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Application of Control Theory in the Efficient and Sustainable Forest Management

Md. Haider Ali Biswas, Munnujahan Ara, Md. Nazmul Haque, Md. Ashikur Rahman

Abstract— This paper focuses on the necessity of forest management using the model of control theory. Recent researches in mathematical biology as well as in life sciences closely depend on control theory. Various popular research papers have been received considerable attentions by engineers and research scholars due to the fact that it has been the central and challenging area of research for its wide range applications in the diverse fields. In this study we have briefly mentioned some of the fields in which these challenges are present, specially sustainable forest management is one of the warming issues in the present cencury. Our main objective in this paper is to investigate the scopes and applications of control theory in real life situation, specially the applications of control theory in the efficient and sustainable forest growth. A particular case of Sundarbans, the largest mangrove forest in the world is discussed with illustrative examples.

Index Terms— Control theory, Malthusian model, Logistic equation, Sundarban mangrove.

—————————— • ——————————

1 INTRODUCTION

OW-a-days the applications of Optimal Control (OC) theory have been increased surprisingly in almost all branches of modern science and engineer-
ing, specially the OC is playing significant, in some cases the dominant roles in the fields of aerospace engineering, medicine, agriculture and economics. See for examples [9], [14]. The theory of optimization continues to be an area of active research not only for the mathematicians but also for the engineers as an indication both of the in- herent beauty of the subject and of its relevance to mod- ern developments in science, industry and commerce. The optimal fuel landings of the space vehicle, the optimal strategies for the drug doses in the chemotherapy of in- fectious diseases along with all others are the fascinating challenges at present in the ongoing development of science and technology [28]. All the achievements in the field of optimal control theory are mostly due to Pontrya- gin. Although several authors made tremendous contri- butions to the further developments in this area, but Pon- tryagin Maximum Principle is still a milestone in OC theory. See for examples [6], [10], [16], [30] and references therein for details study on optimal control theory. How- ever, we are mainly concerned in this paper to discuss the application of OC theory in the sustainable forest man- agement. Many researches have been carried out over the years (see for examples [12], [17] and [37] for the recent development) emphasizing on the optimal strategies for an efficient and sustainable forest management due to the fact that forests are one of the best sources for saving lives of the human being from the poisonous greenhouse gas- es, from many natural disasters like cyclone and so on. European Tropical Forest Research Network (ETFRN) recently pointed out on the similar issues of forest man- agement [20]. The people of the coastal region live de- pending on forests specially for a hunting, wood harvest- ing as well as collecting of honey (see for examples [23], [32]). Accordingly, in the economic point of view, forests
help to enrich the national economy of a country. So steps should be taken so as to manage a sustainable and effec- tive ecosystem where control theory is the essential tool. Different mathematical models were proposed for this purpose. Many studies have been undertaken to deter- mine the optimal forest rotation length under different scenarios since the advent of the modern civilization. Some of these were focused on the optimal rotation length with the consideration of only timber value. Others searched for the optimal rotation length with the inclu- sion of both timber and nontimber benefits (See for ex- amples, [27], [34], [36] and references within). These stu- dies have provided important guidelines on how to man- age the number of existing trees in the even-aged planta- tions. However, their applications in uneven-aged, or natural forests, are limited because age is no longer an appropriate variable under such circumstances. Also, in formulating a forest management plan/policy, particular- ly in the management of a large-scale (such as a regional or national scale) forest resource, it may be more relevant to determine how much timber should be harvested and what level of the forest stock should be maintained than to know when trees should be cut. However, in all the above mentioned models, the authors proposed the op- timal managements of the forests only when the forests are full of trees. All those may bring a good result when the forests will be full of trees. In that case, the Malthu- sian model can give an optimal growth of trees after a certain time. In this study, we have proposed an alterna- tive model discussing the Malthusian model and Logistic equation with illustrative example for the optimal con- trolling of the forest growth. A case study for survival of the existence of the largest mangrove forest in the world will be discussed for the sustainable management.

2 THE MALTHUSIAN MODEL

The Malthusian law of growth enunciated by Thomas

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2 International Journal of Scientific & Engineering Research, Volume 2, Issue 3, March-2011

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Robert Malthusian in his research documentary “An essay on the principle of population” and concluded that “Popula- tion when unchecked, increase, with a geometric ratio”. See for details in [24]. Over the years, the Malthusian growth model is still the granddaddy of all population models, which is simply based on the famous exponential growth law. We begin this section with a simple deriva- tion of the model.
rate. Now we will discuss the solution for following sev- eral cases.

p(t)

r > 0

Let t denote the time and

p(t ) denote the number of p0

r = 0

individuals present at time t . In practice

p(t )

is non
negative integer. We assume that differentiable.

p(t )

is continuously

r 0

The growth rate of a population is the rate at which pop- t0 t

ulation changes. If the population

p(t )

at time t

changes to p(t + 8t ) , the average per capita growth rate at time 8t is

p (t + 8 t ) p (t )

Fig. 1. Graph of the Malthusian model under 3 cases.

Case 01: For r 0 , we have

p(t ) = lim p(t ) = lim p e r (t t0 ) = 0

p ( t ) 8 ( t )

t

t 0

Taking limit 8t 0, we get the instantaneous growth rate per capita at time 8t as
This implies extinction of population. That is if the growth rate is negative, in the long run the population
will be extinct.

lim

8 t 0

p (t + 8 t ) p (t )

p ( t ) 8 ( t )

p (t )

= p(t )

Case 02: For r = 0 , we have
Now let,

p(t ) = lim p(t ) = lim p e r (t t0 )

b =intrinsic birth rate.

= The average number of offspring born per

= lim p0 e

t

= p0

t

t 0

individuals per unit time.

d = intrinsic death rate.

= The fraction of individuals of the population
dies per unit time.
This refers constant population at zero growth rates.
Case 03: For r > 0 , we have

r (t t0 )

r = b d

p(t ) = lim p(t ) = lim p0 e =

t

t

= intrinsic growth rate of the population.
= Excess of birth over death per unit time per
individuals.
Now, we consider a single species of population, the
growth model is then described by

p (t )

= r p (t )

This is the case of unlimited growth.
So the Fig. 1 shows that for a positive growth rate, the
trees of a forest after a certain time can be managed in its
full naturally balanced equilibrum state.

3 THE LOGISTIC EQUATION

p (t ) = rp (t )

with the initial population

p (t0 ) = p0 > 0

(1)
A typical application of the logistic equation (see [15] for
details) is a common model of population growth, origi-
nally due to Pierre-François Verhulst in 1838, where the
rate of reproduction is proportional to:
We have the mathematical model described the growth of single spaces population as
• The existing population
• The amount of available resources

p (t ) = rp(t ), p(t0 ) = p0 > 0

(2)
Let

N (t )

be the population of trees. The Malthusian
The solution of this equation is

p(t ) = ce rt

model assumes a rate of growth proportional to the popu- lation N (t ) = aN(t ) . This gives the exponential growth

For the initial condition

p(t ) = p e rt0 e rt

p(t ) = p e r (t t0 )

p(t 0

) = ce rt0

c = 0

e

(3)

law N (t ) = e at N (0), which is only accurate for relatively small values of N (t ) ; overcrowding and competition for

resources lower the rate of growth. A more realistic mod- el assumes a steadily decreasing, eventually negative
which is known as exponential law of growth or Malthu- sian law of growth.
The behavior of the solutions depend upon the growth

growth coefficient a(N ).

In Malthusian model we assume that the rate in which an organism will reproduce or die remain constant.

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In case of many other populations, such as population which exhibits exponential growth for a limited period

N (t ) = N e


(t h t t )


(10)
ultimately approaches to some steady state. The growth of any population in a restricted environment must even-
This guarantees that

N (t ) = N e

for all

t ?: t if

tually be limited because that individual member com-

cu0 (t

k ) u1 (t ) = 0

for

t ?: t . Target conditions of

plete with each other for the limited living space, natural
resources and food. This forces the growth rate to de-
cline,
the form (9) are called Euclidean; those of the form (10)
are called functional. To see the reason for this name, con-

the model, N (t ) = aN(t ) may be modified to

sider for instance the space C [

h, 0]

of continuous

N (t ) = Nr (N )

functions defined in the interval

h t 0 . Given a

(4)

where r(N ) is some decreasing function of N satisfying

function y(t ) , denoted by

fined by

yt ( ) the section of

y( ) de-

r (N ) 0 and the simplest function regarding r(N ),

yt ( ) = y (t + ),

( h 0)

(11)

having this property, is r(N ) = a

bN , a, b > 0 . Thus

Then the target condition (10) can be written as an ordi-

the model with initial population N (t0 ) = N 0 > 0

nary target condition in the space C [

h, 0] ;

N (t ) = N (a

bN ),

N (t0 ) = N 0

N ( ) = N e

(12)
= (a

bN (t ))N (t )

where

N e denotes the constant function. An optimal net

This is the well-known logistic equation. This model gives
good results for bacteria populations, even for human
population [15] but does not describe accurately pheno- mena such as forest growth because of the fact that, the
profit problem is instance, to maximize the functional

t t

J (t , u1, u2 ) = a I u1 ( )d I u0 ( )d

0 0

inhibiting effects of new trees on the growth rate are neg-
with

a , ?: 0

at some fixed time; the first term
ligible until these have reached a certain “adult” size.
represents the profit from logging and the second, the
Thus, the rate should be a function not only of

N (t ) but

also of N (t

h) for a suitable time delay h > 0 , leading

cost of seeding. Clearly, u0 (t ), u1 (t ) are nonnegative and

to the equation called the delayed logistic equation,
it is reasonable to include upper bounds on both rates;

N (t ) = (a

bN (t

h))N (t )

0 u0 (t ) R, 0 u1 (t ) S

(13)
(5) Similar delay effects are observed in the influence of
overcrowding in human populations, for an elementary exposition of logistic equations with and without delays

Straight maximization of the profit may result in destruc- tion of the forest at time t , thus we supplement the prob-
lem with a (functional) target condition, say

equation (4) and (5) has two equilibrium solutions. One is N (t ) = 0 , whereas the other is


N (t ) Ne

E ,


(t h t t )

(14)

N (t ) = N = a

Ne the equilibrium solution (6), and the terminate seed-

e b (6)

ing at time t

k . If the equilibrium position is stable, this

Assume tree seeds are planted, and trees are logged with

means the forest population will stay near equilibrium
seeding and logging rates

u0 (t ) and

u1 (t ) respectively.

after t . Admissible controls for this problem are
Let k be time it takes a seed to become baby tree. Then the equation becomes
pairs (u0

(t ), u1 (t )) , where

u0 is defined in

k t t

N (t ) = (a bN (t h )) N (t ) + cu0 (t k )

u1 (t )

(7)

and u1 is defined in 0 t t , and satisfying (13) in their

where the coefficient

c for (0 c 1)

accounts for the
respective intervals of definition.
The logistic equation discussed above is in dimensional
function of seeds that actually result in a tree.
To start the equation we need to know the forest popula-
form which have some free parameters, but sometimes the dimensional form is transformed into nondimensional

N (t ) = N 0 (t ) (t

h t t0 )

(8)
results in the numerical treatments. We present here the nondimensional form of the logistic equation,

Now in order to attain the equilibrium population N e at

d11

= 11 (1

11 ), where


= t and 11 = N

a certain optimal time t = t

we have the solution, d t N

N (t ) = N

* *

(t )

and we say that the population is at equilibrium at t = t but not necessarily afterwards. If the population is to stay at equilibrium, the target condition must be

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1.2

1

0.8

0.6

0.4

0.2

0

Logistic Equation

0 1 2 3 4 5 6 7 8

where sustainable forest management is the burning issue worldwide. For details servey on greenhouse effects and climate change issues see [2], [7] and [34].
Forests also provide a large variety of services such as: timber production, recreation and landscape, natural habitat for numerous species, protections of watersheds, and protections of villages from avalanches and landslides and buffering as well. Thus from a socio- economic point of view, the optimal management of fo- rests must take these multiple services into account. All the issues mentioned above indicate that a sustainable forest management is a crying need where the optimal policy strategy is the key.

5 THE SUNDARBAN: A CASE STUDY

In this section, we will discuss the necessaity of forest

Fig. 2. The solution of nondimensional logistic equation

The numerical solution of the nondimensional logistic equation for the initial conditions 0.02, 0.2, 0.5, 0.8, 1 and
1.2 is shown in Fig. 2, where the lowest curve is the cha- racteristic ´S-shape´ usually associated with the solution of the logistic equation. This sigmoid curve appears in many other models. See [4], [13] and [15] for details study on logistic equations and numerical applications.
Another growth model is described by the integrodiffe- rential equation of the form,
management of a particular case of Sundarbans. The Sundarbans, situated both in Bangladesh and India is the world's largest delta, formed from the sediments brought down by three great rivers, the Ganges, the Brahmaputra and the Meghna which converge on the Bengal Basin. The forest consists of about 200 islands, separated by some
400 interconnected tidal rivers, creeks and canals. At
10,000 sq. km, it forms the largest mangrove forest in the
world and it is the only mangrove tiger land on the earth.
In 1947 the whole Sundarban mangroves were divided between India and Bangladesh (formerly East Pakistan), sharing 40% in India and 60% in Bangladesh. Although

(

N ( t ) = I a

( 0

I I b ( ) N (t +

) d I I N (t )

the two parts of the Sundarbans differ considerably in the

nature and extent of investigations, conservations and

I \ h ) I

\ ) (15)

managements due to belonging to the separate indepen-

+ cu0 (t k )

u1( t )

dent countries, the natural resources and beauties attract

taking into account the inhibiting effects of new trees of all sizes on the growth rate. Our further research will be focused on that issue.

4 APPLICATION OF CONTROLTHEORY

Numerous applications of control theory in the diverse fields have created this topic more challenging; specially an efficient and sustainable forest management is a cru- cial issue in the present world of climate change as well as the decay of ozone layer. In recent years, a number of economists and other experts have suggested sequester- ing carbon in forests to help mitigate the accumulation of greenhouse gases in the atmosphere. For more details studies see [1], [2], [21] and [22]. Forests currently store a substantial stock of carbon, amounting to 826 billion me- tric tons in trees and soil [11], and society can potentially remove carbon from the atmosphere by taking steps to increase this pool of carbon. These steps may include in- creasing the amount of carbon stored per hectare through management intensity or rotations ages (see for examples [21] and [37]) or increasing the area of land in forests. Carbon sequestration thus offers the promise of reducing the cost of greenhouse gas mitigation, which could lower the price of carbon and reduce global warming. Recent climate change is the global threat on the environment,

the global attentions simultaneously. As a results the In- dian part of Sundarbans had been declared as a world heritage site in 1987 as Sundarbans National Park and Ban- gladesh part was declared as a world heritage site as The Sundarbans in 1997 by International Union for Conserva- tion of Nature of UNESCO. The whole Sundarbans area was declared as Biosphere Reserve in 1989. See [5], [8], [25] and [32] for the details study about the history of Sundarbans and its local and global importance. However we are mainly concern to discuss the management about the Bangladesh part of Sundarbans. Presently the Sun- darbans is one of the world’s new 7 wonders of nature com- petitors’ which will be declared at the end of 2011. The Sundarbans represents nearly half of the remaining forests of Bangladesh and is dominated by halophytic tree species such as sundori (Heritiera fomes) (from which Sundarbans derives its name), gewa (Excoecaria agallocha), goran (Ceriops decandra), baen (Avicennia officinalis), and keora (Sonneratia apetala). Sun- darban is the habitat of many rare and endangered ani- mals (Batagur baska, Pelochelys bibroni, Chelonia mydas), specially the Royal Bengal Tiger (Panthera tigris) and spot- ted deers. Javan rhino, wild buffalo, hog deer, and bark- ing deer are now extinct from the area. The Sundarbans forest is the home to more than 400 tigers. However, the Royal Bengal Tiger is the king of all animals in the Sun- darbans which have developed a unique characteristic of

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swimming in the saline waters, and are world famous for their man-eating tendencies. Apart from the Royal Bengal Tiger; Fishing Cats, Macaques, Wild Boar, Common Grey Mongoose, Fox, Jungle Cat, Flying Fox, Pangolin, Chital, are also found in abundance in the Sundarbans. However, widespread hunting and forest depletion has reduced the tiger’s range and numbers in Sundarbans. Natural disasters as well as climate change are also affecting this world herit- age. In 2007, Cyclone CIDR hit directly on the coastal areas of Sundarbans and almost destroyed the whole forests and thus destroyed the conservation of the wild animals. However, proper management of the Sundarbans is crucial for the sus- tainable dwelling of wildlife and their prey. The major steps against the unplanned or illegal wood-cutting as well as for increasing the number of trees in the forest should be taken. According to the Rio Convention proposed by the United Nations Conference on Environment and Development (UNCED) Bangladesh has the obligation to accomplish the required of the convention which is ‘timely, reliable and ac- curate information on forests and forest ecosystems is essen- tial for public understanding’. In the meantime, to ensure the protection for wildlife habitat and the management of natu- ral resources, three areas within the forest have been desig- nated as Wildlife Sanctuaries: Sundarbans West (715 sq. km), Sundarbans South (370 sq. km), and Sundarbans East (312 sq. km) by Bangladesh Government. A long term plan named Bangladesh Tiger Action Plan 2009-2017 has been taken by Bangladesh Government for the conservation of tigers and other wildlife recourses as a part of forest management. See for details [3], [5].

5.1 Importance of the Sundarbans

The Sundarbans ecosystem is unique in many respects. The area experiences a subtropical monsoon climate with the annual rainfall of about 1600-1800 mm and several cyclonic storms. The biodiversity includes about 350 spe- cies of vascular plants, 250 fishes and 300 birds, besides numerous species of phytoplankton, fungi, bacteria, zooplankton, benthic invertebrates, molluscs, reptiles, amphibians and mammals. Species composition and community structure vary east to west, and along the hydrological and salinity gradients. Large areas of the Sundarbans mangroves have been converted into paddy fields over the past two centuries, and more recently into shrimp farms. The Sundarbans has been extensively ex- ploited for timber, fish, prawns and fodder.

The Sundarbans is the only largest mangrove forest in the world managed for commercial timber pro- duction and has had a history of scientific management since 1879. In Bangladesh it is now managed by the Sun- darban West Forest Division and Sundarban East Forest Division of the Forest Department, divided into 20 sec- tions each harvested in turn on a 20-year cycle, with the three peripheral wildlife sanctuaries on the coast. Early management consisted of revenue collection by enforcing simple felling rules, subsequent enforcement of which reduced the amount of over-cutting of the four main tim- ber species. A wildlife conservation plan prepared under the joint sponsorship of the World Wildlife Fund and the U.S. National Zoological Park emphasised management
of the tiger and other wildlife as an integral part of sus- tainable forest and coastal management for both timber and the needs of the local population.
Approximately 2.5 million people lived in small villages surrounding the Sundarbans in 1981 which by
1991 had increased to 3 million. At nomination, some
35,330 people worked in the forest, 4,580 of whom col-
lected timber and firewood, 1,350 collected honey and
beeswax and 4,500 harvested the natural resources and
hunted mainly deer, and 24,900 were fisherman and
shrimp farmers. Today, the area provides a livelihood at
some seasons of the year for an estimated 300,000 people.
Some 4,500 people in Bangladesh are employed by con-
tractors in the commercial logging of sundori and other
timber, which is 45% of all that produced in state-owned
forests. As well as construction timber they supply local

Fig. 3. The figure shows the statistics of total visitors visited the Sundarbans during 2005 to 2009 . Source: IPAC, 2009.

newsprint paper, match and board mills (see for examples
[8], [32]).
Local people are also dependent on the forests
and waterways for firewood, charcoal, timber for boats
and furniture, poles for house-posts and rafters, nypa
palm thatch for roofing, grass for matting reeds for fenc-
ing, shells and reptile skins, with deer, fish, crabs and
shrimps taken for food. The season for collecting honey and wax is limited to ten weeks from April 1st. Thou- sands of people, with permits from the Forest Depart- ment, enter the forest for nests. Before Cyclone Sidr,
which has destroyed the fishing industry, more than
10,000 fishermen from as far away as Chittagong camped
along the coast for 3-4 months in winter before returning
home at the start of the monsoon season in April, and as
many or more local people fished year-round [32]. In 1986
the average annual catch was 2,500 tonnes.
The important sector based on Sundarbans is

Tourism Industry. The Sundarbans may be more attractive

to the visitors and tourists due to its natural attractive
beauties as well as the rare wild animals which will play a
significant role both in the national and global economy. In 1996, about 500 foreign tourists plus 5,000 local tourists visited the area, most in the South Wildlife Sanctuary and in recent years most nearly 100,000 local and about 1,500

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foreign tourists per year visited the Sundarbans. Al- though there is no potential for mass tourism but limited eco-tourism from October to April or May is possible.
Recent research (see the Fig. 3) on Sundarbans showed
Solution: Let the required time is T . The exponential law of growth is

( ) r (t t0 )

p t = p0 e

(16)
that the number of national and international visitors and
For the given condition,
tourists are increasing.The figure reflects some variability
in visitor numbers over the last five years, with the high-
est numbers in the year 2008-2009. This indicates that effi-

p (t

0 + T ) = 2 p (t0 )

r t

+T t )

cient and sustainable management of Sundarbans and
availability of logistic facilities are necessary for enriching the tourism industry with strong revenue earning.

2 =

help of (16)]

p (t0 + T )

p (t0 )

( 0 0

p0e = 2

r (t0 t0 )

p0e

[with the

6 SUNDARBAN FOREST MANAGEMENTS

The future of the Sundarbans will depend upon the sus-

e rT = 2

rT = log e

log 2

2 T =

r

tainable management of freshwater resources as much as on the conservation of its biological resources. Considera- ble researches both in nationally and internationally have

After

T = log e 2 times the trees in the forest will be

r

been carried out on the Sundarbans ecosystem and its wildlife (see for examples [18], [25], [26] and [33]). Since
the last few decades several national (see [29] ) and inter- national (see [23], [32]) leading NGOs carried out differ- ent (short-terms and long-terms) research projects on Sundarbans emphasizing on the sustainable manage- ments to meet the future challenges for the next genera- tions. They also emphasized on Collaborative forest man- agement (CFM) which is loosely defined as a working part- nership between the key stakeholders in the management of a given forest—key stakeholders being local forest users and state forest departments, as well as parties such as local gov- ernments, civic groups and nongovernmental organisations, and the private sector [12].
The world heritage Sundarban is the source of natural beauty to the tourists as well as to the environmentalists, where the Royal Bengal tiger is one of the attractions to the visitors. However, the natural disasters like flood, cyclone etc. are very common issues taking place in Bangladesh specially in that region. All most every year the natural disasters hit on this forest and as a result, the forest loses its trees and animals and consequently its beauty. Some- times the unplanned cutting of the trees makes the forest to be destroyed day by day. So it becomes necessary to protect the Mangrove from the unavoided situations by making an optimal design for sustainable management. The above mentioned model for example can be applied for such efficient growth of the trees in the Sundarban Mangrove area. A successful application of the Malthu- sian model for the sustainable management of Sundarban Mangrove may ensure the long-term survival with its full beauty within a certain time applied. For a better under- standing of the model we present here a simple example.

6.1 Example

double in compared to its initial state of the trees of the
forest. This solution is shown in the Fig. 4 considering the
growth rate r = 100.

Graph of T=log(2)./r

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 10 20 30 40 50 60 70 80 90 100

r

Fig. 4. The exponential growth of trees in the forest.

7 CONCLUSION

In Sustainable forest management is the warming issue due to the climate change and increasing the greenhouse gases. Optimal control strategy is the key to such conser- vation of forests. Among the different models for this optimal strategy Malthusian growth model is one which gives a good result where the logistic equation plays the vital role. This study investigates the application of con- trol theory in the light of Malthusian model, which is used in controlling the forest growth and sustainable
Suppose the population of forest of Sundarban

p(t ) giv-

management. An illustrative example is presented taking
en by the exponential growth law. Growth rate r is posi-
into account the special case of the largest mangrove for-
est Sundarban. We claim that this study will play pio-
tive. How much time it will take to increase

p0 to

neer role in exploring further study in the field of control theory in agricultural and life sciences. The proposed
double, where

p0 is the initial trees of forest.

model of controlling the optimal growth of the forests in

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the largest mangrove in world can be applied rigorously which may help to increase the different trees in the fo- rests destroyed by the natural disaster every year in the coastal region and thus can make a natural equilibrium. we believe many more interesting conclusions will be brought under the different assumptions by using the optimal control strategy, and we hope to focus on such management issues in our future research, also we wel- come more researchers to take part in the field.

ACKNOWLEDGMENT

The authors greatly acknowledge the logistic support provided by Mathematics Discipline, Khulna Universi- ty, Bangladesh. We would also like to thank the re- viewers for numerous helpful discussions and con- structive suggestions for the modification of this paper.

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