Natural-anthropogenic regimes in simple models of the global marine fisheries Текст научной статьи по специальности «Биологические науки»

УДК 330.5

DOI: 10.33764/2618-981 Х-2019-3-1-210-222

ПРИРОДНО-АНТРОПОГЕННЫЕ РЕЖИМЫ В ПРОСТЫХ МОДЕЛЯХ ГЛОБАЛЬНОГО МОРСКОГО РЫБОЛОВСТВА

Александр Владимирович Рыженков

Институт экономики и организации промышленного производства СО РАН, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 17, доктор экономических наук, доцент, ведущий научный сотрудник, тел. (383)330-25-46, e-mail: ryzhenko@ieie.nsc.ru; Новосибирский национальный исследовательский государственный университет, 630090, Россия, г. Новосибирск, ул. Пирогова, 1, профессор кафедры политической экономии экономического факультета, тел. (383)363-42-49

Исследование применяет метод системной динамики для развития моделей глобального морского рыболовства Всемирного банка в интересах достижения целей устойчивого развития ООН. Ключевыми переменными выступают глобальный морской рыбный запас, его естественный прирост, а также промысловое усилие и вылов рыбы. Выведены уравнения воспроизводства рыбных запасов в режимах максимально устойчивого вылова, перелова или недолова. Определены временные рамки до коллапса при характерных разновидностях перелова. Предложена новая положительная обратная связь в регулировании глобального морского рыбного запаса для асимптотического обеспечения максимально устойчивого вылова.

Ключевые слова: экосистемный подход к рыболовству, биоэкономическая модель, максимально устойчивые вылов и запасы, подвергающиеся перелову запасы, эксплуатируемые с недоловом запасы, регулирование рыболовства, обратная связь.

NATURAL-ANTHROPOGENIC REGIMES IN SIMPLE MODELS OF THE GLOBAL MARINE FISHERIES

Alexander V. Ryzhenkov

Institute for Economics and Industrial Engineering SB RAS, 17, Prospect Аkademik Lavrentiev St., Novosibirsk, 630090, Russia, D. Sc., Associate Professor, Leading Researcher, phone: (383)330-25-46, e-mail: ryzhenko@ieie.nsc.ru; Novosibirsk National Research State University, 1, Pirogova St., Novosibirsk, 630073, Russia, Professor, Chair of Political Economy the Faculty of Economics, phone: (383)363-42-49

The study applies the system dynamics method in upgrading the World Bank's global marine fisheries models in the interest of achieving the UN goals of sustainable development. The key variables are a global marine fish stock, its natural growth, as well as the fisheries' effort and catch. The equations for the reproduction of the fish stock in the regimes of maximum sustainable yield, overfishing or undercatch are derived. Determined are collapse time frames in result of typical overfishing varieties. A new positive feedback loop has been proposed in regulating global marine fish stock to ensure maximum sustainable yield asymptotically.

Key words: ecosystem approach, bioeconomic model, maximum sustainable yield, maximally sustainably fished stocks, overfished stocks, underfished stocks, fishery management, feedback loop.

Introduction

The state of global marine fishery resources, based on FAO's monitoring of assessed marine fish stocks, has continued to decline [1, p. 6]. This requires substantial improvement of fishery management in the framework of ecosystem approach to renewable natural resources.

Substantial struggles are needed for achieving a target (14.4) set for marine fisheries among the UN Sustainable Development Goals [2, p. 24 (35)]: "By 2020, effectively regulate harvesting and end overfishing, illegal, unreported and unregulated fishing and destructive fishing practices and implement science-based management plans, in order to restore fish stocks in the shortest time feasible, at least to levels that can produce maximum sustainable yield as determined by their biological characteristics."

The reader can explore [3, pp. 6-8, 60] as a valuable source for detailed explanation of the terminology and the established models in the field. Thereby the terms "depletion" and "depleted" describe respectively the process of decline in biomass and the state of a stock driven well beyond its level of maximum productivity (referred to as the maximum sustainable yield - MSY - level). The term "overfished" is a root term for "overexploitation", "depletion", "collapse" and may overlap with "rebuilding" and "recovery".

Depletion is graded depending on severity. Collapses occur because of excessive fishing, natural climate-induced calamities or both. Acute depletion may lead to collapse and even to extinction as a possible extreme outcome if rebuilding is not undertaken in time.

The terms "recovery" and "rebuilding" have been used interchangeably in the stock assessment literature. In the aggregated research approach of the present paper, the rebuilt or restored state is achieved whenever an overfished stock has been reestablished to the rebuilding target level (particularly, Xc in Table 3 below).

In disaggregated studies, the term "rebuilding" is more demanding: it means a more comprehensive re-establishment of particular depleted stocks, including age structures, evolutionary mechanisms, and population traits. Rebuilding is assumed to require much more time to be achieved than time needed to recover or restore over-fished stocks.

According to the control theory, open-loop control is completely determined at the initial instant t0; here, the integration of the equation (or equations) of motion for fixed initial conditions defines the phase trajectory x(t) of the states of the system [4]. Closed-loop control (with feedback) assumes the definition of control as a function of phase coordinates and time (ibid.). These concepts have wide theoretical and applied significance for bioeconomic theory and practice, in particular.

Science-based management plans are at the core of an adaptive management process that includes regulative feedback loops at different time scales based on past and present observations and experiences. This requires equilibrium and, especially, non-equilibrium system dynamics models of global marine fisheries. Such models have to propose policies (harvesting control rules) that can bring global marine fisheries from worrying disequilibrium closer to a state that supports MSY.

A World Bank's complete bioeconomic model (WB model) allows assessing the sustainability of the global marine fishery [5]. It contains endogenous variables involved in feedback loops: biomass, gross and net biomass growth, harvest. Several economic variables are not engaged in feedback loops: fishing effort, fish price, costs, revenues and net benefits. This model is used, in particular, for comparing how fast fisheries' recovery can be expected depending on the fishery's governance structure supported by interwoven feedback loops for endogenous variables.

This brief paper analyses a reduced WB model that abstracts from fish price, costs, revenues and net benefits. This model, supported by real data and by application experiences, has a potential for significant modification. The serious disadvantage of this model (as well as of the complete WB model) is exogenous fishing effort to be healed in an upgraded model. The notion of MSY is supplemented by notion of fitting sustainable effort (FSE) in this paper.

W-1, W-2, W-3 and finally W-4 are acronyms for the models considered. A higher number corresponds to a higher degree of sophistication measured by a quantity of feedback loops involved. Therefore, each subsequent model generalizes the preceding one. The reduced WB model is denoted as W-3 hereby.

The author uses the concept of aggravation modes (regimes), investigated in different contexts, in particular, in [5-9]. Especially dangerous are those arising from dominance of the positive feedback connecting biomass and the rate of its net change when the biomass diminishes faster and faster. On the other hand, dominance of such positive feedback loop can foster recovery of the depleted stock up to the point when a maximal positive growth rate of biomass is reached. Afterwards the growth rate of biomass asymptotically declines to zero when the biomass approaches stock that can open-endedly support MSY as in W-3 and W-4 below.

Natural net growth of biomass in W-1

Fish hatch (give birth), grow to maturity, lay eggs and die. Fish death rate is the number of fish per year that die from causes other than fish harvesting. Factors of fish population simple growth are reflected by the Pella-Tomlinson net biomass growth function as a specific non-linear autonomous differential equation [5, 10 and 11]:

x = ^ (x) = ax - px y. (1)

The initial assumptions for (1) are as follows: the rate of reproduction of the population is proportional to its current level; the second term of the equation reflects intraspecific competition for resources, which limits the growth of the population, or, in plain words, the death rate increases as crowding increases.

If y = 2, the Pella-Tomlinson function becomes the well-known logistic function [12], and as y approaches unity, it converges to the Fox biomass growth function [13].

Tables 1 and 2 reflect variables and parameters of W-1 and subsequent models.

Table 1

The variables of the biomass models

Variable Notation Measurement unit

Catch y, c fish mln tons /year

Fish stock (biomass) x fish mln tons

Carrying capacity X (a / ß)1/( ^ fish mln tons

Birth rate ax fish mln tons / year

Death rate ßx^ fish mln tons / year

Net change of fish stock x fish mln tons / year

The growth rate of fish stock x l/year

The growth rate of catch y l/year

Table 2

The parameters and base-year quantities

Characterization Values How obtained

Biological coefficients

Intrinsic growth rate a 1.644 Calculated

P (component of death rate) 0.45 Calculated

Pella-Tomlinson exponent y 1.188 Estimated

Bioeconomic coefficients

Catchability q 1.76 Calculated

Schooling parameter b 0.71 Estimated

Base-year (2012) quantities

Fishing effort e(20l2) 1 Normalized

Biomass x(20l2), fish mln tons 214.9 Calculated

Landed quantity _y(20l2), fish mln tons / year 79.69 Estimated

The derivative of the natural net change is defined as

^ = a - Pyx y-1. (2)

The stationary states are found from the condition that the right-hand side of (1) is equal to zero. They differ qualitatively and quantitatively.

On the one hand, carrying capacity x1 = X is asymptotically stable node, since i i

(xi) = (X) = a(l - y) < 0 for y > 1, on the other hand, x2 = 0 is unstable node, as ^x(x2) = a>

The population growth is S-shaped. The biomass tends to X that can sustain most of random external shocks except huge calamities. W-l is structurally stable.

Exogenous catch in W-2

W-2 additionally assumes a harvesting control rule implying that human fishing activities reduce the increase in the fish population by catch amount y = c = const > 0:

x = f (x) = ^(x) - c. (3)

Next equation defines the rate of growth of the stock

x = a - Pxy-1 - c / x, (4)

where the hyperbolic element is potent of biomass extinction through aggravation mode for some c > 0. A birth of the aggravation mode results from the transition from

dominant negative feedback x->x ^ x to dominant positive feedback

2 2

x ^ x ^ x at a tipping point, when the sign of dx / dx = -P(y -1)xy- + c / x < 0

turns into its opposite. Quite dramatically for x ^ 0 x ^-ro and dx / dx ^ Notice that x = 0 reflecting extinction is not literally mathematical equilibrium.

W-2 can possess two, one or no stationary states depending on the parameters values. Consider one distinct stationary state at first.

i

Lemma 1. The line y = c > 0 is tangential to the curve of ^(x) at Xc with = 0

ii i

and < 0 - here net increment ^(x) is globally maximal. Besides > 0 for x < Xc i

and < 0 for x > Xc. Maximal catch is Yc = ^(Xc) (Table 3).

Table 3

Equilibriums for contrast natural-anthropogenic regimes in the models

Variable W-2 (saddle), W-3 (stable node) and W-4 (stable node) W-3 (saddle) and W-4 (unstable node)

Stock X 1 f « V-1 Xc = — = 391.98 IPYJ Maximally sustainably fished in W-3, W-4 1 Ta(1 -b) Vl _ _ xc = —-- = 68.68 l_P(y-b) J Unsustainable

Catch Yc = « Y-1 Xc=102 y MSY in W-3 and W-4 yc = ^ xc = 44.42 1-b Unsustainable

y

Fishing effort e E «y-1 X 1-b 0 8354 Ec = Xc = 0.8354 q y FSE in W-3 and W-4 ec = P(y-1) xy-b = 1.2528 q(1 - b) c Unsustainable

Proposition 1. The stationary state for maximal catch Yc is Xc. Proof. Apply Lemma 1 and notice that f(Xc) = 0.

Corollary l. No stationary state exists if y = const > Yc. Available biomass x decreases from x0 to its elimination through aggravation mode.

For example, for x0 = X = 980 and c = 110 > Yc = 102 > y0 it takes 45 years until extinction through aggravation mode.

Pay attention to two distinct stationary states.

Proposition 2. Let 0 < c < Yc. The stationary states are defined as 0 < x2 < Xc < xi. Proof. Thanks to the properties of ^(x) the line 0 < y = c < Yc intersects the curve of function ^(x) twice in stationary states 0 < x2 < xl.

i

Corollary 2. Lower stationary state x2is unstable node, since fx(x2) > 0, while

i

higher stationary state xi is stable node, since fx(xi) < 0.

Biomass falls from x0 > 0 to bottommost through aggravation mode whenever x0 < x2. Biomass decreases from x0 to xl if x0 > xl, it increases to xl if x2 < x0 < xl.

Let y = y0 < Yc. There are two equilibriums: unstable node x2 = 175.33 < x0 < xl = 648.22. The latter is stable node.

Biomass available in 2012 was not destined for collapse under these assumptions. Fixing global catch y0 would facilitate the biomass growth up by factor of 3 asymptotically. On the other hand, extinction, for example, could result from 0 < x0 = 174 < x2 = 175.33 after 18.9 year.

It is easy to see that both stationary states merge into one Xc if y = Yc. There is a catastrophic change in the system's regime in response to a smooth change of this control parameter as in the Schaefer - Arnold model [12, 14 and 15].

Proposition 3. For y = Yc = a(y -1)Xc / y, a saddle-node bifurcation takes place.

This saddle-node is unstable for x < Xc and is stable for x > Xc.

Proof. The necessary and sufficient conditions for the saddle-node bifurcation are fulfilled [16, pp. 84-86]: the fusion of the nodes with the conversion into the saddle is confirmed by the inversion of the derivative at the critical point to zero

i ii

fx(Xc,Yc) = 0 in the absence of degeneracy in it, fx(Xc,Yc) ^ 0, and it is addition-

i

ally supported by transversality condition fy (Xc ,Yc) = -l ^ 0 satisfied.

i

For the lower (unstable) branch of solutions x<Xc derivative fx(x,Yc) > 0,

i

whereas for the upper (stable) branch of solutionsx > Xc derivative fx(x,Yc) < 0. In other words, Xc is an attractor for x > Xc and a repeller for x < Xc.

Consider saddle equilibrium and saddle-node bifurcation for the given parameters magnitudes: y = Yc > y0, Xc = 391.98 > x0. There are stable branch for x > Xc and unstable branch for 0 < x < Xc. The constant catch that is only 28 per cent higher

than the observed one in 2012 would inevitably entirely deplete the global marine fish biomass within about 6.5 years.

Catch dependence on biomass in reduced WB model W-3

Now a harvesting function, or a harvesting control rule, is defined as

y = E (x) = qexb (5)

with fishing effort e = const. If b = 0 then abnormal W-3 is identical to W-2. W-3 differs from W-2 substantially for 0 < b < 1 as it includes new negative feedback loop B2 (Table 4). This structural change is stabilizing for global marine fisheries, as the reader will soon see.

Table 4

Three feedback loops in W-3

Loops descendant from W-1 and W-2 New loop

R1 of length 1 Stock x ^ Birth rate B2 of length 1 Stock x ^ Catch y

B1 of length 1 Stock x ^ Death rate

There is non-linear dependence of the rate of growth of the stock on itself

x = a-Pxy-1 -qexb-\ (6)

where hyperbolic element -qexb-1is potent of aggravation mode leading to extinction. Indeed, X ^ -ro for x ^ 0 as b < 1 and dX / dx = -p(y -1)xy-2 + (1 - b)qexb-2 ^+ro for x ^ 0.

W-3 can possess one, two, or three stationary states depending on the parameters values. Of course, only for xs = 0 and xs = X it is true that ys = 0 and es = 0. Otherwise, for given stationary state xs there can be one or two ys and co-responding one or two es for each ys. This disjointedness prohibits direct extension of above Propositions from W-2 to W-3.

Notice that in W-3, contrary to W-2, for es > 0, xd = 0 is broadly asymptotically

i

stable (BAS) stationary state as f (0) = 0 and fx(x) ^ -ro for x ^ 0 . Similarly, xs = X

i

is BAS stationary state as fX) = 0 requires es = 0 and fx(X) = a(1 - y) < 0.

Stationary catch effort es >0 can be uniquely defined for known stationary biomass xs > 0 by (5). However, the reverse is not true: for the same stationary catch effort es > 0 there can be one, two or three stationary biomass magnitudes

xs > 0 .

Lemma 2. The line 0 < es = ec is tangential to the curve of function es(xs) at its global maximum in xc (Table 3).

Proof. Function es (xs) achieves in xc its global maximum ec (xc ) = ec. Indeed,

e 'c = 0 when a(l - b)/[p(y- b)] = xcy-i, besides that e ''c < 0 and e's > 0 for xs < xc, e's < 0 forxs > xc.

Corollary 3. No strictly positive equilibrium exists if e > ec (Table 3) since line es = const > ec goes wholly strictly above the curve of function es(xs). The only equilibrium is globally asymptotically stable node xd = 0. The available biomass declines from x0 to zero. A biomass collapse through aggravation mode is the consequence of a persistent over-stretched effort.

For example, for x0 = 214.9, e0 = l.26 > ec = 1.253 it takes about 200 years for extinction of the fish despite parallel decline in catch y from y0 = 100.4 to zero.

Proposition 4. For 0 < es < ec there are three stationary states: BAS node xd = 0, unstable node 0 < x2 < xc and stable node xc < xi < X.

Proof. The line 0 < es = const < ec intersects the curve of function es(xs) in two

i i points corresponding to xi with fx( xi) < 0 and to lower x2with fx( x2) > 0.

For example, take es = Ec < ec (Table 3). There are three equilibriums: xd = 0 is

BAS node, x2 = 2.7l is unstable node, x2 < xc < xl = Xc, xl is stable node. We see Xc,

Yc and Ec are also uniquely defined. If the effort in 2012 and in subsequent years was

fixed at Ec, the biomass would grow up by factor of l.82 asymptotically from x0.

Quantitatively the same Xc that is saddle equilibrium in W-2 becomes broadly

stable node xl instead thanks to negative FB loop B2 (Table 4).

Proposition 5. For e = ec, a saddle-node bifurcation takes place. There are two

equilibriums for ec in W-3: stable node xd = 0 and saddle x = xc (Table 3).

i

Proof. Indeedf(0) = 0 and fx(x) < 0 for x close to xd = 0.

The necessary and sufficient conditions for the saddle-node bifurcation are fulfilled [16, pp. 84-86]: the fusion of the nodes xl and x2 with the conversion into the saddle xc is confirmed by the inversion of the derivative at the critical point to zero

i ii

fx(xc,ec) = 0 in the absence of degeneracy in it, fx(xc,ec) = a(l-b)(l- y)/xc < 0

' b

and with satisfied transversality condition 0 > fe (xc, ec) = -qxc ^ 0.

This saddle-node is unstable for x < xc and is stable for x > xc. For the lower

i

(unstable) branch of solutions x < xc the derivative fx( x, ec) > 0, whereas for the upi

per (stable) branch of solutions x > xc the derivative fx( x, ec) < 0.In other words, xc is an attractor for x > xc and a repeller for x < xc.

Compare properties of saddle xc in W-3 to those of saddle Xc in W-2 (Table 3).

l - b l

Lemma 3. If b < l, y > l then -< —.

y-b y

Corollary 4. The following subordination is true: xc < Xc and yc < Yc. Corollary 5. The equality Xc = xc is not possible in W-3.

Proof. Indeed,

a

Py

y-i _

a(l - b)

y-i

if b = 0 (as in W-2) or b > 0 and

0 < xs < X,

then

P( y- b)

y = 1 that is excluded in W-3.

Corollary 6. If x = xd = 0, then ys = 0, es = 0; if

ys = axs - Pxsy, es = ys / qxsb >0. In particular, ys(X) = 0, es(X) = 0.

Thus, we have considered peculiarities of the three cases of stationary biomass magnitudes for given stationary catch effort: one equilibrium case, two equilibriums case, and three equilibriums case in W-3. New feedback loop B2 enhances sustain-ability of maximal catch Yc in particular, through transformation of corresponding saddle Xc in W-2 into stable node xl = Xc higher than new saddle xc in W-3.

Effort dependence on target biomass in W-4

l

l

W-l and W-3 are from [5], W-2 and W-4 as their modifications are developed in this paper. Notice that the base-year fishing effort e exceeds FSE, biomass x is overfished; catch y is lower than MSY (Table 3).

Let recovery of biomass capable of robust maximal sustainable yield Yc be the goal of stabilization policy. Robustness means that even if the global fishing effort e0 equals ec the biomass will not plunge to xc from x(20l2) as is inevitable in W-3 but will climb to Xc in modified model W-4. Parameter fishing effort e is transformed into variable e. This means that W-4 generalises W-3 as its special form.

The causal loop structure of W-4 augments causal loop structure of W-3 with new positive feedback loop that includes new variable e (Table 5).

Table 5

Four feedback loops in W-4

Loops descendant from W-3 New loop - positive

Rl of length l Stock x ^ Birth rate R2 of length 2 Stock x-> Effort e ^ Catch y

Bl of length l Stock x ^ Death rate

B2 of length l Stock x ^ Catch y

This structural upgrading, not suggested in [5], enables simultaneously needed destabilization of former W-3 saddle xc by turning it into W-4 unstable node and - at the same time - by keeping W-3 stable node Xc as stable node in W-4 too.

Next harvesting control rule is a reasonable substitute for (5) in W-3:

y = F (x) = qexb = qsxh+b, (7)

where variable effort e, new constants s and h are determined as

e = e0( x / x0)h =sxh, (8)

s = e0 / x0h > 0, (9)

h = ln(Ec / e0)/ln(Xc / x0) < 0. (10) The growth rates of biomass, effort and catch are defined next

x = a-Pxy-1 -qsxh+b-1, (11)

e = hx, (12)

y = (h + b)x. (13)

Hyperbolic element -qsxh+b-1 in (11) is potent of aggravation mode. Indeed, as h + b < 1 consequently x ^ -ro for x ^ 0 and Sx / Sx ^ +ro for x ^ 0 .

Lemma 4. Let e0 = ec, y0 = yc and x0 = xc. Then in agreement with (10) hc = ln(Ec / ec) / ln(Xc / xc) < 0.

Proposition 6. For these parameters, there are three equilibriums in W-4: BAS node xd = 0, unstable node x2 = xcand stable nodex1 = Xc. Corresponding effort and catch are e = 0, y = 0 for xd, ec and yc - for x2, finally Ec and Yc - for x1.

Proof. Check stationarity and stability (instability) of these three points: f(0) = 0,

i

fx( x) ^ -ro for x ^ 0 therefore xd is BAS node as in W-3; fxc) = 0 by definition of i

xc for e = ec, fx(xc,ec) > 0 therefore xc is unstable node contrary to W-3; fXc) = 0 i

and fx( Xc, Ec) < 0 due to Lemma 4 therefore Xc is stable node as in W-3.

For the given data, sc = 3.351 and hc =-0.2326. If initially e(0) = ec = 1.253, x(0) = 68 < xc = x2, the biomass plunges to extinction accompanied by relentless growth of effort e; for x(0) = 70 > x2 the biomass climbs to stable node x1 = Xc while effort e diminishes to Ec (Tables 3 and 6).

If non-equilibrium biomass, catch and effort observed in 2012 are used instead then s = 4.989, h = -0.2993 and x2 = 106.4. In the satisfying solution, biomass x and catch y will climb for 40 years asymptotically from 54.8 to 98.6 per cent of Xc and

from 78.1 to 99.4 per cent of Ycwhereas effort e will asymptotically decline from 119.7 to 100.4 per cent of Ec. This targeted transition is moderate in pace.

Table 6

Specific equilibriums in W-2, W-3 and W-4

Equilibrium W-2 W-3 W-4 (for hc)

Stock xd (BAS node) (not present) (not present) 0 0 0

Stock x2 (unstable node) 175.33 Overfished x(2012)= 214.9 Overfished 7.26 Overfished 13.61 Overfished xc = 68.68 Overfished

Stock x1 (stable node) 648.22 Underfished 594.31 Underfished 283.14 Overfished x(2012)= 214.9 Overfished Xc = 391.98 Maximally sus-tainably fished

Catch y for x1 y(2012)= 79.69 (const) < MSY 87.7 (const) < MSY 96.93 < MSY 87.7 < MSY Yc = 102 MSY

Effort e for x1 (implicit y0/(qx1b)) 0.457 < FSE (implicit y0/(qx1b)) 0.535 < FSE e(2012) = 1 (const) > FSE 1.1 (const) > FSE Ec = 0.8354 FSE

Conclusion

This study augments W-3 by a revitalising positive feedback loop in W-4. This loop implies endogenously reduced fishing effort to rebuild the overfished fish stock.

The harvesting control rules are developed, firstly, for avoiding extinction of the global marine fish through typical aggravation modes in W-2 and W-3 and, secondly, for attaining maximal sustainable yield (MSY) and fitting sustainable effort (FSE) asymptotically in W-4.

The disequilibrium initial state of the 2012 year has had margins of safety (Table 6) favorable for recovery of the overfished stock under the proposed harvesting control rules. Catching more fish with the same or less fishing effort could happen due to the organized increases in the biomass in W-2, W-3 and W-4.

Control solutions with jump discontinuity in fishing effort for a most rapid path based on Pontryagin's maximum principle [4, 5, 17 and 18] will be substituted by smooth proportional and derivative control in original two-dimensional predator -prey models in line with [15]. Transition time from y(2012) to vicinity (99 per cent) of MSY could be reduced under urgency from about forty years in the satisfying solution in W-4 to roughly ten years through parametric policy optimization in the predator - prey models. Social and political costs associated with altering the way fisheries are operated will be taken into fuller account than in the neo-liberal policy proposals grounded on mainstream ("neoclassical") models.

A later work will also address system dynamics aspects of fish price, costs, revenues and net benefits touched in [5] still without mentioning transnational corporations as key actors in marine ecosystems. Choking this omission requires deepening the simple functional relationships between fish replenishment and human fishing activities.

This study has been carried out with the plan of research work of IEIE SB RAS; project "Innovative and environmental aspects of structural transformation of the Russian economy in the new geopolitical reality ", no. AAAA-A17-117022250127-8.

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