Shedding Light on the Harvesting Control Rules in Abstract Bioeconomic Models
@©Alexander V. RYZHENKOV
Economic Faculty
Novosibirsk State University
1 Pirogov street Novosibirsk 630090 Russia
Institute of Economics and Industrial Engineering
Siberian Branch of Russian Academy of Sciences
17 Academician Lavrentiev Avenue Novosibirsk 630090 Russia
E-mail address: ryzhenko@ieie.nsc.ru
Abstract
The theory of bifurcations and catastrophes is applied to the development of modifica-
tions of the Schaefer fishery model. The key variables are the stock of the bioresource,
its natural net change, as well as the Man harvesting activity. The global and local
analysis reveals quantitative and qualitative characteristics of open or closed loop con-
trol. Especially dangerous are the aggravation regimes arising from the dominance of
the positive feedback connecting the biomass and the rate of its net change. The equa-
tions for excessive or sparing harvesting are derived. The time frames for collapses
have been determined. This paper facilitates creation of more complex and realistic bi-
oeconomic models and enhances Harvesting Control Rules.
Key words: renewable resource, depletion, maximum sustainable yield, harvest-
ing control rule, aggravation mode, saddle-node bifurcation, catastrophe theory
Introduction
As well established by system dynamics research over decades, reserves of fish and
other resources of flora and fauna, due to their natural reproductive capacity, can
grow, contributing to the preservation and increase of natural capital [1, 2]. However,
according to the World Bank experts [3, 4], a decrease in the biomass of global fish
stocks, as a result of their excessive catch, created a threat to sustainable fishing.
“Global marine fisheries are in crisis. The proportion of fisheries that are fully
fished, overfished, depleted, or recovering from overfishing increased from just over
60 percent in the mid-1970s to about 75 percent in 2005 and to almost 90 percent in
2013.” [4: 1]. These conclusions are shared by OECD analysts [5] who believe that
the current rate of resource utilizationisfar in excess of what is sustainable
in the long run.!
' Unsustainable management of renewable resources can lead to their permanent depletion in much
the same way as the finite extraction of nonrenewable resources. Stagnant or declining (even slight-
ly) catches can accompany a long-term decline in fish stock. If left unchecked, harvesting could de-
stroy the fisheries that would become biologically or commercially extinct over time.
There is a need for a transition to fundamentally more favourable natural-
anthropogenic regimes. This transition should be based on in-depth studies of con-
trasting regimes of ecological and economic interaction based on system-dynamic
models, starting with engaging ones such as Fish Banks Game developed by D.
Meadows and his colleagues [2]. A great constructive role in clarification of such
models and in their further development belongs to the mathematical control theory
[6] with strong footage in mathematical analysis and theory of differential equations.
According to the control theory, open-loop control is completely determined at
the initial instant to; 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 [7].
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 economic theory and economic practice.
To simplify exposition of economics of renewable resources we will keep in mind
their rich diversity and consider non-farming fish as their representative. Peculiarities
of specific types of these resources are not considered on this stage of investigation.
Then according to existing conventions, biomass is total amount of fish re-
sources, biomass net change is due to natural processes and harvesting by Man.
Hereby harvest equals yearly catch. Table 1 lists model variables and their units of
measurement. It may be a prompt on variables of differential equations below.
Table 1. The main variables of simplified biomass models
Variable Notation | Measurement unit
Catch y,c fish/year
Fish stock (biomass) x fish
Carrying capacity lla. fish
Birth rate Bx fish/year
Death rate Bax fish/year
Net change of fish stock x fish/year
The growth rate of fish stock g l/year
The growth rate of catch y l/year
The reader sees that global marine fish stock is considered as a scalar. This per-
mits application of single equation technique akin to methods developed in the re-
search on mineral resources and proved stocks (see references and critique in [12]).
1. Simplified Verhulst — Schaefer — Arnold models
1.1. Verhulst's textbook model M-1
The logistic equation, also known as the Verhulst equation (named for the first time
formulated by a Belgian mathematician), originally appeared when considering the
model of wild population growth. Denoting by x the population size, by t > 0 time,
the model can be represented by a non-linear autonomous differential equation
x = (x) = Bx(1-ax). (1)
where parameter 8 characterizes the potential rate of growth (multiplication) in the
absence of intraspecific competition, and « — the reciprocal of the supporting capaci-
ty of the environment (that is, the inverse of the maximum possible population size).
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 depicted on Figure 1.
Birth vale Death rate
x0
Figure 1 — The Vensim diagram of the Verhulst logistic model M-1
The initial assumptions for the derivation of the equation when considering pop-
ulation dynamics are as follows: the rate of reproduction of the population is propor-
tional to its current level; the second term of the equation reflects intraspecific com-
petition for resources, which limits the growth of the population, or, in plain words,
the death rate increases as crowding increases.
The derivative of the natural net change is defined as
by '=B— 20x. (2)
When $,'=0, net increment (x) is maximal for x,=1/(2a).
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, x; = 1/a is an asymptotically stable node, since $, (x1) =-B
<0, on the other hand, x. =0 — unstable node, as $y (xy) =B >0.
The population growth is S-shaped. Neither open nor closed loop control of the
wild population by Man is active. The size of the population tends to dynamic
equilibrium at the maximum number that can sustain most of random external
shocks except huge calamities. M-1 is structurally stable.
3
1.2. Simplified Schaefer — Arnold model M-2
The model [8] supplements the assumptions of the logistic growth of biomass by the
assumption that human fishing activities reduce the increase in the fish population by
the catch amount y, the amount of which linearly depends on the available biomass
without delay:
x =f(x) = Bx(l—ox) —y, (3)
where y = kx, 1 >k=const > 0 (Figure 2).
OQ Stock x ae
Birth rate Death rate
+ (R1) (B1) +
x0
Figure 2 — A causal structure of M-2 without information delay in catch y
Table 2. Three feedback loops in M-2
Loops descendant from M-1 New loop
R1 of length 1
Stock x > Birth rate B2 of length 1
B1 of length 1 Stock x > Catchy
Stock x — Death rate
Without loss of generality, let 0 = 1 and B =1 [9: 98-99]. Then
x=(1-k-x)x, (4)
where Xp > 0 fortp=0, fy (x) =1—k-2x.
For the linear harvesting control rule (HCR) there is a negative linear depend-
ence of the growth rate of the stock on its magnitude like in M-1 (fork = 0):
K=1-k-x. (5)
The rate of change in catch and stock is, contrary to M-1, the same:
f=% =1-k-yhk. (6)
Let us consider the properties of stationary states more closely. The first of these
is a stable nodexy}=1—k>0 as f, (x) =k—1 <0. The value of x;smoothly depends
on control parameter k, the latter’s changes in the specified boundaries do not affect
the established mode qualitatively. The necessary and sufficient condition for domi-
nance of the negative feedback x > x is fulfilled in M-2 se =~ 1 <0. This
X
property gives the equilibrium global asymptotic stability.
In addition, there is a second steady statexy= 0. It is unstable node since
f, (x2) =1-k >0.
If X) >0, x, =1 -k, a solution to (6) is
1X9
x= ——_ a __. (7)
Xg - e FM (x, —%)
This formula generalizes its particular case (1) for k = 0, y = 0 in M-1. Similarly to
the former, M-2 is structurally stable for 0<k <1.
Next equation determines catch
v1Yo
Y=— Th (8)
yo-@ (yyy)
Proposition 1. For t>0 y—y,=k(1-k). Maximum sustainable catch MSY
Ys =Cs = 0.25 is achieved at k =k, = 0.5, when the biomass volume is determined by
the conditions of a stable node x; =x; =1—k, =0.5.
The general biological overexploitation of fish stocks beyond the biomass level
corresponding to MSY inevitably leads to subsequent reduction in catches. Timely
reduced fishing efforts can allow depressed fish stocks to recover.
As pointed out in [10: 6], “from a biological point of view the concept of MSY
is simply not sufficient. Nevertheless, it should be stressed that it provides a valua-
ble rough index of production potential. As a first rough cut at management policy
for major commercial species, MSY is probably acceptable. But ones the level of
MSY is attained, it should be expected that it may not be sustained.”
The simplified Schaefer — Amold models represent social production in very (if
not extremely) abstract form. They are enhancement for thought experiments on the
rocky and hard way from abstract to concrete. Still teaching experience demonstrates
that these models strongly stimulate interest of students and newcomers to the system
dynamics field discovering how their mathematical knowledge can be applied for bet-
ter understanding of acute — local and global — sustainability issues.
In more complex models, this deficiency is overcome by explicitly taking into
account the goals of capitalist production and the methods for achieving them [1, 2-5,
11-13]. For economic reasons, economic entities that maximize profits and / or rents
tend to choose k # ks. Technological capabilities, property relations, as well as fea-
tures of competition, narrow the boundaries of the choice of k and y.
According to [4], “stocks are defined as fully or overfished if their biomass is at
or below the level that supports maximum sustainable yield (MSY). Maximum eco-
nomic yield (MEY), which maximizes the sustainable net benefits flowing from the
stocks, occurs at a stock size that is larger than that at MSY level.”
Harvesting control rule in M-2 is not satisfactory from the control theory stand-
point. It can be dangerous for low stock x to harvest it with rate kx if random fluctua-
tions are taken into consideration. It violates to an extent - dangerous under some
typical circumstances - the precautionary principle in the renewable resources man-
agement.
1.3. Modified Schaefer — Arnold model M-3
Guided by the precautionary principle the author transforms M-2 into M-3, preserv-
ing the logistic natural increase in the bioresource (1), but replacing the linear de-
pendence of the catch on stock by a quadratic one:
y =mx’, (9)
where m > 0.
The net increase in biomass is now defined as
x =f(x) =x[1—(1 +m)x], (10)
where Xp > 0 for to=0, fy (x) =1-2(1 +m)x.
Similarity to M-1 and M-2, for quadratic HCR the growth rate of the stock has a
negative linear dependence on its positive magnitude as well
¥=1-(1+m)x. (11)
However, now the product (1 + m)x has replaced the algebraic sum k + x in (5). Fig-
ure 3 presents causal-loop structure of M-3 that is very similar to that of M-2 (Figure
2).
Catch y lea
©)
Stock x 0 aoe
Birth rate 0 (on $4 rate 0
+ A (RI) (eof
v
<x0>
Figure 3 — A causal structure of M-3 without information delay in catch y 0
Still unlike M-2, the rate of increase in catch is now twice the rate of increase in
stock. This brings about a non-linear negative dependence of this rate on the catch
magnitude:
jy =2% =2-20+m), |. (12)
m
The negative derivative & =~ (1 +m) <0 guarantees the dominance of the sta-
bilizing negative feedback x ——> £ > x. The dominant negative feedback is deeper
in M-3 than that in M-2. This deepening accelerates restoration of equilibrium after
disturbance. Thanks to this deepening the threat of overshooting which is present for
k> 1 in M-2 has disappeared in M-3 that is more structurally stable than M-2.
Let us consider stationary states for m > 0 explicitly.
Proposition 2. The system has two equilibrium states. One of them is a stable
node x; =1/(1 +m) >0, because f, (x,) =—1 <0. The other is unstable node x7 =0,
because fy (x2) =1>0.
Corollary. Stock x depends smoothly on control parameter m, the latter’s chang-
es do not affect the steady state qualitatively.
The solution to (9) is the logistic function
XX0
X= (13)
XQ +e (xy — Xo)
This formula is valid for the Verhulst model (1) above, as a special case, in which
m=0.
The catch is given by
———— (14)
aver{ [2 a)? .
Yo
Proposition 3. Catch y> yy = a 7
(1+m)
for too. Maximum sustainable yield
My
(catch) ys = 5
0.25 is achieved at m; = 1 and x, =x, =1/(1 +m,) =0.5.
(1+ m,)
For low Xo, the integral (cumulative) catch in M-3 is lower than that in M-2 over
short segments (few years) and higher when integrating over longer segments of 5-10
years (Figure 4).
Initial stock x) =0.1 Initial stock xp =1
54 1
054 1
an om
a7 om
0 048
0 048
0123 45 67 8 9 1 012 3 4 5 6 7 8 9 WO
Tine (Ye) Tine (Yea)
X x
x0 x0
Stock x in M-2, x 0 in M-3 Stock x in M-2, x 0 inM-3
Catch y in M-2, y 0in M-3
Catch y in M-2, y 0 in M-3
Cumulative y, y 0
Cumulative catch y in M-2, y 0 in M-3
Cumulative catch y in M-2, y 0 in M-3
Figure 4 — Stock, catch and integral (cumulative) catch under two HCRs (k =0.5,
m = 1, respectively) over 0-10 years for xo = 1 on the left and for xp = 0.1 on the
right, blue curves — for linear HCR, red curves — for quadratic HCR
Table 3. Average stock, catch and net change of stock
under three harvesting control rules for x) = 1 over 0-10 years
HCR Stock x Catch y Net change x
Linear 0.570 0.285 -0.051
Quadratic 0.536 0.294 -0.053
Enhanced in S-1 0.516 0.297 —0.053
Table 4. Average stock, catch and net change of stock
under three harvesting control niles for x) = 0.1 over 0-10 years
HCR Stock x Catch y Net change x
Linear 0.341 0.171 0.039
Quadratic 0.419 0.189 0.040
Enhanced in S-1 0.433 0.192 0.040
Transition from linear to quadratic HCR strengthens precaution in nature man-
agement, due to its adherence to longer-term efficiency, which helps overcome "quar-
terly capitalism", aimed at immediate profit.
2. Aggravation modes and catastrophes in Arnold’s model M-4
The author uses the concept of aggravation modes, investigated in different contexts,
in particular, in [4, 11-12, 14, 15].
Let the natural increase in the biological resource be determined as before,
whereas the catch y is redefined as constant c [9: 98]. Then open loop control deter-
mines HCR in M-4, whereas the net increase in stock is given as
¥=f(x) =x—x’-y =6(x)-c, (15)
where c >0, X90 >0, fy (x) =1—2x.
The analysis reveals non-linear dependence of the rate of growth of the stock on
itself $= ~=1-x- where the last hyperbolic element is potent of an aggrava-
x x
tion mode. Indeed, >-cofor x > 0.
A birth of the aggravation mode results from the transition from dominant nega-
tive feedback x——-+%-— Xto dominant positive feedback x > %— Xat a tipping
point, when the sign of ¢ =-1 — < 0 tums into its opposite. Quite dramatically
Xx x
dg
be
The conditions for emergence of an aggravation (exacerbation) regime and its
unfolding are described in detail below. This aggravation mode arises when misman-
agement in M-4 destroys structural stability present in M-1, M-2 and M-3.
For brevity, the author defines auxiliary parameter
a= /[cs —c|. (16)
Proposition 4. The stationary state forc =c, =0.25 and a =0 is x, =0.5. The sta-
tionary states forc <c, are defined as
>oforx>0.
10
0 <x,2=0.5+a, (17)
A lower stationary state x2 =; a is an unstable node, since f,'(x,) = 2a >0,
while the higher stationary state x, = ; +aisastable node, since fy (xy) =-2a <0.
Maximum catch y, =c, = 0.25 requires stock x.. The quantity c; is critical, or bi-
furcational: jump-like changes in dynamic regimes are generated by infinitesimal
changes in this parameter’s magnitude in M-4.
Proposition 5. Let 0 <c <c, and Xp < X2 <x;. There is a monotonous decrease in
biomass down to complete exhaustion; a solution to (15) is
%-% XQ — Xo g2at
1- 2 0 g2at
x — Xo
Exhaustion x = 0 occurs at the moment
n= Znl(2=2]/ 2] (19)
2a X, X2 —X9
For example, if c =0.2499, a =0.01, T; =23.54 for xp =0.46 <x) =0.49 <x, =0.51.
Proposition 6. Let c > c;. There is no stationary state. There is a monotonous de-
crease in available biomass up to its elimination; a solution to (15) is
a*tg(—at) +a(Xg — Xz)
. 20
a —tg(—at)(Xq — X,) (20)
X=X,+
Exhaustion x = 0 occurs at the moment
T= 4] sretg( 222} erca(%)] (21)
a a a
For example, for x) =0.52 and c = 0.2501, T. = 265.81.
It is easy to see that both stationary states merge into one x, if a = 0. There isa
catastrophic change in the system's regime in response to a smooth change of this
control parameter.
Proposition 7. For a = 0, a saddle-node bifurcation takes place. This saddle-node
state is unstable for x < x, and is stable for x > X,.
Proof. The necessary and sufficient conditions for the saddle-node bifurcation
are fulfilled [16: 84-84]: the fusion of the nodes with the conversion into the saddle is
11
confirmed by the inversion of the derivative at the critical point to zero
f, '(Xg,Cc) =1—2x, =0 in the absence of degeneracy init, f, "(X,,Cs) =-2 #0, and
itis additionally supported by transversality condition f, '(x,,c,)=—1 # 0 satisfied.
For the lower (unstable) branch of solutions x<x,the derivative
f, (x,C,) =1-2x > 0, whereas for the upper (stable) branch of solutions x > x, the de-
rivative f, '(x,c,.) =1-2x <0. In other words, x, is an attractor for x > x, and a repel-
ler for X<Xg.
If a = 0 and xg < Xg, depletion of the resource occurs on the hyperbolic curve; a
solution to (15) is shaped as
X= Xp + t I (22)
t+
Xo — Xs
Complete extermination of the bioresource occurs at the moment
Ty=2%)——. (23)
0.5- Xo
For example, for x) =0.2 anda = 0 T3=1.33.
Proposition 8. A steady steady-state regime with attractor x, dies, colliding with
an unstable regime with a repeller x2, and at the moment of collision, the conver-
gence rate is infinite.
Proof. Forc — Cg there are ne and —4=—— .
Oc 2a Oc 2a
The author would like to turn the reader attention to the notion of characteristic
return time T, near a threshold [17, 18]. This notion rests, in my view, on the concept
of the first order delay in the system dynamics literature.
Proposition 9. For x > x2 and values of the parameter c, increasingly close to
the critical value c,, the delay time T,, which characterizes the asymptotic approxima-
tion of x to aso far stable node, increases unlimitedly.
The proof for the present case follows a more general proof [17, 18: 244]. Near
x, stock adjustment process of the first order takes place
X=A(x1—x), (24)
1
where 4 = — = 2a.
T
As aresult of integration (24), I obtain
x= % — (4 —xye**. (25)
12
For c->Cs, A — 0 holds, therefore unbounded growth of the delay in over-
coming the initial deviation of x from the attractorx, happens: T, 00. Thus, the
characteristic return time T, stretches to infinity near threshold c, of the parameter c.
Let us explain this important aspect. The excessively growing time interval re-
quired to eliminate imbalances can be a precursor for a catastrophe.” Catastrophic re-
gimes are possible even with a catch locally close to MSY c, with a minor excess.
The collapse will happen relatively faster if the author assumes that the catch is
more or less steady growing in M-5. On the contrary, timely and accurate reduction
of catch allows avoiding collapse. Table 5 and Figure 5 illustrate this principal differ-
ence in the evolutionary patterns in M-4 and in M-5. Notice that M-5 has the same
equations and parameters as M-4 except those that directly affect catch y.
Table 5. Effects of HCR on fishery time frame for the same xp = 1 and c = 0.2501
in M-4 and M-5
Model Catch y Catch growth ratey | Time left until
full depletion of
stock x
M-4 y =c =const 0
310.2
M-5 Growing y = ce" 0.02 12.65
M-5 Declining y = ce” —0.02 Infinity
2 In other bioeconomic models, saddle-node bifurcations and hysteresis occur with the initial
presence of three rather than two, as in our case, stationary states. Examples are [19-21]. They have
properties similar to those presented in the eighth and ninth Propositions.
13
Growth rate of catch y = 0.02 in M-5 Growth rate of catch y =—0.02 in M-5
034 0.26
0.34 0.26
0123456789 WN 2 BY 0 8 156 233 3iL
‘Tae (Yea) Time (Year)
y
yo yo
Catch y in M-4, y 0in M-5 Catch y in M-4, y 0in M-5
1 1
| 1
05 05
05 05
0 0
0 0
0123 45 67 8 9 WN 2B 0 78 156 233 311
‘Time (Year) ‘Time (Year)
: x
x0 x0
Stock x in M-4, x 0 in M-5 Stock x in M-4, x 0 in M-5
Figure 5 — Catch y and stock x under HCRs for xp = 1: blue curves for constant har-
vest in M-4 with collapse at t = 311, red curves for exponentially increasing catch on
the left with stock depletion at t = 12.65 and exponentially declining catch with stock
recovery on the right
Being severely or moderately harvested, fish stock x is completely depleted at
very different moments separated by roughly three hundred years (in the year 12.65
in M-5 or in the year 311 in M-4). Fish stock is allowed to recover after the initial
plunge to a high sustainable level if catch is reduced exponentially in M-5.
The increased overfishing makes collapse in fisheries closer and faster. To re-
verse the decline in fish stocks, catch y has to become lower than the natural net in-
crement x — x” > 0. Such a reduction promotes the ability of depleted stocks to recov-
er from otherwise a dangerously low level.
The rigorous mathematical control theory maintains these rather clear sustaina-
bility rules and assists in their further development. The author proceeds to a quite
more elaborated closed loop control than developed by my predecessors in M-2 and
14
even than newly proposed for M-3 in this paper earlier. HCRs will be upgraded as
well.
3. MSY-centred stabilization in predator-prey model S-1
An appropriate stabilization policy for proved mineral reserves for avoiding their de-
pletion has been proposed in the system dynamics literature [12]. This section elabo-
rates a stabilization policy for a renewable resource that improves economic efficien-
cy and maintains bioresource sustainability in the middle- and long-term.
The recent World Bank and FAO studies have identified lack of prudent control
as one of the main factors detrimental for the global renewable resources [3, 4]:
“\..the state of governance worldwide varies greatly and, despite some encouraging
successful approaches, is in dire need of improvement [4: 17].”
3.1. Enhancing harvesting control rule
Transforming the previous models into a predator-prey model is the necessary step
for designing more reliable and efficient HCR than considered above. Catch c be-
comes the new phase variable in addition to stock x.
The urgent question arises: how do we turn the hyperbolic element c/x as a foe
of sustainability under open loop control as in M-4 or M-5 in its champion? In real
life similar transformation of certain quality into the opposite are ubiquitous: for in-
stance, an explosive gas that destroys homes accidentally faster than a blink of eye
can be thoughtfully applied for home construction purposes instead.
The new phase variable c has become a subject of proportional and derivative
control as Table 6 and Figure 6 demonstrate. Thereby catch c is targeted at MSY c,,
and stock x is targeted at corresponding optimal level x,.
Table 6. Four feedback loops in S-1
Loops descendant from M-3 New loops
A1 of length 1 B1 of length 2
x—>x c—>%6
A2 of length 2 B2 of length 4
x— 5% >x x>¢—>c— 5k x
Note. Only a negative partial derivative and partial derivative with alternating algebraic sign are
explicitly shown with N and A, respectively, immediately above arrows. All other first partial deriv-
atives are positive.
15
Stock x
Net change xdot
(Al)
Sey / x0 (B2)
ke)
Rate of change xhat 4@n)
Catch c Ag ~y
Net change cdot
c0
q Pp x MSY
Figure 6 — A condensed causal loop structure of S-1; total number of feedback loops
—4, among them: 1“ order— 3 (1 — negative, 2 — alternative), 2" order—1 (negative)
The positive feedback x > — Xx is transformed into one with altemating po-
larity A2 of length 2 (Table 6) that is much safer indeed. A mathematical analysis
that follows maintains this expectation.
The above causal loop diagram is the basis for the 2™ order system of ODEs
x =f(x) =x-x’-c (26)
¢ =p(x;—x) +q% =Plxs—¥) +g(1-x-S), (27)
where p < 0 and q>0.
For this system the Jacobi matrix is defined as
1-2x -1
Js= | tq tas -qi<0
x x
. (28)
The reader sees S-1 can belong to predator (c) — prey (x) models whenever
= >0as = <0 is always satisfied. There is predator intra-specific competition as
C.
= <0. Preys co-operate with each other if x < 0.5 and = > 0 or compete with each
oO
other if x >0.5 and <0 , aneutral case is forx =x, =0.5.
Oo
The above system has the non-trivial stationary state:
E,= (Xs, Cs) (29)
with corresponding Jacoby matrix
0 -1
Js= -p>0 —2q <0 . (30)
Proposition 10 (a). The stationary state E, (29) is locally asymptotically
stable.
Proposition 10 (b).If 0>p2-q’ the stationary state E; (29) is stable node;
if p<-q’ <0 itis stable focus. In both cases, it is hyperbolic.
Proof (applying the Routh-Hurwitz stability criterion).
The necessary and sufficient conditions for asymptotically local stability of (29) are
satisfied:
Js|=-p >0 (31)
as p < O and
Trace(J,) =—2q <0 (32)
as q>0.
For gaining additional information consider a characteristic equation that is writ-
ten as
17+ 2qgh—p =0. (33)
It has one real root or two roots
Mi2=-atya7+p. (34)
For having one negative real root A,» = —q there must be
p=-q’. (35)
Roots (34) are real and negative if p>-q’. They are complex-conjugate with a
negative real part if p<-q’.
A stable focus arises for p <—q? with
Mi2= -qtiy-(q’ + p). (36)
The period of converging fluctuations is then
T= Qn
= .
+p
The author has proved that at q = 0 sufficient requirements for Andronov — Hopf
bifurcation are satisfied. Y et this case in not immensely relevant for selecting appro-
priate HCL and it is skipped therefore.
Explicit solutions to the linearized at the stationary state system are different for
two distinct negative 4,2, on the one hand, and for negative 4, =”, =— q for p =—q’,
on the other hand.
(37)
17
3.2. Policy optimization
The author has carried out two parameters policy optimization for co and q for one
stable node with p =—q’. The optimization criterion is mostly grasped as cumulative
catch over 0-T years. Besides this, Penalty for negative c is added in Pay-off:
T
Penalty = [dt (38)
0
where 5 = 0, if c > 0, 6 =—100, ifc <0.
Formally, this policy optimization in Vensim is based on a restricted dynamic
optimization problem:
T T
Max [ott jat| , (39)
0 0
subject to (26) and (27)
with Zp = (Xo, Co),
initially: O<co =O <1 andl<q=1<3.
A solution for the stable node depends on x» (Table 7).
Table 7. Policy optimization results depending on xp in S-1
Run’s No. Xo Co q Dp
2 0.1 0 2.261 —5.112
1 1 1 3 ~9
The convergence of biomass x and catch c to its distant attractor E, (29) is al-
most monotonous in the both runs. Results for this HCL judged by average catch c
and integral catch [ict | over years 0-10 are better now than for aforementioned M-
0
2 and M-3 with linear and quadratic HCL correspondingly when all other conditions
are the same (Tables 3 and 4, Figure 7).
18
Stock x Stock x
(green in M-2, red in M-3, blue in S-1) (green in M-2, red in M-3, blue in S-1)
forx) =1 for x) =0.1
X X
! 1
1 | !
1 \ !
“| oi
5 05
4 05
05 :
0
! - 0 1 2 3 4 5 6 7 8 9 10
01 2 3 4 5 6 7 8 9 1 Tine (Mont)
Time (Month)
x optimal
x opine x qqacatio
Xquactatc hear
i
Catch y =c Catch y =c
(green in M-2, red in M-3, blue in S-1) (green in M-2, red in M-3, blue in S-1)
forx) =1 for x) =0.1
12 026
12 026
12 026
\|
07 i 3
a7 |\ 03
07 | \ 023
02 0
02 0
02 0
01 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 § 1
Time (Month) Tine (Month)
y optimal yoptinal
yquadratic Yeqattac
y near yhhex
19
Cumulative catch Cumulative catch
(green in M-2, red in M-3, blue in S-1) (green in M-2, red in M-3, blue in S-1)
forx) =1 for xX) =0.1
32 1
32 1
32
16 iT
16 1
16 1
0
y 0
0
012 3 4 5 6 7 8 9 WO 0123 65 67 8 9 10
Tie (Mon Tie (Mont
hare y opi
yoqatc ¥ matt
Figure 7 — Comparison of HCL results in M-2, M-3 and S-1 for years 0-10
for Xo = 1 on the left and for x) = 0.1 on the right
The upgraded closed loop control in S-1 gives a bit more time for the biore-
source to grow from low Xo by choosing cy = 0 initially. Quite contrary this control
sets Co = 1 initially for high Xo.
Conclusion
Typical modes of renewable resource management are considered for open-loop or
closed-loop control. Using the theory of bifurcations and catastrophes, the policies of
improving bioresource catch and renewal, with raised long-term effectiveness in rela-
tion to the policy proposed in the simplified Schaefer — Arnold bioeconomic model
M-2, are elaborated in M-3 and S-1.
The obtained results related to the compared modes of nature-use are not only
local, as is often the case in the applications of catastrophe theory, but also global in
nature (particularly, in M-4 with open loop control). For all the considered regimes in
one-dimensional models (except M-5), the original formulas of integral curves for
stocks and catches are derived. Still the analytical results for the proposed two-
dimensional predator-prey model S-1 are mostly local, they are extended to broader
areas thanks to Vensim simulations.
20
A more concrete presentation of the ecological and economic reproduction and
its current global crisis is expected to be carried out in further studies with detailed
elaboration of technological and institutional aspects.
The transition from the above simplified analysis of sustainability to the study of
the evolutionary ecological stability of interacting bioresources is promising [22].
The research should also enhance the probabilistic approach to bioeconomic model-
ling.
However, there is no doubt that depletion of bioresources is Damocles sword for
the world economy. Management of reproduction based on scientific foresight be-
comes more and more pressing necessity. Only this path, traditionally favoured by
advanced system dynamics, opens the rich opportunities for overcoming the global
crisis of nature management.
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