) Extensive h behavioural divergence following colonisation of the freshwater essentially similar way. environment in three spine sticklebacks. PLoS One 9:e98980 Remark 3 It is possible to analytically derive a theoretical Ebeling W, Schweitzer F (2001) Swarms of particle agents with harmonic interactions. Theory Biosci 120:201–224 convergence estimate of the first algorithm. By (58), the Flecker AS, McIntyre PB, Moore JW, Anderson JT, Taylor BW, Hall Ã sequence jjwr À w ¼ cr satisfies RO Jr (2010) Migratory fishes as material and process subsidies Ã in riverine ecosystems. In community ecology of stream fishes: crþ1 ¼ jjwrþ1 À w concepts, approaches, and techniques. Am Fish Soc Symp Ã ¼ jjFðwrÞÀFðw Þ for r 0 73:559–592 Gueron S, Levin SA (1995) The dynamics of group formation. Math yÀ1 Ã ð68Þ h jjwr À w Biosci 128:243–264 hyÀ1c ; Guse B et al (2015) Eco-hydrologic model cascades: simulating land ¼ r use and climate change impacts on hydrology, hydraulics and habitats for fish and macroinvertebrates. Sci Total Environ which leads to 533:542–556 c hðyÀ1Þrc \hðyÀ1Þr r ; Hinch SG, Cooke SJ, Healey MC, Farrell AT (2005) Behavioural r 0 for 0 ð69Þ physiology of fish migrations: salmon as a model approach. Fish since c ¼ jj1 À wÃ \1. The convergence estimate which Physiol 24:239–295 0 Jager HI, Efroymson RA, Opperman JJ, Kelly MR (2015) Spatial does not explicitly depend on is derived as follows: design principles for sustainable hydropower development in ðyÀ1Þr river basins. Renew Sustain Energy Rev 45:808–816 cr\h for r 0: ð70Þ Kabalak A, Smirnova E, Jost J (2015) Non-cooperative game theory in biology and cooperative reasoning in humans. Theory Biosci 134:17–46 Katz Y et al (2011) Inferring the structure and dynamics of interactions in schooling fish. 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