数理解析研究所講究録
第 1663 巻 2009 年 170-175

170

Density-vague dynamics of biological populations
水産工学研究所丹羽洋智 (Hiro-Sato Niwa)

National Research Institute of Fisheries Engineering

1

Introduction

Biological populations fluctuate in abundance over time. These fluctuations are affected bothby extemal environmental factors independent of population density, and by density-dependentfactors. The long-term average about which a population fluctuates will ultimately be set by thedensity-dependent factors. The nature of the negative feedback relationship between population
growth rate and density is at the heart of population ecology. The existence and detection ofdensity dependence in real data sets has been, however, the subject of much controversy forseveral decades. The consensus emerging from this prolonged debate is now that, when sufficiently long mns of data are available, ecologists should be able to detect density dependence
from population time-series [1-91. The duration of a time series necessary to distinguish a regulated population trajectory from a random walk is still uncertain [8, 10, 11]; how long of a timeseries is required to be able to disentangle deterministic (density-dependent) signals from external noise in the ecological systems? When plotting data on the form of the density-dependent
relationship, ecologists have been confounded by considerable noise around each relationship,
i.e. ”density vagueness” [12]; explaining this bivariate scatter has remained a largely unrealized goal [13, 14]. The vagueness in the density-dependent relationship concems uncertainty
in locating the population equilibrium. The issue of indeterminacy of the density-dependentrelationship, which has not been analyzed quantitatively [15], is set in a formal mathematicalframework associated with the number of years in the census. I analyze the population renewal
process of marine exploited fishes in the North Atlantic. Data are extracted from the 208 working group reports of the Intemational Council for the Exploration of the Sea (ICES) prepared
for advice on living resources and their harvesting provided by the ICES Advisory Committee[16]. The analysis is based on the 38 commercial stocks, which are the same as the fish stocks
analyzed by Sparholt et al. [17].

2Stochastic population dynamics

For most fish species the breeding population consists of several age groups, and the mostuseful measure is usually the spawner abundance (biomass in tonnes). During year an
amount $Y(t)$ is harvested with a fraction to the harvestable adult population, $1-e^{-F(t)}$ , where$F(t)=\ln[1+Y(t)/S(t)]$ denotes the fishing mortality. Harvesting leaves a population size
$S(t)$ , i.e. the escapement of adults from the harvest. The biomass grows as a result of, not only
growth of escapees in body weight, but also of recmitment of offspring;is designated the$S$$t$$\Xi(t)$171

recruits to the adult population (counts of surviving progeny) in year . All density dependence
is assumed to be exerted by the adult population density [18]; this would apply ifjuveniles do
not compete with adults. The system is described by a discrete difference equation$t$$S(t+1)=\Lambda(S(t))S(t)+w_{0}\Xi(t+1)-Y(t+1)$(1)

with a growth-survival factor dependent on , whereis body weight at entry to the harvestable adult population. Choose as geometric means, the equilibrium escapement , recmitment- and yield . In a stationary state, Eq.(l), log-transformed, gives $\ln(1-w)+F_{\star}=$$\ln A(S.)$ with the stationary fishing mortality $F_{*}=\ln(1+Y_{\star}/S_{\star})$ , whereis the stationary ratio of each year’s recruitment (in biomass) to the harvestable adult population. Denoting the relative deviations from equilibrium point as $n(t)=\ln(S(t)/S_{\star}),$ $\xi(t)=$$\ln(\Xi(t)/--\star$and $f(t)=F(t)/F_{\star}-1$ , linearizing the (log-transformed) equation (1) yields$S$$\Lambda$$w_{0}$$S_{\star}$$=\star$$Y_{\star}$$w=w_{0-\star}^{-}-/(S_{\star}+Y_{\star})$’$n(t+1)-n(t)=-\gamma n(t)+w\xi(t+1)-F_{\star}f(t+1)$(2)

with constant coefficient $\gamma=1-(1-w)(1+$ dln dln
evaluated at equilibrium. The
successive difference $r(t)=n(t+1)-n(t)$ approximates the annual rate of increase of a
population, i.e. the per-unit-abundance growth rate. Stochasticity enters the population dynamics in two ways [19], both as environmental variation mirrored by recmitment variability
and as variation in the fishing mortality $f(t)$ . These namral and human-caused forces$(t)=w\xi(t)-F_{\star}f(t)$ are considered as extemal disturbances.
$\Lambda/$$S)_{*}$$\xi(t)$

3 Population-dynamic equilibrium
Figure lA shows the empirical variance trajectories of fish populations in the North Atlantic,
demonstrating the increase in temporal population variability over time scales as long as decades,
which are often used to describe temporal pattems of population fluctuations [20-23]. Recruitment and fishing rate are presumed to track population flucmations. Adjacent values in the timeseries of recruitment and catches-to-escapement ratio are correlated [24]; as the consequence
the delayed effect of density dependence appears in population dynamics [25]. For generating temporally structured noise,
and $f(t)$ are both assumed to be described as a first-order
autoregressive, AR(1), process [19, 21, 25, 26] with serial correlation coefficients
and .
AR(1)
The
coefficients are estimated for the examined time-series of length years, giving
and $\beta_{f}<e^{-1/L}$ which implies that both the processes ofand $f(t)$ contain constantmeans, i.e. there exist stationary equilibrium values for recruitment
and fishing mortality .
Consequently, the population is anticipated to exhibit a retum tendency to the equilibrium point. The population process with multiple decay-time constants $(-1/\ln(1-\gamma),$,and $-1/\ln\beta_{f})$ is considered to determine the total decay-time constant, i.e. the asymptotic
equilibration time.
The necessary calculations of the variance of a population with autocorrelated environment
were done more than three decades ago [26]; the result is generalized to connect the threeparameters
and to the population flucmations, the mean square successive difference 2and the variance in population size , the variances and of recruitment and fishing rate
. On the assumption that and are inter-independent, the impact of fishing on the population$\xi(t)$$\beta_{\xi}$$\beta_{j}$$L$$\xi(t)$$|\beta_{\xi}|$$—\star$$F_{\star}$$S_{\star}$$-1/\ln|\beta_{\xi}|$$T_{eq}$$\gamma,$$\beta_{f}$$\beta_{\zeta}$

$r$

$n2$

$f$

$n$

$\xi$

$f$

$\xi 2$

$f2$

$\xi$

172

A

$B$

Time lag (coarse-grained unit)

Time lag [years]

op 1: Variance trajectories. (A) The mean squaret-year difference of population size, $\sigma^{2}( t)=$$E[\{n$ ( $t+$t) $-n(t)\}^{2}]$ , increases with the census period. The solid circles are the average for theexamined populations, which, fitted to a power law (black line), increases as $( t)^{2H}$ with $H=0.55\pm$
0.01. The spawning biomass is walking a geometric random-walk [23] at short periods $t<10$ years.
(B) The coarse-grained plots of variance trajectories (the same as the data in panel A). The mean square-year differenceis plotted at $j=1,2,$ .
$\sigma^{2}(j\lceil T_{eq}\rceil)/2\sigma_{n}^{2}$$j\lceil T_{eq}\rceil$$\ldots$

dynamics is separable from the effects of environmental variability. Let
andwith A $=1-\gamma$ . Taking the mean squares over Eq.(2) and the-year difference of population size, $n(t+\triangle t)-n(t)$ , with AR(1) disturbance terms mediates
the following two relations, respectively,$\wedge\epsilon^{=w^{2}}2\xi 2/(1-\lambda\beta_{\xi})$$\wedge f2=F_{\star}^{2}f2/(1-\lambda\beta_{j})$$\Delta t$(3)$r2/2n2_{=\gamma-\sum_{k\in\{\xi,f\}}\beta_{k/}}\wedge 2kn2$

with subscript$k$representing$\xi$

or

$f$

, and(4)$\gamma\sum_{\Delta t=0}^{\infty}\rho(\Delta t)=1+n-2\sum_{k\in\{\xi_{2}j\}}\wedge k2\beta_{k}/(1-\beta_{k})$with the autocorrelation function$\rho(\Delta t)=(1-\gamma)^{\Delta t}+\rho_{1}(\Delta t)$, where the correction term
(5)$\rho_{1}(\Delta t)=n-2\sum_{k\in\{\xi_{J}j\}}\wedge k2\beta_{k}\frac{\lambda^{\Delta t}-\beta_{k}^{\Delta t}}{\lambda-\beta_{k}}$is due to memory effects in the variable recmitment and fishing mortality. From the recursion,$n(t)= \sum_{j=0}^{\infty}\lambda^{j}(t-j)$ , a corollary to Eq.(4) is obtained:$n2_{=\sum_{k\in\{\xi_{1}f\}}(1+\lambda\beta_{k})_{k}/(1-\lambda^{2})}\wedge 2$

Eqs.(3) and (6) yield the equation for.

(6)$\gamma$(7)$\gamma^{2}n2+(1-\gamma)r2_{=}\eta 2$

with$\eta 2=w^{2}\xi 2+F_{\star}^{2}j2$. According to the relation (Eq.7) connecting with population fluctu$\gamma$ations, Eq.(2) yields$\underline{n(t+1)-n(t)}r=-\gamma\underline{n}rn\underline{n(t)}+\sqrt{1-\gamma+\gamma^{2}\frac{n2}{\prime 2}}^{\underline{(t+1)}}\eta$’(8)173

implying that, if these populations are govemed by undercompensatory dynamics, extemalnoises mask signals for evidence of deterministic behavior of the systems. Unless the population
exhibits perfect- or over-compensation, the relationship between $r(t)[=n(t+1)-n(t)]$and $n(t)$ is plagued by large variance in growth rate; most of the changes in population size occur in a density-independent manner.
The average relaxation time of population fluctuations defines the equilibration time:$(\gamma\geq 1)$$T_{eq}$$\sum_{\Delta t=0}^{\infty}\rho(\triangle t)=\sum_{\Delta t=0}^{\infty}\exp(-\Delta t/T_{eq})$,(9)

whereis estimated with Eq.(4). After the time
memory of the initial conditions is lost:the population unbalance (i.e. deviation from the equilibrium size) is expected to decay away inan exponential fashion with a time constant given by the equilibration time . The asymptotic
elasticity with respect to change in population size measures the strength $D$ of total density
dependence [18]:(10)$E[n(t+1)-n(t)|n(t)=n]=-Dn$ ,$T_{eq}$$T_{eq}$$T_{eq}$i.e. the expectation of change in population size given $n(t)=n$ is equal to $100D$ percent
negative feedback. Since the total density dependence is the asymptotic multiple growth rateof population per year, $-\ln(1-D)$ reads the asymptotic decay constant. Therefore, the total
density dependence is estimated as$D=1-e^{-1/T_{eq}}$.

(11)

Accordingly, Eq.(10) is iterated to give the conditional mean of$T_{eq}$-year difference:$E[n(t+T_{eq})-n(t)|n(t)=n]=-(1-e^{-1})n$ .

(12)

I now apply the above theory to examine the density dependence by using the fish population
time-series in the North Atlantic. Since the length of each time-series data is longer than theequilibration time, $L>T_{eq}$ , the examined populations flucmate about the time-series average;
the population changes in the-year interval level out (Fig. lB). The relationship of $r(t)/D$ to$n(t)$ contains much noise in population growth (Fig. $2A$ ). There is no or negligible correlation
between $r(t)$ and $n(t)$ . On the other hand, at large temporal scales, it is expected thatthe time series is mean-reverting and that the negative relationship (Eq. 12) is testable. These
are indeed found: when $y=n(t+T_{eq})-n(t)$ are plotted against $x=n(t)$ , the data fall aroundthe line $y=-x$ as shown in Fig. $2B$ . The result quantifies the fact that the number of years
monitored has a strong positive influence on the evidence for density dependence.
$T_{eq}$$\Delta t>T_{eq}$4 Indeterminacy
When one performs some observations for actual years , the uncertainty in locating the population equilibrium,
SD , i.e. the standard deviation of the empirical mean, propagates into the density dependence. The variance of is wnitten as [18, 21],
$L$$\Delta\overline{n}=$$L^{-1} \sum_{t=1}^{L}n(t)$$\overline{n}=$$[\overline{n}]$$\overline{n}$

174

A

Population unbalance

Population unbalance

2: Population growth diagram on fine- and coarse-grained scales. (A) Aggregated plots of annual
growth rate (divided by $D$ ) against population abundance. The coefficient of determination $R^{2}=0.05$ .
x-and y-axes are scaled by . The solid line represents $y=-x$ . $(B)$ Relationship between-year
growth rate and population unbalance from equilibrium. $R^{2}=0.44$ .
$\lceil T_{eq}\rceil$$\sigma_{n}$Var. Making use of the explicit form for the autocorrelation func$=[2\sum_{\Deltat=0}^{\infty}\rho(\Deltat)-1]n2L^{-1}+O(L^{-2})$ , giving, by virtue of Eq.(9), ation yields Var[fi]time-equilibrium uncertainty relation$n2L^{-2} \sum_{i_{\theta}=1}^{L}\rho(i-j)$$[\overline{n}]=$(13)$\Delta\overline{n}/n=\sqrt{L_{c}}/L\pi+\mathcal{O}(L^{-3/2})$

with $L_{c}=$given by$1+2/(e^{1/T_{q}}\cdot-1)\pi$ ,$\epsilon L^{-3/2}\sqrt{\pi}/4L_{c}$where $O(L^{-3/2})$ denotes a remainder term of order $L^{-3/2}$

with.

$\epsilon=-\frac{2\lambda(1-\lambda^{L})}{(1-\lambda)^{2}}(1+\sum_{k\in\{\xi_{1}f\}}\frac{\beta_{k}\wedge k2}{\lambda-\beta_{k}n2})-\sum_{k\in\{\xi_{1}f\}}\frac{2\beta_{k}(1-\beta_{k}^{L})\beta_{k}\wedge 2k}{(1-\beta_{k})^{2}\lambda-\beta_{k}n2}$(14)

Any observation over a short duration is associated with large indeterminacy. The population
equilibriumchosen as the sample mean and the number of years in the census are complementary: the uncertainty of population equilibrium is decreased by increasing the durationof observation.
For normally distributed with $E[n|n\gtrless 0]=\pm n\sqrt{2}/\pi$, the inequality $L_{c}<L$ is translated
into. It is then expected that the negative relationship between mean
growth rate and mean abundance, conditioned on a current population size that is larger (orsmaller) than the equilibrium size, is visible. It is difficult to judge whether the system isheading toward the equilibrium point in the time series of length less than the complementary; then $E[n|n\gtrless 0]$ are in the uncertain range oftime. Though,when plotting amual growth rate against population size, the relationship well looks like a
shotgun pattem (Fig. $2A$ ), the negative relationship with marked correlation can be seen as anaverage in the time-series data over $L>L_{c}$ years (Fig. $3A$ ); after averaged over sizes $n>0$$(or n<0)$ in the data comprising $L<L_{c}$ years, the dependence of $E[r|n\gtrless 0]$ on $E[n|n\gtrless 0]$
remains vague (Fig. $3B$ ).
$\ln S_{\star}$$n$$\Delta\overline{n}<\pm E[n|n\gtrless 0]/\sqrt{\pi}$$L_{c}[=2\pi T_{eq}+\mathcal{O}(T_{eq}^{-1})]$$\pm\sqrt{2}\Delta\overline{n}$

175

A

$B$

$\Pi_{\pm}$(scaled)

図 3: Diagrams (scaled by) of mean growth rate$\sqrt{2}\overline{n}$$\Pi_{f}$$\overline{[r]}_{\pm}$(scaled)(divided by $D$ ) versus mean abundance, conditioned on a current population unbalance $n\gtrless 0$ . lhe negative feedback only becomes
statistically visible with an observation series of length $L>L_{c}$ years panel $A;R^{2}=0.62)$ ; the bivariate
plots fall around the black line $y=-x$ . For $L<L_{c}$ , the diagrams are plagued by much scatter (panel ;$\overline{[n]}_{\pm}$$($$R^{2}=0.35)$.

References[1] Hassell MP, Latto J, and May RM. J. Anim. Ecol. 58, 883 (1989).
[2]Hanski I. Phil. Trans. R. Soc. Lond. B330,141 (1990).

[3] Solow AR and Steele JH. J Anim. Ecol. 59, 1073 (1990).
[4] Godfray HCJ and Hassell MP. Nature 359, 673 (1992).
[5] Holyoak M and Lawton JH. Oecologia 91, 425 (1992).
[6] Woiwod IP and Hanski I. J. Anim. Ecol. 61, 619 (1992).

[7] Dennis B and Taper ML. Ecol. Monogr 64, 205 (1994).
[8] Shenk TM, White GC, and Bumham KP. Ecol. Monogr 68, 445 (1998).
[9] Brook BW and Bradshaw CJA. Ecology 87, 1445 (2006).

Connell JH and Sousa WP. Am Nat. 121, 789 (1983).
Hixon MA, Pacala SW, and Sandin SA. Ecology 83, 1490 (2002).
[12] Strong DR. Trends Ecol. Evol. 1, 39 (1986).
[13] Krebs CJ. Phil. Trans. R. Soc. Lond. B357, 1211 (2002).
[14] Sibly RM and Hone J. Phil. Trans. R. Soc. Lond. B357. 1153 (2002).
[15] Murdoch R. Ecology 75, 271 (1994).
[16] ICES. ICES Advice 2008, Report of the ICES Advisory Committee, Books 1-10 (2008).
[17] Sparholt H, Bertelsen M, and Lassen H. ICES J. Mar Sci. 64, 707 (2007).
[18] Lande R, et al. Am. Nat. 159, 321 (2002).
[19] Jonz6nN, Ripa J, and Lundberg P. Am. Nat. 159, 427 (2002).
[20] Pimm SL and Redfeam A. Nature 334, 613 (1988).
[21] McArdle BH. Nature 338, 628 (1989).
[22] Inchausti P and Halley J. Science 293, 655 (2001).
[23] Niwa HS. ICES J. Mar Sci. 64, 496 (2007).
[24] Niwa HS. Aquat. Living Resour 19, 195 (2006).
[25] Royama T. Ecol. Monogr. 51,473 (1981).
[26] Roughgarden J. Am Nat. 109, 713 (1975).
[10][111$B$