Matsuda, Hiroyuki

Matsumiya, Yoshiharu

Takeuchi, Yasuhiro

Wada, Tokio

Kyoto University Research Information Repository

Title Cyclic Advantage Hypothesis for the Species Replacement ofPelagic Fishes.(Mathematical Topics in Biology)Author(s)Matsuda, Hiroyuki; Wada, Tokio; Takeuchi, Yasuhiro;Matsumiya, YoshiharuCitation数理解析研究所講究録 (1994), 870: 173-183Issue Date1994-05URL http://hdl.handle.net/2433/84011RightType Departmental Bulletin PaperTextversionpublisherKyoto University173Cyclic Advantage Hypothesis for the Species Replacementof Pelagic Fishes.Hiroyuki Matsuda, Tokio Wada \dagger , Yasuhiro Takeuchi \Sand Yoshiharu Matsumiya\PDeparrment of Biology, Kyushu University, Fukuoka 812, Japan\dagger Hokkaido National Fisheries Research Institute, Kushiro 085, Japan\Spaculty of Engineering, Shizuoka University, Hamamatsu 432, Japan\PFaculty of Bioresource, Mie University, Tsu 514, Japan: 3( ) ( )( ) ( )Abstract:We consider a mathematical model which incorporates two factors affecting long-tern fluctuationof planktotrophic pelagic fish: environmental fluctuation and interspecific competition. Long-term catch data of pelagic fishes in Japan suggest that the chub mackerel (labeled A) wasreplaced by the sardine (B), $B$ was replaced by the anchovy, Pacific saury and horse mackerel,and these species (labeled group C) were replaced by A. If species A defeats $B,$ $B$ defeats $C$ and$C$ defeats A in interspecific competitive ability, the mathematical model predicts that theabundance of these three groups fluctuate forever and dominate in the cyclic order. We call thiscyclic advantage hypothesis for species replacement. In this model, environmental fluctuationgreatly affects when the next replacemtnt occurs, whereas cyclic relashionship in competitiveability determines what is the next dominant.870 1994 173-183174It is known that the catch abundance fluctuation of planktotrophic pelagic fishes is enormouslylarge (fig. 1). This fluctuation qualitatively reflects the stock abundance fluctuation from year toyear. The sardine (Sardinops melanostictus) was abundant in $1930s$ and in $1980s$ . Theanchovy(Engraulisjaponica), Pacific saury (Calorabis saira) and horse mackerels (Trachunsjaponica and Decapterus muroadsi) were abundant in ca. 1960 and they are becoming dominantin 1990. The chub mackerel (Scomberjaponicus) was dominant in $1970s$ and is at a very lowstock level.The purpose of this paper is to elucidate what is a major factor on such fluctuations.Some considered that the fluctuation of environmental condition is important. When the sardineincreased rapidly in early $1930s$ and ca. 1980, the water temperature at the spawning area wassignificantly high (Sugimoto pers.comm.). Other peOple considered that interspecificrelationships between these pelagic fishes are important. When the sardine stock declined in$1940s$ and now, the anchovy, Pacific saury and horse mackerel increased. When these 3 speciesdecreased, in ca. 1970, the chub mackerel increased. When the chub mackerel decreased in1980, the sardine became the dominant. It is called the species replacement of the pelagic fishes.Some other people considered that both environmental fluctuation and species interactionare important for the stock fluctuation. I agree to this argument. I will show that the speciesinteraction is a major factor on which species is the next dominant, and that the short-termfluctuation of environment determines when the next replacement occurs. Our hypothesis iscalled “the cyclic advantage hypothesis“ for the species replacement of the planktorrophic pelagicfishes.We have three questions: (1) Can we predict which fish will be the next dominant? Ouranswer from the cyclic advantage hypothesis is “Yes.“ After the sardine began to decline, theanchovy and other two species become to be the dominant. After the anchovy, the chub mackerelis the dominant. Finally, the sardine comes back to be the dominant after the chub mackerel.175$\underline{\approx\infty}$$.\underline{-}$$\overline{\overline{O}}$$\frac{\sum_{O}}{\circ\infty}$510 20 30 40 50 60 70 80 901905-1991Figure 1. The long-term fluctuation of catch amount of pelagic fishes inJapan (Kawai&Takahashi 1983; Chikuni 1985, see also Matsuda et al.$1992b)$The second question is: (2) can we predict when the next species replacement occurs?Our answer is said to be “No“ if the environmental fluctuation is not predictable. This decidedlydepends on the short-term fluctuation of environmental conditions. Our final question is: (3) canwe make an optimal harvesting policy for each fish independent of other fishes? Our answer is$|No$ .“ The optimal harvesting policy for each fish depends on which fish is currently dominant.When the sardine was dominant about several years ago, the future of chub mackerel was verypessimistic. However, when the anchovy was dominant as recently, the future of chub mackerelis a little bit optimistic.There are at least four characteristic features of the long-term stock fluctuation of thesardine: (1) This has a long $period|\dagger$ of fluctuation. About 50 years passed from the last peak(ca. 1930) in the sardine stock to th$e$ last peak (ca. 1980), in spite of the fact that th$e$ generationtime of the sardine is two or three years. (2) The fluctuation has a long history, at least for about176300 years. This suggests the stock fluctuation of the sardine is not induced by overexploitation.(3) There is a huge variation of the peak to the bottom in the stock abundance. According to thecatch amount data, the ratio of the peak (1988: 4,490 kt) to the bottom (1967: 10 kt) is about450fold. This, however, seems to be underestimation. According to the estimated annual eggproduction, the peak (1986: 8985 billion $eggs/year$) is about 9,000 times larger than the bottom(1970: 1 billion $eggs/year$) as shown in fig. 2. The forth characteristic feature is that (4) bothhigh-density phase and low-density phase are long. These continue for several generation time.year (1950-89)Figure 2. The long-term fluctuations of estimated annual eggproduction of the sardine and chub mackerel in the Pacific $oc$ean offJapan (compiled by the Fisheries Agency of Japan, see also Matsuda etal. $1992a$)From the viewpoint of population ecology, there are several dynamic models whic$h$ canexhibit a permanent oscillation of stock abundance. The simplest model is the single populationdynamic model. It is known as Ricker’s equation in fisheries science:$N_{t+1}=N_{t}\exp[r\langle 1-N_{t}/K)]$ , (1)where $N_{t}$ and $N_{t+1}$ are respectively the stock abundances at the present and the next generations,$r$ and $K$ respectively denotes the intrinsic rate of increase and the carrying capacity. If $r$ issufficiently large, this system exhibit a chaotic behavior with a permanent oscillation. The low177density phase continues for several generations, increases rapidly and collapsed drasticaUy (fig.$3a)$ . Figure $3b$ illustrates the reproduction curve of this system with the same parameter value asin fig. $3a$. Note that, in this system, a high density phase can not continue for more than onegeneration. Although we know a typical example of this dynamics in nature: Lemming dynamics(Pitelka 1972), this does not explain the sardine‘s fluctuation.(a) (b)Figure 3. $(a)A$ simulated fluctuation and (b) the re-production curveof the $Ricker^{I}s$ eqn (1). $r=4.5,$ $K=10$ .Although somebody consider the two species model for the sardine‘s dynamics. TheLotka-Volterra competition model is written as:$dN_{1}/dt=(r_{1}-a_{11}N_{1}-a_{12}N_{2})N_{1}$ (2a)$dN_{2}/dt=(r_{2}-a_{21}N_{1^{-}\% 2}N_{2})N_{2}$ (2b)where $N_{i}$ denotes the stock abundance of species $i,$ $r_{i}$ is the intrinsic rate of increase of species $i$ ,$a_{ij}$ is the competition coefficient of species $j$ from $i$ . This system, however, does not produce apemlanent oscillation: the two species coexist at an equilibrium state (fig. $4a$) or either specieswins (figs. $4b$ and $4c$).178$N_{1}$ $N_{1}$$N_{1}$Figure 4. Phase diagrams of competition model (2): (a) two speciescoexist at an equilibrium, (b) either species goes to extinction, and $(c)$species 2 always wins. Closed and open circles are respectively stable andunstable equilibria.Lotka-Volterra prey-predator dynamics is written by$dN/dt=(r-aN-bP)N$ (3a)$dP/dt=(-\delta+bN)P$, (3b)where $N$ and $P$ respectively denote the prey and predator abundances, $r,$ $a,$ $b$ and $\delta$ are positiveconstants. It is well known that prey-predator dynamics produces permanent oscillations ofboth, not either, prey and predator populations. However, we have no data of long-termfluctuation of the prey or enemy of the sardine.Thus this does not explain the sardine’sfluctuation either.A system with two prey and one predator is written by$dN_{1}/dt=(r_{1}-a_{11}N_{1}-a_{12}N_{2}-b_{1}P)N_{1}$ (4a)$dN_{2}/dr=(r_{1}-a_{11}N_{1}-a_{12}N_{2}-b_{2}P)N_{2}$ (4b)dI)/dt=(-\delta $b_{1}N_{1}+b_{2}N_{2}$)$P$ (4c)where $N_{1},$ $N_{2}$ and $P$ respectively denote the prey abundance of species 1, 2 and the predatorabundance. It is known that this system $c$an produce permanent oscillations of two prey, but thepredator density does not fluctuate very much (Fujii 1977). The parameter region which exhibitssuch a pernanent fluctuation is very small.179We propose the cyclic advantage model which consists of 3 competing species written as:$N_{1}’=c_{1}+N_{1}\exp[r_{1}-a_{11}N_{1}-a_{12}N_{2}-a_{13}N_{3}]$ (5a)$N_{2^{=c_{2}+N_{1}\exp[r_{2}-\% 1^{N_{1}-\% 2^{N_{2}-a_{23}N_{3}]}}}}’$ (5b)$N_{3}’=c_{3}+N_{1}\exp[r_{3}-a_{31}N_{1}-a_{32}N_{2}-a_{33}N_{3}]$ (5c)where $N_{i}$ and $N_{j}’$ respectively denote the stock abundance of species $i$ at the present and the nextgenerations, $c_{i}$ is a positive constant representing a constant immigration from outside, $r_{l}$ is theintrinsic rate of increase of species $i$ and $a_{ij}$ is the competition coefficients of species $j$ from $i.$ . Ifspecies 1 defeats 2, 2 defeats 3 and 3 defeats 1, these 3 species can fluctuate permanently (Gilpin1975). In mathemati$c$al terms, these cyclic advantage conditions in competitive ability $wh$en $c_{i}$ issufficiently small are written $by;the$ following 6 inequalities:$a_{12}r_{2}>a_{11}r_{1}>a_{13}r_{3}$ ($N_{2}\Uparrow,N_{3^{\Downarrow}}$ at $N_{1}=r_{1}/a_{11}$ ) (6a)$\% 3^{r_{3^{>}\% 2^{r}2^{>}\% 1^{r_{1}(N_{3}\Uparrow,N_{1^{\Downarrow}}}}}$ at $N_{2}=r_{2^{/}\% 2)}$ (6b)$a_{31^{\gamma}1^{>}\% 3^{\Gamma}3^{>a_{32}r_{2}(N_{1}\Uparrow,N_{2^{\Downarrow}}}}atN_{3}=r_{3\% 3}/$) (&)The first two inequalities shown in the first line implies that species 2 increases and 3 does notincreases when species 1 is dominant and nearly reaches its carrying capacity, $r_{1}/a_{11}.Th\overline{e}$ secondline implies that species 1 does not increase but 3 increases when 2 is dominant. The last lineimplies that species 2 does not increase but 1 increases when 3 is dominant.We consider such a cyclic advantage relationship actually exists in pelagic fishes. Thethree species are the sardine, anchovy and chub mackerel. In fig. 1, we summarized theanchovy, Pacific saury and horse mackerel into a single group, because the $c$orrelationcoefficient in annual catch fluctuation of any pair of these three species are high (Table 1).180In the previous models, we implicitly ignored the environmental fluctuation. However,there is a large fluctuation of environmental conditions in nature. The environmental fluctuationaffects on the survival rate of pelagic fishes at an early life stage. For simplicity, we consider thatthe rate of increase fluctuates due to environmental fluctuation from time to time, whereas thecompetitive coefficients are assumed to be constant. Random fluctuation of environment isexpressed by$r_{i}=r_{i^{*}}+f_{i}Z_{i}(t)$ , (7)where $r_{i^{*}}$ is the average value of $r_{i},$ $f_{i}$ ig the amplitude of variation of $r_{i}$ , and $Z_{i}(t)$ is the time-dependent random variable with no auto-correlation, which is called white noise. In computersimulations, we choose $Z_{j}(t)$ as the unifolm random variable between-0.5 and 0.5.Figure 5 illustrates resultant stock fluctuations from simulations with the same parametervalues of $r_{i^{*}}$ and $a_{\dot{\tau}j}$. Under a constant environment which corresponds a case where $f_{i}=0$ and $r_{i}$$=r_{i^{*}}$ , the dynamic$a1$ behavior would be recognize$d$ as a limit cycle. The period of this cycle isabout 51 time units. The species replacement occurs in cyclic order, ABCABC... Under arandomly fluctuating environment, the stock abundance unpredictably varied with time. Even a181(a) constant environment$\Phi N$’$\#O$$\circ$$\overline{\overline{\underline{\alpha}}}$time unit(b) random environment$\Phi N$$\dot{\overline{\infty}}$$\frac{4OQ}{\infty}$$\omega\succ$$\overline{\approx v}$time $un\ddagger t$Figure 5. Simulations of the cyclic advantage model under (a) constantenvironment and (b) randomly fluctuating environment. Parameter values are:$(r_{1}, r_{2}, r_{3})=(0.9,0.8,0.7),$ $(a_{11},$ $a_{12},$ $a_{13},$ %1’ %2’ %3’ $a_{31},$ %2’ %3 $)$ =$(0.2, 0.4, 0.1, 0.1, 0.2, 0.4, ().4,0.1,0.2),$ $c_{j}=0.02,$ $f_{i}=0$ for (a) and $f_{i}=1$for (b).182case when the survival rate at an early life stage (corresponding the value of $r_{i}$ ) fluctuates 10times from year to ye$ar$ , the species replacement is infrequently and occurs in still cyclic order.Th$e$ “period” of stock fluctuation is not constant; often falls in 6$()$ to 90 time units. That is whywe can not predict when the next replacement occurs but we can predict which is the nextdominant as mentioned above.We have three major conclusions: (1) Th$e$ cyclic advantage hypothesis predicts which isthe next dominant. The sardine increases after the chub mackerel is dominant. The anchovyincreases after the sardine. The chub mackerel increases after the anchovy. This is testable. If thesardine increases just after the anchovy is dominant, this hypothesis falls into error. This istestable in the future in spite of the fact that long years must be required. (2) Cyclic advantage$h$ypothesis $d$oes not predicts when the next replacement occurs, which depends on the short-tern fluctuation of the environment. The environmental factor affects on the species replacementas a tngger.Furthermore, (3) species replacement of pelagic fishes is inevitable. We should notexpect a constant catch amount of each species every year. We should make a fish market whichdemands the currently dominant fish species. Canadian and American people can consume thePacific saury when this is abundant. The mortality from exploitation is not significant when thefish is dominant, but is critical when it is rare. Fishermen should focus their catch effort on thecurrently dominant fishes.$Y$We would like to express sincere thanks to Dr T. Kishida and Ms T.Takahashi forvaluable comments an$d$ kindly helping throughout the study.183ReferencesChikuni, S. (1985) The fish resources of the north-west Pacific. $Fao$ Fish.Tech. Paper 266: 1-190.Fujii, K. (1977) Complexity-stability relationship of two-prey-one-predator species systemmodel: local and global stability. J.Theor.Biol. 69:613-623.Gilpin,M.E. (1975) Limit cycles in competition communities. $Am$ .Nat. 109: 51-60.Kawai,T. an$d$ Takahashi,T. (1983) Changes of species composition and diversity of Japanesefishes in the 20th century by each region and living layer. $Dat$. Col. Tokai $Reg$ . Fish. Res.Lab. No.ll: 1-128. (in Japanese).Matsuda,H., Wada,T., Takeuchi,$Y$ and Matsumiya,$Y$ (1991) Alternative models of speciesreplacement of pelagi$c$ fishes. Res.Popul.Ecol. 33:41-56.Matsuda,H. Kishida,T. and Kidachi,T. (1992a) Optimal harvesting policy for chub mackerel(Scomberjaponicus) in Japan under a fluctuating environment. Can.J.Fish.$Aq$ .Sci.49: 1796-1800.Matsuda,H., Wada,T., Takeuchi,$Y$ and Matsumiya,$Y(1992b)$ Model analysis of th$e$ effect ofenvironmental fluctuation on the species replacement pattern of pelagic fishes underinterspecific competition. Res. Popul. Ecol. 34:309-319.Pitelka, F.A. (1972) Cycle pattem in lemming populations neax BEUrOW, Alaska. Proc. 1972Tundra Biome Symposium (Univ. of Washington), pp.132-135.

Cyclic Advantage Hypothesis for the Species Replacement of Pelagic Fishes.(Mathematical Topics in Biology)

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Cyclic Advantage Hypothesis for the Species Replacement of Pelagic Fishes.(Mathematical Topics in Biology)

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