Optimal Harvesting Policy for the Logistic Growth Model

W O R K I I G P A P E R

OPTIMAL HARYESTING POLICY FOR T H E LOGISTIC GROWTH MODEL

V . F e d o r o v Y . P l o t n i k o v C.S. Binkley

J u l y 1985 N'P-65-40

I n T e r n a T t o n a l l n s t ~ t u t e for Appl~ed Systems Analys~s

YOT FOR QUOTATION WITHOUT PERMISSIO?:

OF THE AUTHOR

OPTIXAL HAEWESTING POLICY FOR THE LOGISTIC GROWTH MODEL

V. Fedorov Y. Plotnikov C.S. Binkley

Juiy 1985 WP-05-40

Working Papers a r e interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only lim- ited review. Views o r opinions expressed herein do not neces- sarily r e p r e s e n t those of t h e Institute o r of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

CONTENTS

INTRODUCTION

1. OPTIMAL HARVESTING POLICY The Right Boundary

The L e f t Boundary

Equation f o r the Boundary P o r t i o n

2 . THE GLOBAL OPTIMALITY OF THE SOLUTION GIVEN BY THE MAXIMUM PRINCIPLE FOR THE ABOVE PROBLEM

3. THE DUAL PROBLEM 4. CONCLUSIONS REFERENCES

OPTIMAL HARVESTING POLTCY FOR THE LOGISTIC GROWTH MODEL

V. Fedorov, Y. Plotnikov and C.S. Binkley

INTRODUCTION

Logistic growth functions h a v e b e e n widely used t o study optimal management of f i s h e r i e s (e.g. C l a r k , 1976), f o r e s t s (Kilkki a n d Vaisanen, 1969; Andersson a n d Lesse, 1984) a n d mammal populations ( s e e f o r i n s t a n c e S p e n c e , 1973): Although t h i s simple model d o e s not c a p t u r e many of t h e important elements of biological dynamics, i t d o e s p o s s e s s t h e c r i t i c a l ele- ment of s a t u r a t i o n , t h e slowing of biomass accumulation as a " c a r r y i n g capacity" i s r e a c h e d . Thus t h e t h e o r e t i c a l r e s u l t s based on t h i s v e r y sim- ple growth model a r e useful in understanding t h e r e s u l t s of more complex a n d more realistic optimal management problems. In t h i s light, t h i s p a p e r makes f o u r contributions.

F i r s t , w e i n t r o d u c e c o n s t r a i n t s on t h e c o n t r o l v a r i a b l e (Heaps and N e h e r , 1979). R a r e l y if e v e r are h a r v e s t l e v e l s in r e a l i s t i c problems com- pletely unconstrained, s o t h i s f i r s t complication of t h e t r a d i t i o n a l bionomic model provides a n important a d d e d d e g r e e of realism.

Second, w e study t h e situation where t h e boundary conditions of t h e state v a r i a b i e are s p e c i f i e d . Generaliy t h e r e s o u r c e manager i s not free t o c h o o s e t h e initial r e s o u r c e conditions (indeed much of r e s o u r c e manage- ment is c o n c e r n e d with decisions when t h e s e conditions are judged t o b e somehow u n d e s i r a b l e ) so t h i s complication of t h e model is important. Termi- nal conditions are sometimes specified by l a w o r administrative direction.

F u r t h e r m o r e , i t i s f r e q u e n t l y computationally infeasible t o soive r e a l i s t i c planning models f o r a n infinite time horizon. Examination of t h e system b e h a v i o r n e a r a specified terminal value i s consequently useful t o under- standing applied management problems.

Third, we show t h a t t h e solution derived by applying Pontryagin's max- imum principle is indeed globally optimally. Other treatments of this prob- lem do not attend t o t h i s important detail of sufficiency.

Finally, w e state and solve t h e dual optimization prcblem. V e r y often t h e duals of many complex management problems a r e far e a s i e r t o solve than' a r e t h e primals. In o u r c a s e t h e dual problem a l s o h a s a v e r y c l e a r i n t e r p r e t a t i o n f o r management: maximize t h e terminal inventory stock sub- ject t o t h e condition t h a t h a r v e s t levels should n e v e r fa1 below a p r e s c r i b e d level.

The conclusion outlines a somewhat more r e a l i s t i c model where t h e bio- logical system i s c h a r a c t e r i z e d by a n age-class model, and indicates t h e kind of issues which a r e interesting in t h a t context.

1. OPTIldAL

HARVESTING

POLICY

S y s t e m

where

K ,

T a r e given positive numbers,

z

and

u

a r e t h e s t a t e and control v a r i a b l e s correspondingly. The point above t h e c h a r a c t e r stands f o r t h e time derivative.

B o u n d a r y C o n d i t i o n s

z ( 0 ) = z o > O

,

z ( T ) = z T > O

C o n s t r a i n t s

The p a i r z

(t

),u

( t

), which satisfies t h e conditions

(1)-(3),

will b e called admissible.

O b j e c t i v e f u n c t i o n a l

I t is r e q u i r e d t o find out a n admissible p a i r

z O ( t ),u '( t

) such t h a t t h e value I ( x

' ( t ),u '( t

⁾⁾ i s minimal among t h e values I f o r all admissible p a i r s

z , u .

Maximum Principle for the Above Problem

If t h e optimal p a i r z O ( t ) , u O ( t ) e x i s t s , t h e n one c a n t r y t o find i t by applying t h e traditional technique of t h e maximum principle of Pontryagin [1962]. F o r t h e a b o v e problem i t involves t h e introduction of t h e Hamil- tonian function

where $(t ) i s t h e adjoint function satisfying t h e equation:

a H

4 = -1 _az =

^{0 o =}

^-$[K

- 2 K r 0 ( t ) - u O ( t ) ] - ~ ~ u ~ ( t ) e - ~ ~

.

⁽⁶⁾

By t h e maximum principle t h e optimal p a i r z O ( t ),u '(t ) h a s a p r o p e r t y t h a t f o r any t E [O,T]

M(t)

h -

H ( t , $ ( t ) , z 0 ( t ) u o ( t ) )

=

max ~ ( t , $ ( t ) , z ~ ( b ) . u ) ,

osu sii (7)

where ($(t ), Xo) is not z e r o v e c t o r a n d M(t ) i s a continuous function of t a n d f o r t E (to ,T) h a s t o s a t i s f y t o t h e equation:

T a H

M(t)

=

H ( T , $ ( T ) . Z ~ ( T ) , ~ ~ ( T ) )

-

~ K ( ~ . ~ ( i ) . z O ( r ) . u O ( r ) ) d r

t

Equation (7) c a n b e used t o define f i r s t l y u O as a function of t , z , $

.

u O = u ( t , z , $ ) ,

Since

-at) H ( t , $ ( t ) , z ( t ) , u ( t ) )

=

-$Kz(l - 2 ) + u z ( - $ + Aoe 1

from (7) and positiveness of z (t ) i t follows t h a t

[Z

.

when ( + ( t ) + ~ ~ e - ~ ' ) > O us , when (-$(t)

+

~ , e - ~ ' )

=

0

O(t ^{l z} ")

= I

^{(so ?ailed} singular c o n t r o l ) , (9)

0 , when (-$(t)

+

~ ~ e - ~ ' )

<

0

Then o n e c a n eliminate t h e v a r i a b l e u from equations (1) a n d (6) and come t o t h e i r solution as t o t h e two-point boundary value ( t p b v ) problem. This solution, if successful, w i l l give z '(t ) and u '(t ). T h e r e is n o r e g u l a r way t o solve t h e tpbv problem if i t is nonlinear ( a s o u r s ) . The following i s a n attempt t o g e t t h i s solution f o r all possible combinations of boundary condi- tions (2).

Singular part of the solution

If t h e r e i s a n i n t e r v a l f o r which

@ ( t

)

= Xoe

- 6 t , t h e n within i t as follows from (6), and from ( 7 ) ,

K - d

z , ( t ) = z 0 ( t ) = - _2K ⁼

^const,

( 1 0 )

A t t h e same i n t e r v a l

1

-- 6' M ( t ) = X o e - 6 t ~

^lY2

4

Since

@ ( t ) = ho e - 6 t ,

and

( @ ( t ) , h o )

is nonzero i t implies t h a t

ho > 0

and c a n b e t a k e n as

ho =

1 f o r example.

In g e n e r a l

z o

^f

z , ( 0 )

a n d Z T

+ z , ( T ) .

T h e r e f o r e , t h e optimal p a i r

z O ( t ) , u O ( t )

cannot consist only of

z , ( t )

and

u , ( t ) .

A t l e a s t in t h e vicinity of

t =

0 and

t =

T t h e optimal p a i r should generally d i f f e r from

z , ( t > , u , ( t > .

The Structure of the Optimal Solution

Knowing t h e singular p a r t of t h e optimal solution on t h e i n t e r v a l

[ti ,t,], 0 < ti

S

t , < T ,

o n e c a n show t h a t for all possible boundary condi- tions (values z

,,

^and

^{z T )}

t h e optimal c o n t r o l

u ' ( t

) consist of t h e s a t u r a t i o n portions n e a r boundaries, spanned by t h e singular c o n t r o l in between.

To show this, o n e should c h e c k n e a r boundaries t h e e x i s t e n c e of a function

@ ( t

) f o r which

u ' ( t ,@,z

) g e n e r a t e s t h e admissible t r a j e c t o r y

z ( t

).

W e will d o this by studying t h e admissibility of t r a j e c t o r y for t h e r i g h t and l e f t boundary conditions

( 2 ) ,

s e p a r a t e l y .

The Right Boundary

The Case ZT

> z , ( T ) .

To t h e boundary condition

z~ > z , ( T )

one c a n

"ascend" f r o m

z , ( t )

with c o n t r o l

u ( t ) =

0 ,

t > t ,

Time of d e p a r t u r e

t ,

from

z , ( t )

c a n b e chosen from t h e condition t o "hit"

zT

at time

T .

If this c o n t r o l i s optimal h e r e , t h e n f r o m

( 9 )

t h e corresponding

$ ( t )

should b e such t h a t

@ ( t ) > ~ ~ e - ~ ~ , ( 1 3 )

The Case

_{K + d}

ZT

< z, ( T ) . -

To t h e boundary condition

zT < z ,

( T ) when

iT>- u

2

^{, and}

z~ >

1

-

^-one c a n "descend" s t a r t i n g f r o m t h e singular k

level at

t = t ,

with t h e c o n t r o l

u ( t ) = C.

If t h i s i s t h e optimal control, t h e n from

( 9 )

t h e function

@ ( t ) ,

corresponding t o i t , should b e such t h a t

@ ( t ) <hoe-" . ( 1 4 )

T h e Left Boundary

T h e C a s e

z o <

r,(O). From this boundary condition one can "ascend"

with control u

( t

⁾

=

0 t o t h e singular level. For this ascent t o be the p a r t of t h e optimal t r a j e c t o r y i t is necessary t h a t

-

+ ( t )

>

hoe-6t ,O s

t <

ti , ( 1 5 )

T h e C a s e z

> z,

( 0 ) . From this boundary condition one can "descend"

with control u ( t )

= u

^(when

- +' ^{< a)}

t o t h e singular level. For this des- 2

cent t o b e p a r t of t h e o p t i m a l t r a j e c t o r y i t is necessary (as follows from ( 9 ) ) , that

To prove t h a t t h e inequalities (13)-(16) are fulfilled f o r t h e chosen controls w e introduce t h e new variable p by t h e formula

+ ( t ) =

A, e-6t p ( t ) ( 1 7 )

The inequalities

+ ( t ) >

^ho

e

-dt and +(t )

<

will become equivalent t o p ( t )

>

^{1 and p ( t )}

<

1 correspondingly. On t h e singular p a r t p ( t )

=

1. From ( 6 ) we can get t h a t

p

=

-p(K

-

^d - 2 K z )

+

u ( p -1). ( 1 8 ) This equation is simpler than ( 6 ) and will more easily bring us t o o u r goal.

C a s e u G O 0 ( z T > z , ( T ) a n d z o < x s ( 0 ) ) For u = 0

b = - p ( K - d - 2 & ) x = K x

-&'

From t h e s e two equations and t h e f a c t t h a t in t h e "singular" interval p

=

¹

one can find

It proves t h e optimality of

u O ( t ) =

0 a t t h e boundaries f o r t h e c a s e s

Z T

>

^{Z s} ( T ) and Z o

<

^{X ,} ( 0 ) .

W eu

' ( t

)

= u ( x T < z,

( T ) and

z , > z ,

( 0 ) )

When

u ( t ) = < ,

then by substitution

z = -

^KY _K ^{y t o ( 1 8 ) and (1) we}

come t o the equation

w i t h y > l . y ,

= z z s .

K

I t follows from ( 2 0 ) and p(y,)

=

1 t h a t

From this and

(20)

one c a n conclude t h a t p is d e c r e a s i n g (being positive, see p. 7-8) when

y < y ,

and

y

i s decreasing and when

y > y,

and

y

is increasing

That gives a l s o t h e r e q u i r e d proof f o r optimality of

u O ( t ) = u

a t t h e boun- d a r i e s f o r t h e cases

z~ < z , ( T )

and

z , > z , ( 0 ) .

By t h e s e f o u r possible boundary conditions t h e s t r u c t u r e of optimal solution w a s p r o v e n valid.

Equation for the Boundary Portion

For t h e i n t e r v a l s with

u ' ( t

)

= 0

and

u ' ( t

)

= iT

t h e state equation h a s t h e form

with a

= K

o r

K - ^C

correspondingly. The solution f o r

t

2

t o

and

a +

^{0 is}

z ( t ) =

Q

( a z -l(t o ) - ^K)e

^{-cr(t +}

and when a

=

0

z ( t ) = ( z - ' ( t o ) + K . ( t -to)-'.

For given boundary conditions

z o

and

z T

t h i s solution c a n b e used t o define t h e values of

ti

and

t , (ti < t ,

)-moments of time f o r joining t h e boun- d a r y portions of optimal solution with t h e singular arc

(10).

If f o r t h e given boundary conditions

t , s ti

t h e n t h e optimal solution h a s no portion with t h e singular arc and consists of only two conjuncted boundary portions.

2. THE GLOBAJ., OPTIHALITY OF THE SOLUTION GIVEN BY THE lUXIMlJM PRINCIPLE FOR THE B O W PROBLEX

To p r o v e t h i s w e will use t h e a p p r o a c h t o global optimality developed by Krotov

(1962, 1963).

In t h i s a p p r o a c h one can p r o v e t h e global optimal- ity of

z O ( t ) . u O ( t )

from t h e discussed problem by constructing t h e function

where

\k(t , z )

i s t h e so-called Krotov's function, and by checking for t h i s function t h e fulfillment of t h e following condition

R ( t , z o ( t ) . u o ( t ) ) =

m a x

R ( t , z , u ) , OSt ST

u ,Z

s u b j e c t t o O s u

S c .

Let

\k(t , z

)

= $(t )z

and let

$(t

) b e t h e adjoint v a r i a b l e from t h e dis- cussed problem.

With such Krotov's function t h e maximum of

R ( t , z , u )

with r e s p e c t t o

u

i s r e a c h e d along

u = u ' ( t

) since

where H ( t ,$,z , u ) is t h e Hamiltonian f o r o u r problem, which r e a c h e s its maximum with r e s p e c t t o u along u

=

u

' ( t

)

.

Since R ( t , z , u ) is t h e quadratic function of z , we will check t h e vali-

aR a2R

dity of ( 2 3 ) with r e s p e c t t o z by calculating

-

^and

- ^.

a 2 a 2

and due t o (6) with X o

=

1.

-

^*R

⁼

^-2K$.

If $ ( t ) r 0 then ( 2 4 ) holds and t h e global optimality f o r

ax2 z O ( t ) , u O ( t )

^{is pro-}

ven. Let us check this.

On t h e singular a r c

$(t

⁾

=

Xoe -"

>

0 . Then f o r the' cases with boun- dary conditions

z~ > z ,

^{( T ) and}

z o <

z,(O) i t was shown t h a t

For t h e cases

z , > z ,

( 0 ) and

zT < z ,

( T ) due t o ( 1 7 ) t h e positiveness f o r

$ ( t )

follows from p ( t )

>

0 . This inequality is t r u e because a s follows from ( 7 ) , ( 7 a ) , (8) f o r z

(t

)

> z ,

( O ) ,

t < ti

,

p ( t )

> -

^L

uz

- K

. z

( l - z )

This proves t h e global optimality f o r

z O ( t

) , u O ( t )

3. THE

DUAL

PROBLEM

System and Boundary Conditions S e e ( 1 ) and

( 2 ) .

Constraints

Objective functional

The optimization problem

(I), ( 2 ) , ( 2 5 ) ,

(26) c a n b e t r a c t e d as a maximi- zation of a n inventory stock at t h e given moment

T

with t o t a l harvesting no less than t h e p r e s c r i b e d volume W. The straightforward elaboration led t o t h e same conditions of maximum principle f o r t h i s problem as in Section 1 with t h e following additions:

$ ( T )

r 0 a n d t h e sign of X o is initially not specified now.

This means t h a t o u r considerations concerning t h e s t r u c t u r e of optimal c o n t r o l f o r t h e f i r s t problem a r e applicable h e r e too and will produce t h e same conclusions as before: namely, f o r all possible initial conditions (values of

z o )

n e a r boundaries t h e optimal control u

' ( t )

consists of t h e saturation portions ( u

' ( t

)

=

0 o r u

( t

)

= c )

spanned in between by t h e same

0

K f 6 , t i S t S t , .

singular control u s

(t

⁾

= - ₂

If

z o > z s ( 0 )

, then

u O ( t )

= O ,

t

S

ti

, if

z o < z , ( O )

^{, then}

u O ( t )

= u

-

,

t

S

ti

1 - -

6

After

t = t i

t h e optimal t r a j e c t o r y

z O ( t ) = z , ( t ) =

2

till t h e time

t = t , <

T which i s defined by t h e moment when

H e r e f o r t ,

< t S T u O ( t ) = O , a n d z O ( ~ ) > z , ( T ) .

If

t , > T

then

t ,

time of "departure"

Fs

from

z , (t

) i s defined as

a n d i n this c a s e f o r

rs < t S T u O ( t ) = c

a n d z O

( T ) > z , ( T )

4. CONCLUSIONS

Bioeconomic models based on logistic biological dynamics are widely used in t h e f i s h e r i e s , f o r e s t r y and renewable r e s o u r c e Literature. W e p r e s e n t a r a t h e r complete solution to t h e r e s o u r c e management problem with logistic growth, showing t h e effect of c o n t r o l c o n s t r a i n t s and a r b i t r a r y boundary conditions as w e l l as demonstrating t h e sufficiency of t h e maximum principle solution a n d solving t h e dual problem. These solutions have some utility in t h e i r own r i g h t for p r e s c r i b i n g optimal management policies.

Furthermore, t h e y suggest how a more r e a l i s t i c system might behave.

How would one complicate t h i s model t o c a p t u r e t h e n e x t d e g r e e of realism? Let us c o n s i d e r t h e case of forest growth. In many p a r t s of t h e world forests r e g e n e r a t e a f t e r e i t h e r n a t u r a l c a t a s t r o p h i c d i s t u r b a n c e s (e.g. f i r e , windthrow o r insect defoliation) o r anthropogenic o n e s (timber harvesting, a g r i c u l t u r a l abandonment). The dynamics of t h e resulting even-aged f o r e s t s c a n b e c h a r a c t e r i z e d by t h e aging of individual stands and t h e r e g e n e r a t i o n of new s t a n d s through t h e h a r v e s t of old ones.

Optimal c o n t r o l c a n b e studied in t h i s context.

Heaps (1984) and h e a p s and N e h e r (1979) examined t h e continuous time p r o c e s s . While some of t h e c h a r a c t e r i s t i c s of t h e solutions have been derived, o t h e r s have not. In p a r t i c u i a r , t h e temporal asymptotic behavior i s not w e l l understood: u n d e r what circumstances d o e s t h e rate of h a r v e s t converge? What i s t h e n a t u r e of t h e asymptotic a g e s t r u c t u r e of t h e f o r e s t ? While t h e logistic model provides some insight into t h e development of a renewable r e s o u r c e . i t o b s c u r e s t h e answer t o some of t h e s e interesting questions.

REFERENCES

Andersson A.E., P . Lesse. 1984. Renewable r e s o u r c e s economics

-

^optimal

rules of thumb. WP-84-84, October 1984. IIASA.

Clark C.W. 1976. Mathematical bioeconomics: t h e optimal management of renewable r e s o u r c e s , Wiley.

Heaps, T. 1984. The f o r e s t r y maximum principle. Journal of Economic Dynamics and Control 7:131-151.

Heaps, T. and P.A. Neher. 1979. The economics of f o r e s t r y when t h e r a t e of harvest is constrained. Journal of Environmental Economies and Management 6297-319.

Kikki, P.. and V. Visanen. 1969. Determination of t h e optimal policy f o r f o r e s t stands by means of dynamic programming. Acta Forestalia Fer- mica 102:lOO-112.

Krotov V.F. Methods f o r solution of variational problems based on suffi- cient conditions of absolute minimum. Automation and Remote Control (a) 1962, 12; (b) 1963, 5.

Pontryagin. L.S. et al. 1962. The mathematical theory of optimal processes, Wiley-Interscience,

N.Y.

Spence, M. 1973. Blue whales and applied control theory. Tech. r e p o r t No.

108, Stanford University, Institute f o r Mathematical Studies in Social Sciences.