Renewable resource management in a seasonally fluctuating environment with restricted harvesting effort

Mathematical Biosciences xxx (xxxx) xxx–xxx

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Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs

Renewable resource management in a seasonally fluctuating environment with restricted harvesting effort ⁎

Srinivasu D.N. Pichika , Simon D. Zawka Department of Mathematics, Andhra University, Visakhapatnam, 530003, India

A R T I C L E I N F O

A B S T R A C T

Keywords: Bioeconomics Binding constraint Blocked interval Optimal periodic solution

This paper presents bio-economics of a renewable resources in a seasonally changing environment in which the resource exploitation is subjected to restrictions on harvesting effort. The dynamics of the resource is assumed to be governed by the logistic equation. Seasonality is incorporated into the system by choosing the coefficients in the growth equation to be periodic functions with the same period. A linear optimal control problem involving binding constraints on the control variable has been considered. As a result the concept of blocked interval plays a key role in the construction of optimal solution. In view of the periodicity associated with the considered problem, we first construct an optimal periodic solution. The optimal solution is established using the most rapid approach path to the said optimal periodic solution. The global asymptotic stability property of the optimal periodic solution enables construction of a suboptimal solution. The optimal solution is found to be periodic after some finite time and suboptimal solution approaches the optimal periodic solution asymptotically. Key results are illustrated through numerical simulation.

1. Introduction Optimal exploitation of a renewable resource has been a common concern for researchers, owners of the resource as well as open access users. The books authored by Clark [2,3] offer an excellent exposure to various aspects associated with bio-economics of resources such as optimal resource management and sustainable exploitation of resources. Some other works in this area can be found in [7,11,12,18,23,25,27,29,32]. Enormous contributions made on the optimal exploitation of resources pertaining to autonomous systems have driven the researchers to expand the domain to include non-autonomous systems. One of the vital concepts considered for inclusion in this regard is the influence of environmental fluctuations on the resource, in particular, seasonal variations. Incorporating seasonality into the resource dynamics makes the mathematical model more reliable and takes the study closer to reality. Profound influence of the seasonal variations on the dynamics of renewable resources such as fisheries, naturally guide the investigators to model the optimal exploitation problems in a periodic environment. This is to ensure that the study mimics the real world as closely as possible. We expect that outcomes of such studies will be in a position to offer solutions to some practical problems encountered by the exploiters. One of the recent works which is appropriate in this framework is Hasanbulli et al. [5] and references therein. Some recent and relevant

⁎

work on seasonally varying environment can be found in [8,10,14–17,20–22,24,26]. In this context, the work presented in Castilho and Srinivasu [9] becomes significant and it contains several references pertaining to the dynamics and management of renewable resources in periodically fluctuating environment. It presents a method to solve a linear optimal control problem wherein the dynamic constraints are periodic differential equations. Solutions were constructed using Pontryagin’s Maximum principle [34]. Thus an optimal periodic singular control and optimal periodic singular stock path are obtained for the considered optimal harvest problem. In practice we come across situations where it may not be possible to follow the singular stock path due to some restrictions on the fishing effort such as maximum/minimum number of vessels to be used for the harvesting activity. The major driving force for restricting the capacity of fishing vessels is to overcome the over exploitation of the resource and avoid resource depletion. On the other hand the minimum possible number of vessels or the minimum possible fleet size may also be considered in some fisheries management activities due to socioeconomic conditions. The study presented in [13] can be considered as a good example for restriction on vessels to overcome over exploitation. The book by Barkin and DeSombre [4] highlights the need to reduce fishing capacity and to decrease the number of people and amount of capital employed in the industry so that the reduction of over

Corresponding author. E-mail address: [email protected] (S.D.N. Pichika).

https://doi.org/10.1016/j.mbs.2017.12.008 Received 16 December 2016; Received in revised form 16 December 2017; Accepted 26 December 2017 0025-5564/ © 2018 Elsevier Inc. All rights reserved.

Please cite this article as: Pichika, S., Mathematical Biosciences (2018), https://doi.org/10.1016/j.mbs.2017.12.008

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A

B

Es (t)

Es (t) Emin Emax

Es (t)

Es (t) Emin Emax

Effort E

Effort E

Es (t)

Es (t) Time t

t0

Time t

t0 + T

D

C Es (t)

Effort E

Effort E

Es (t)

Es (t) Emin Emax

Es (t) t0

t0 + T

Es (t) Emin Emax

Es (t) t0

Time t

Time t

t0 + T

Fig. 1. This figure presents four distinct relations that can exist between Es (t ), Es (t ), Emax and Emin where Es (t ), Es (t ) are the maximum and minimum values of the periodic singular effort Es(t) and Emax , Emin are the maximum and minimum effort levels respectively. Frame A depicts the situation where Es(t) is not influenced by the bounds Emin and Emax i.e., Emin ≤ Es(t) ≤ Emax . This particular case is considered and studied in [9]. Frame B depicts the situation where Es(t) is influenced by only Emin i.e., Es (t ) < Emin < Es (t ) ≤ Emax (Case I). Frame C depicts the situation where Es(t) is influenced by only Emax i.e., Es (t ) < Emin < Emax < Es (t ) (Case III).

(Case II). Frame D depicts the situation where Es(t) is influenced by both Emin and Emax i.e.,

exploitation of marine fisheries becomes more effective. The studies presented in [19], Hegland [6] discuss various fisheries management systems for controlling fishing effort leading to improved fishery management. These observations bring forward an interesting and practical situation where either a part of the singular control or the whole of it cannot be tracked due to restrictions on the control variable. Hence studying the problem considered in [9], wherein the singular control is influenced by the binding constraints, is fairly important as a real world application. In this case, the singular control fails to be optimal, and the influence of the binding constraints calls for modification of the singular control in order to obtain the optimal control. Here the constraints on the control variable make it infeasible to follow the singular control on some intervals that occur periodically. This forces the control to leave the singular arc temporarily and follow appropriate binding constraints. Consequently, optimal solution is forced to leave singular solution on an interval of positive length which is called as blocked interval [1]. To be more specific, an interval during which constraints of a linear control problem prevent optimal solution from following singular solution is termed as blocked interval [2]. Optimal control problem involving binding constraints was first addressed by Arrow [1] wherein the concept of blocked interval was introduced. It is observed that, whenever a blocked interval is encountered, the optimal control switches from singular control to extremal control prior to the time at which the constraint on the control variable becomes binding. This phenomenon is referred to as premature switching principle for blocked intervals [2]. A few significant applications of the concept of blocked intervals for problems pertaining to optimal exploitation can be found in [2]. For other work related to this area, one can refer [28,30,31,33]. In this article we intend to solve an optimal harvest problem associated with the logistic equation with periodic coefficients. In this study we address the case where the binding constraints on the control variable influence the singular control. In Section 2, we introduce the problem and recall some relevant results that are previously established. In the subsequent section, optimal periodic harvest policies for

various significant cases are derived and the corresponding stock paths are constructed. Taking the initial condition and its deviation from the initial state of optimal stock path into consideration, optimal and suboptimal harvest policies are designed in Section 4. Section 5 presents numerical illustration of some important outcomes in the study. Discussion is presented in Section 6.

2. The problem Consider the optimal harvest problem

max

I (E ) =

Emin ≤ E ≤ Emax

∫0

∞

e−ρt (pqx − c ) Edt

(1)

dx = x (a (t ) − b (t ) x ) − qEx , dt

(2)

x (0) = x 0 > 0

(3)

where the coefficients a(t) and b(t) are T- periodic functions, ρ is the instantaneous discount rate, p is price of unit catch, c is the unit cost of effort, q is the catchability coefficient and Emax (Emin ) represents the maximum(minimum) possible fishing effort in the harvesting activity. We wish to determine the optimal harvest policy denoted by Eo(t) that maximizes (1) and the corresponding stock path denoted by xo(t). In this work ρ, p, q, c, Emax , Emin are assumed to be constants. It is noteworthy that the problem (1)–(3) has been studied earlier [9] to some extent and the following are a few significant outcomes. The singular control Es(t) and the corresponding singular solution xs(t) for the problem (1)–(3) are T- periodic functions, and they are given by

Es (t ) = and 2

pqa′ (t ) + cb′ (t ) − 2pqb′ (t ) x ⎤ 1⎡ a (t ) − b (t ) x − 4pqbx − cb (t ) − pq (a (t ) − ρ) ⎥ q⎢ ⎣ ⎦

(4)

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xs (t ) =

b (t ) c + pq (a (t ) − ρ) +

(b (t ) c + pq (a (t ) − ρ))2 + 8pqbρc 4pqb

3.1. Case I: Es (t ) < Emin < Es (t ) ≤ Emax

.

Let [a, b] be an interval in the positive T- axis such that Es(t) < Emin in (a, b) and Es (a) = Es (b) = Emin . Due to the constraint E ≥ Emin , the optimal harvest policy cannot be singular on [a, b]. Hence it forces optimal stock path to leave xs(t) at a time t1 such that t1 < a and the control turns to the extremal control Emin , resulting in a blocked interval that starts at time t1. Let the end point of this blocked interval be t2, where optimal periodic stock path switches to xs(t) with b < t2. The periodicity requirement on the optimal solution ensures t2 to satisfy b < t2 < t1 + T . If t0 is any point of t-axis such that t0 ∈ (t2 − T , t1), then optimal periodic harvest policy Eop(t) on the interval [t0, t0 + T ] becomes a combination of Emin and Es(t) and it is defined by

(5) The earlier study in [9] was limited to the case where Es(t) satisfied Emin ≤ Es(t) ≤ Emax . This is a special case where the singular control is not influenced by binding constraints. Here we intend to study the cases where at least one of the constraints Emin ≤ E or E ≤ Emax has influence on the singular control Es(t). Thus we solve the problem (1)–(3) for which the singular control Es(t) satisfies one of the following cases 1. Case I: Es (t ) < Emin < Es (t ) ≤ Emax 2. Case II: Emin ≤ Es (t ) < Emax < Es (t ) 3. Case III: Es (t ) < Emin < Emax < Es (t ),

Eop (t ) =

where Es(t) and Es (t ) are minimum and maximum values of the Tperiodic function Es(t). Pictorial representation of these cases is presented in Fig. 1. Due to the linearity of the problem (1)–(3) in the control variable E, the optimal harvest policy shall be a combination of bang-bang and singular controls. In view of the periodicity associated with considered problem and the binding constraints being trivially T- periodic, we first look for an optimal solution that is T- periodic in nature by ignoring the initial condition (3). Thus we concentrate on a suitable interval [t0, t0 + T ] over which the periodic boundary condition x (t0) = x (t0 + T ) is satisfied for a t0 ≥ 0. Henceforth, we concentrate on the following problem

max

I (E ) =

Emin ≤ E ≤ Emax

∫t

t0 + T

e−ρt (pqx − c ) Edt

0

x op (t ) =

x (t0) = x (t0 + T ) > 0.

(8)

t0 + T

0

(10)

where xm(t) denotes the solution of (7) with E (t ) = Emin and x m (t1) = xs (t1) . Now, we have

∫t

PV[t0, t0+ T ] =

t1

e−ρt (pqxs (t ) − c ) Es (t ) dt

0

+

∫t

t2

1

∫t

t0 + T

2

e−ρt (pqx m (t ) − c ) Emin dt e−ρt (pqxs (t ) − c ) Es (t ) dt .

(11)

From the optimality of xop(t) it follows that t1 maximizes (11). This critical value t1 can be identified as follows. By continuity of xop(t) we have

Note that the optimal control and the corresponding optimal stock path of the above problem (6)–(8) are defined on [t0, t0 + T ]. Clearly, the solution of (6)–(8) enables us to design optimal periodic control and corresponding optimal periodic stock path for the control problem (1) and (2) defined on the infinite horizon. This is achieved by extending the optimal control and optimal stock path of (6)–(8) periodically on the entire positive T- axis, which we represent by Eop(t) and xop(t) respectively. We call these functions as optimal periodic harvest policy and optimal periodic stock path respectively. Let

∫t

⎧ xs (t ), if t0 ≤ t ≤ t1 x m (t ), if t1 ≤ t ≤ t2 ⎨ ⎩ xs (t ), if t2 ≤ t ≤ t0 + T ,

+ (7)

(9)

and optimal periodic stock path xop(t) becomes

(6)

dx = x (a (t ) − b (t ) x ) − qEx , dt

PV[t0, t0+ T ] =

⎧ Es (t ), if t0 ≤ t < t1 Emin, if t1 ≤ t < t2 ⎨ ⎩ Es (t ), if t2 ≤ t ≤ t0 + T

xs (t1) = x m (t1)

(12)

and

x m (t2) = xs (t2).

(13)

From (12) to (13) we observe that t2 can be expressed as a function of t1. Thus (11) can be viewed as a function of t1 alone. Now treating t1 as a parameter we solve the equation

dPV[t0, t0+ T ]

e−ρt (pqx op (t ) − c ) Eop (t ) dt

dt1

represent the present value of net revenues along the optimal solution on [t0, t0 + T ]. In the light of the Proposition 3.2 in [9], the following lemma is relevant for the discussion to follow.

= 0,

(14)

where

dPV[t0, t0+ T ] dt1

= e−ρt1 (pqxs (t1) − c ) Es (t1) − e−ρt1 (pqx m (t1) − c ) Emin

Lemma 2.1. Let 0 ≤ E1(t) < E2(t) be two T- periodic functions such that T 1 a − E2 = T ∫0 (a (s ) − E2 (s )) ds is positive. If x E1 (t ) and x E2 (t ) are the nontrivial globally asymptotically stable T- periodic solutions of (2) with E (t ) = E1 (t ) and E (t ) = E2 (t ) respectively, then they satisfy 0 < x E2 (t ) < x E1 (t ) .

+ e−ρt2 (pqx m (t2) − c ) Emin

dt2 dt1

∂x m (t , t1) ⎞ dt ∂t1 ⎠ dt − e−ρt2 (pqxs (t2) − c ) Es (t2) 2 dt . dt1 + pqEmin

∫t

t2

1

e−ρt ⎛ ⎝

⎜

⎟

Let xmp(t) and xMp(t) be the nontrivial globally asymptotically stable T- periodic solutions of (2) with E (t ) = Emin and E (t ) = Emax respectively. Then by Lemma 2.1, we have 0 < xMp(t) < xmp(t).

Simplifying (14) we obtain

3. Characteristics of optimal periodic harvest policy

e−ρt1 (pqxs (t1) − c )(Emin − Es (t1)) + e−ρt2 (pqx m (t2) − c )(Es (t2) dt − Emin ) 2 dt1

For the sake of clarity let us assume that the T- periodic function Es(t) admits at most one crest and one trough in any interval of length T. Below we discuss the characteristics of optimal solution of (6)–(8) for the three cases mentioned in the previous section.

= pqEmin The term 3

dt2 dt 1

∫t

t2

1

e−ρt ⎛ ⎝ ⎜

∂x m (t , t1) ⎞ dt . ∂t1 ⎠ ⎟

(15)

in (15) can be evaluated using (12) and (13). Thus (12),

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(13) and (15) can be used to identify the values of t1 and t2. Here, following Lemma 2.1, we have xMp(t) ≤ xop(t) ≤ xs(t).

⎧ Es (t ), ⎪ Emin, ⎪ Eop (t ) = Es (t ), ⎨ ⎪ Emax , ⎪ Es (t ), ⎩

3.2. Case II: Emin ≤ Es (t ) < Emax < Es (t ) Suppose that we have an interval [c, d] such that Es(t) > Emax in (c, d) and Es (c ) = Es (d ) = Emax . Following similar arguments as in Section 3.1, we can find a blocked interval [τ1, τ2] such that τ1 < c and d < τ2 < τ1 + T . If τ0 ∈ (τ2 − T , τ1), then the optimal periodic harvest policy Eop(t) on [τ0, τ0 + T ] becomes

Eop (t ) =

⎧ Es (t ), if τ0 ≤ t < τ1 Emax , if τ1 ≤ t < τ2 ⎨ ⎩ Es (t ), if τ2 ≤ t ≤ τ0 + T

(21)

and the corresponding stock path xop(t) takes the form

⎧ xs (t ), ⎪ x m (t ), ⎪ x op (t ) = xs (t ), ⎨ ⎪ xM (t ), ⎪ xs (t ), ⎩

(16)

if if if if if

t0 ≤ t ≤ t1 t1 ≤ t ≤ t2 t2 ≤ t ≤ τ1 τ1 ≤ t ≤ τ2 τ2 ≤ t ≤ t0 + T .

(22)

Here we have

and the stock path xop(t) takes the form

PV[t0, t0+ T ] =

⎧ xs (t ), if τ0 ≤ t ≤ τ1 ⎪ x op (t ) = xM (t ), if τ1 ≤ t ≤ τ2 ⎨ ⎪ xs (t ), if τ2 ≤ t ≤ τ0 + T , ⎩

t0 ≤ t < t1 t1 ≤ t < t2 t2 ≤ t < τ1 τ1 ≤ t < τ2 τ2 ≤ t ≤ t0 + T

if if if if if

∫t

t1

0

e−ρt (pqxs (t ) − c ) Es (t ) dt t2

∫t e−ρt (pqxm (t ) − c) Emin dt τ + ∫ e−ρt (pqxs (t ) − c ) Es (t ) t τ + ∫ e−ρt (pqxM (t ) − c ) Emax dt τ +

1

1

(17)

2

2

where xM(t) denotes the solution of (7) with E (t ) = Emax and xM (τ1) = xs (τ1) . By a parallel argument to that presented in the Section 3.1, the times τ1 and τ2 can be computed using the relations

e−ρτ1 (pqx

s (τ1)

− c )(Emax − Es (τ1)) +

e−ρτ2 (pqxM (τ2)

1

+

∫τ

τ2

1

e−ρt ⎛ ⎝ ⎜

∂xM (t , τ1) ⎞ dt , ∂τ1 ⎠ ⎟

(18) (19)

(20)

From Lemma 2.1, we have xs(t) ≤ xop(t) ≤ xmp(t) . 3.3. Case III: Es (t ) < Emin < Emax < Es (t ) Observe that the binding constraints in this case are nothing but the binding constraints involved in the Sections 3.1 and 3.2 discussed above. From the property of Es(t), we can find two disjoint intervals [a, b], [c, d] contained in an interval of length T, say [t0, t0 + T ] such that the behaviour of Es(t) on the intervals [a, b] and [c, d] is as described in Sections 3.1 and 3.2 respectively. Further, with no loss of generality, we assume that b < c. Below we list the four subcases that arise due to the relations that exist among t1, t2, τ1 and τ2. These subcases are given by 1. 2. 3. 4.

Subcase Subcase Subcase Subcase

(23)

3.3.2. Subcase ii: τ1 < t2 and τ2 − T < t1 Here, we have [t1, t2]⋂[τ1, τ2] ≠ ϕ . Clearly, optimal periodic stock path cannot switch to xs(t) in (b, c) and the extremal control Emax cannot be employed at τ1. The reason being, enforcing Emin at t1 with initial stock level xs(t1) forces the resulting stock path to fall below xs(t) at t = τ1 and continuity of xop(t) does not permit implementation of Emax at τ1 with stock level xs(τ1). Thus the optimal periodic harvest policy in (a, d) shall be a combination of Emin and Emax only. Consequently, there exists a switching point t1* in (b, c) at which the control switches from Emin to Emax and this control follows Emax until the time t2* in (d, t1 + T ), where xop(t) switches to xs(t). If t0 is any point in (t2* − T , t1), then the optimal periodic harvest policy Eop(t) on the interval [t0, t0 + T ] takes the form

and

xM (τ2) = xs (τ2).

e−ρt (pqxs (t ) − c ) Es (t ) dt .

From the optimality of xop(t), the parameters t1 and τ1 maximize (23). Clearly, computations of the time sets {t1, t2} and {τ1, τ2} are independent of each other due to the fact that t2 < τ1 and τ2 − T < t1. Hence these time sets can be obtained directly using the relations (12), (13), (15) and (18)-(20) respectively.

− c )(Es (τ2)

xs (τ1) = xM (τ1)

t0 + T

2

dτ − Emax ) 2 dτ1 = pqEmax

∫τ

⎧ Es (t ), if t0 ≤ t < t1 ⎪ Emin, if t1 ≤ t < t1* Eop (t ) = ⎨ Emax , if t1* ≤ t < t2* ⎪ ⎩ Es (t ), if t2* ≤ t ≤ t0 + T

(24)

and the corresponding stock path xop(t) is

i: t2 < τ1 and τ2 − T < t1 ii: τ1 < t2 and τ2 − T < t1 iii: t2 < τ1 and t1 < τ2 − T iv: τ1 ≤ t2 and t1 ≤ τ2 − T .

⎧ xs (t ), if t0 ≤ t ≤ t1 ⎪ x m (t ), if t1 ≤ t ≤ t1* x op (t ) = ⎨ xM (t ), if t1* ≤ t ≤ t2* ⎪ ⎩ xs (t ), if t2* ≤ t ≤ t0 + T .

In all these subcases, the optimal periodic stock path xop(t) satisfies the relation xMp(t) ≤ xop(t) ≤ xmp(t). Here we discuss subcases i and ii. The remaining two cases viz., iii and iv are presented in online appendix associated with this article.

(25)

Now, we have

PV[t0, t0+ T ] =

3.3.1. Subcase i: t2 < τ1 and τ2 − T < t1 In this case, we observe that [τ1 − T , τ2 − T ]⋂ [t1, t2] = ϕ = [t1, t2]⋂[τ1, τ2]. This implies that the optimal periodic stock path shall be xs(t) in the intervals [τ2 − T , t1] and [t2, τ2]. In fact, for any t0 ∈ (τ2 − T , t1) the optimal periodic harvest policy Eop(t) on the interval [t0, t0 + T ] becomes

∫t

t1

0

+

e−ρt (pqxs (t ) − c ) Es (t ) dt

∫t

t1*

1

+

t2*

∫t *

e−ρt (pqx m (t ) − c ) Emin dt

e−ρt (pqxM (t ) − c ) Emax dt

1

+

t0 + T

∫t * 2

4

e−ρt (pqxs (t ) − c ) Es (t ) dt .

(26)

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By continuity of xop(t) we have

xs (t1) = x m (t1)

(27)

x m (t1*) = xM (t1*)

(28)

xM (t2*) = xs (t2*).

(29)

E (t ) = Emax , where x0 > xop(0). If tM is such that xM (tM , 0, x 0) = x op (tM ), then we have the optimal harvest policy Eo(t) on the positive t-axis to be

Emax , if 0 ≤ t < tM Eo (t ) = ⎧ E (t ), if tM ≤ t < ∞ ⎨ ⎩ op

(34)

From the above relations we observe that t1* depends on t1 and t2* depends on t1* . Hence both t1* and t2* can be viewed as functions of t1. Thus, we have

and the corresponding stock path becomes

dPV[t0, t0+ T ]

xM (t , 0, x 0), if 0 ≤ t ≤ tM x o (t ) = ⎧ x (t ), if tM ≤ t < ∞. ⎨ ⎩ op

dt1

= e−ρt1 (pqxs (t1) − c ) Es (t1) − e−ρt1 (pqx m (t1) − c ) Emin dt * + e−ρt1* (pqx m (t1*) − c ) Emin 1 dt1 t1*

A suboptimal stock path xso(t) can be obtained by following the optimal periodic harvest policy Eop(t) right from the initial time t = 0 with initial state x0. This suboptimal stock path xso(t) approaches the optimal periodic stock path xop(t) asymptotically.

∂x m (t , t1) ⎞ dt ∂t1 ⎠ dt * + e−ρt2* (pqxM (t2*) − c ) Emax 2 − e−ρt1* (pqxM (t1*) dt1 dt1* − c ) Emax dt1 + pqEmin

∫t

1

t2*

e−ρt ⎛ ⎝

⎜

⎟

∂xM (t , t1) ⎞ dt ∂t1 1 ⎠ dt * − e−ρt2* (pqxs (t2*) − c ) Es (t2*) 2 . dt1

+ pqEmax

∫t *

e−ρt ⎛ ⎝

⎜

5. Numerical simulations This section provides illustrative examples for the construction of Eop(t), xop(t), Eo(t), xo(t) and xso(t) for the subcases i and ii (of Case III) discussed in Section 3. Numerical study associated with subcases iii and iv (of Case III) can be found in online appendix associated with this article. Numerical study associated with Subcase i (of Case III) directly guides the construction of optimal solution for cases I and II. The examples represent the dynamics of a fish population in a periodically fluctuating environment where the coefficient functions and parameters considered are related to actual values one might have in a fishery as considered in [7]. Now let us consider the problem (1)–(3) with r . The specific values assigned to the a (t ) = r and b (t ) = 2πt

⎟

(30)

Simplifying (30) and equating it to zero, we obtain

e−ρt1 (pqxs (t1) − c )(Emin − Es (t1)) = e−ρt1* (pqx m (t1*) − c )(Emin dt * − Emax ) 1 dt1 dt * ρt − * + e 2 (pqxM (t2*) − c )(Emax − Es (t2*)) 2 dt1 + pEmin + pqEmax

t2*

∫t * 1

The terms

dt1* dt 1

and

e−ρt ⎛ ⎝

⎜

dt2* dt 1

∫t

t1*

1

e−ρt ⎛ ⎝ ⎜

⎞ K − σK sin ⎛ ⎝ cycle ⎠

involved parameters and their significance in the model are presented in Table 1. Table 2 presents some significant results obtained through numerical computation.

∂x m (t , t1) ⎞ dt ∂t1 ⎠

∂xM (t , t1) ⎞ dt . ∂t1 ⎠

⎟

Example 1. (Illustration pertaining to Subcase i). Let us consider the problem (1)–(3) with associated data as given in Table 1 for Subcase i. The computed values of t1, t2, τ1 and τ2 (which are presented in Table 2) confirm that the considered problem satisfies Subcase i. The T-periodic functions Es(t) and xs(t), which are defined by (4) and (5), are respectively shown in frames A and C of Fig. 2. The blocked intervals [t1, t2] and [τ1, τ2] in this case are [14.25, 31.473] and [44.26, 63.458] respectively. Using (21) and (22) the optimal periodic harvest policy Eop(t) and the corresponding optimal periodic stock path xop(t) are computed and presented in frames A and C of Fig. 2 respectively. Since the given initial state x 0 = 2.7 × 106 is different from that of xop(t), which is 2.09 × 106, initial adjustment is required that drives the stock from its initial state to xop(t) optimally. This optimal approach path takes the time tM = 9.81(years) to reach xop(t) under the extremal control Emax = 450 (vessels) and this can be seen in Frame D of Fig. 2. Using (34) and (35), the optimal harvest policy Eo(t) is given by

⎟

(31)

can be evaluated using (27)–(29). Therefore, (31)

together with (27)–(29) define t1, t1* and t2*. 4. Optimal and suboptimal harvest policies In the previous section, we have solved the optimal control problem (6)–(8) and hence constructed optimal periodic harvest policy and the corresponding optimal periodic stock path for (1) and (2) which are given by Eop(t) and xop(t) respectively. Observe that xop(t) need not necessarily satisfy the initial condition prescribed in the optimal control problem (1)–(3). Hence, in order to obtain the solution of (1)–(3), there is a necessity for an initial adjustment that drives the state from its initial position to xop(t) optimally. In view of the linearity of the considered problem with respect to the control variable E, the needed initial adjustment can be achieved by the most rapid approach path. Let xm(t, 0, x0) denote the solution of (2) and (3) with E (t ) = Emin , where x0 < xop(0). If tm is such that x m (tm, 0, x 0) = x op (tm), then we have the optimal harvest policy Eo(t) on the positive t-axis to be

Emin, if 0 ≤ t < tm Eo (t ) = ⎧ E (t ), if tm ≤ t < ∞ ⎨ ⎩ op

Table 1 Details of various parameters and constants pertaining to the subcases i and ii. Parameters Carrying capacity Intrinsic growth rate Unit cost of effort Unit price of the catch Catchability coefficient Amplitude of K fluctuation Environmental cycle Discount rate Minimum harvest effort Maximum harvest effort

(32)

and the corresponding optimal stock path is given by

x m (t , 0, x 0), if 0 ≤ t ≤ tm x o (t ) = ⎧ if tm ≤ t < ∞. x (t ), ⎨ ⎩ op

(35)

(33)

Similarly, let xM(t, 0, x0) denote the solution of (2) and (3) with 5

Symbol K r c p q σK T ρ Emin Emax

Subcase i

Subcase ii 6

3.5 × 10 0.36 4.5 × 104 150 0.0004 1.5 × 106 55 0.05 300 450

6

3.5 × 10 0.20 4.5 × 104 150 0.0004 1.5 × 106 55 0.05 230 295

units ton 1/year $/vessel/year $/ton 1/vessel ton years 1/year vessels vessels

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S.D.N. Pichika, S.D. Zawka

Example 2. (Illustration pertaining to Subcase ii). Consider the problem (1)–(3) with associated data as given in Table 1 for Subcase ii. The computed values of t1, t2, τ1 and τ2 (which are presented in Table 2) confirm that the considered problem satisfies Subcase ii. Thus, using the relations (31) and (27)–(29) the blocked interval [t1, t2*] to be [10.8, 64.81] where the control switches from Emin to Emax at a time t1* = 46.05. Using (24) and (25), optimal periodic effort Eop(t) and the corresponding optimal periodic stock xop(t) are evaluated and shown in frames A and C of Fig. 4. Since the given initial state x 0 = 2.5 × 106 is different from that of xop(t), which is 1.93 × 106, the initial adjustment is required that drives the stock from the given initial state to the path xop(t). This is implemented in the period [0, 12.32], where the initial adjustment effort and the corresponding optimal approach path are shown in frames B and D of Fig. 4 respectively. Thus, using (34) and (35), the optimal harvest policy Eo(t) is given by

Table 2 Table presents results of various computations pertaining to the subcases i and ii. Subcase i t2 < τ1& τ2 − T < t1

Subcase ii τ1 < t2 & τ2 − T < t1

units

t1 t2 τ1 τ2 Emax to Es(t) Es(t) to Emin Emin to Es(t) Es(t) to Emax Emax to Emin Emin to Emax PVo[0,2T ]

14.25 31.473 44.26 63.548 τ2-T t1 t2 τ1 — — 5.1485 × 108

10.8 47.04 46.9 64.97 64.81 − T t1 — — — 46.05 2.9482 × 108

years years years years years years years years years years $

PVso[0,2T ]

5.1475 × 108

2.7994 × 108

$

Emax , if 0 ≤ t < 12.32 Eo (t ) = ⎧ E (t ), if t ≥ 12.32, ⎨ ⎩ op

Emax , if 0 ≤ t < 9.81 Eo (t ) = ⎧ E (t ), if t ≥ 9.81, ⎨ ⎩ op

where the optimal periodic harvest policy Eop(t) on one cycle [0, T] is given by

where the optimal periodic harvest policy Eop(t) on one cycle, [0, T] is given by

⎧ Emax , ⎪ Es (t ), ⎪ Eop (t ) = Emin, ⎨ ⎪ Es (t ), ⎪ Emax , ⎩

if if if if if

⎧ Emax , ⎪ Es (t ), Eop (t ) = ⎨ Emin, ⎪E , ⎩ max

0 ≤ t < 8.548 8.548 ≤ t < 14.25 14.25 ≤ t < 31.473 31.473 ≤ t < 44.26 44.26 ≤ t ≤ T .

A

B

520

Es (t) Emin(= 300 vessels) Emax (= 450 vessels) Initial adjustment effort

E(vessels)

E(vessels)

Es (t) Es (t) Emin (=300 vessels) Emax (=450 vessels)

225 0

Time t(years) 6

x 10

T

225 0 9.81

2T

6

C

2.85

xs (t) xs (t) xm (t) xM (t)

1.1 0

x 10

T

Time t(years)

2T

D xs (t) xm (t) xM (t) Optimal approach path

x(ton)

x(ton)

2.85

0 ≤ t < 9.81 9.81 ≤ t < 10.8 10.8 ≤ t < 46.05 46.05 ≤ t ≤ T .

The optimal effort function Eo(t) and the corresponding optimal stock xs(t) are presented in frames B and D of Fig. 4. Note that premature switching also occurs in Subcases ii and iii (which is presented in online appendix associated with this article), and it is similar to what is presented in Fig. 3 for Subcase i. We know that, in all the four subcases, a suboptimal solution xso(t) can be obtained by implementing the optimal periodic effort function Eop(t) from t = 0 . This is presented in Fig. 5 (for subcases i and ii), where the asymptotic approach of the suboptimal stock path xso(t) to xop(t) is

The optimal effort Eo(t) and the corresponding stock path xo(t) are respectively presented in frames B and D of Fig. 2. We know that premature switching takes place whenever the optimal control switches from the singular control Es(t) to any of the extremal controls. Hence we have occurrence of premature switching in Subcase i, and it is highlighted in Fig. 3.

520

if if if if

Time t(years)

T

1.1 0 9.81

2T

T

Time t(years)

2T

Fig. 2. This figure presents illustrations pertaining to Subcase i. In frames A and C, the continuous curves made up of dotted blue and solid blue represent the singular effort Es(t) and the corresponding singular stock path xs(t) respectively. The discontinuous periodic function in Frame A, which is made up of magenta, solid blue and green segments, represents the optimal periodic effort Eop(t). The continuous curve in Frame C, made up of magenta, solid blue and green segments, represents the optimal periodic stock path xop(t) corresponding to Eop(t). Since the initial value of the stock (2.7 × 106) is different from that of xop(t) (2.09 × 106), an optimal approach path to xop(t) is constructed to obtain the needed optimal solution (Eo(t), xo(t)) and it is presented in the frames B and D. Frame B presents Eo(t) which is made up of initial adjustment effort (Emax , shown as a black segment) in the period [0, 9.81], followed by Eop(t). Frame D presents optimal path xo(t) (corresponding to Eo(t)) consisting of the optimal approach path (shown as a black curve) in the period [0, 9.81] followed by xop(t). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Es (t) Es (t) Emin (=300) Emax (=450)

A 300

450 Effort E

14.25

14.26

450

300

Fig. 3. This figure presents a closer view to Eop(t) (pertaining to Subcase i) to highlight the premature switching that is taking place whenever the optimal control switches from Es(t) to any of the extremal controls. The blowup in Frame A shows the premature switching that is taking place at t1 = 14.25 prior to the time t = 14.26, where the constraint in the control variable becomes binding. The blowup in Frame B shows the premature switching that is taking place at τ1 = 44.26 prior to the time t = 44.28, where the constraint in the control variable becomes binding. Note that the optimal control switches from singular control Es(t) to the extremal controls Emin and Emax at t1 and τ1 respectively.

B

0

14.25

31.473

44.26

44.28

44.26

T

Time t(years)

63.458

A

B

350

E(vessels)

E(vessels)

350

Es (t) Emin (= 230 vessels) Emax (= 295 vessels) Initial adjustment effort

Es (t) Es (t) Emin (=230 vessels) Emax (=295 vessels)

100 0

Time t(years) 6

x 10

100 0

2T

12.32 6

C

2.65

xs (t) xs (t) xm (t) xM (t)

Time t(years)

T

2T

D

x 10

xs (t) xm (t) xM (t) Optimal approach path

x(ton)

x(ton)

2.65

T

1 0

Time t(years)

T

1 0

2T

12.32

T

Time t(years)

2T

Fig. 4. This figure presents illustrations pertaining to Subcase ii. In frames A and C, the continuous curves made up of dotted blue and solid blue represent the singular effort Es(t) and the corresponding singular stock path xs(t) respectively. The discontinuous periodic function in Frame A, which is made up of green, magenta and solid blue curves, represents the optimal periodic effort Eop(t). The continuous curve in Frame C, made up of green, magenta and solid blue curves, represents the optimal periodic stock path xop(t) corresponding to Eop(t). Since the initial value of the stock (2.5 × 106) is different from that of xop(t) (1.93 × 106), an optimal approach path to xop(t) is constructed to obtain the needed optimal solution (Eo(t), xo(t)) and it is presented in the frames B and D. Frame B presents optimal effort Eo(t) which is made up of initial adjustment effort (Emax , shown as a black segment) in the period [0, 12.32], followed by Eop(t). Frame D presents optimal stock path xo(t) (corresponding to Eo(t)) consisting of the optimal approach path (shown as a black curve) in the period [0, 12.32] followed by xop(t). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6

2.85

x 10

6

A

2.5

x 10

B xop (t)

xop (t)

xso (t)

x(ton)

x(ton)

xso (t)

1.1 0

Time t(years)

T

0

2T

Time t(years)

T

2T

Fig. 5. This figure demonstrates the global asymptotic stability of the optimal periodic solution xop(t) for subcases i and ii (of Case III). Frames A and B represent the subcases i and ii respectively. Here it is shown that the suboptimal solution xso(t) approaches the optimal periodic solution xop(t) asymptotically. In both the frames, the red and blue curves stand for xso(t) and xop(t) respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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S.D.N. Pichika, S.D. Zawka 6

3.55

x 10

6

A

2.5

x 10

B xop (t) xmp (t) xMp (t)

0.8 0

x(ton)

x(ton)

xop (t) xmp (t) xM p (t)

Time t(years)

T

0.9 0

2T

Time t(years)

T

2T

Fig. 6. This figure demonstrates Lemma 2.1 for subcases i and ii (of Case III). Frames A and B represent the subcases i and ii respectively. In both the frames, the continuous green (magenta) curve denotes the globally asymptotically stable solution xmp(t) (xMp(t)) of (2) with E = Emin (E = Emax ) and the continuous blue curve (which is sandwiched between the green and magenta curves) represents the optimal periodic stock path xop(t). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

bang and singular controls, as expected for any linear optimal control problem. While the optimal periodic harvest policy and optimal harvest policies happen to be discontinuous, the corresponding stock paths are continuous in nature. In addition to deriving optimal harvest policy, suboptimal harvest policies are also constructed for the considered problems. When only one of the binding constraints was considered to be restricting the singular control, the corresponding optimal periodic control came out to be a combination of the singular control and the extremal control that is restricting the singular control. When both the binding constraints were restricting the singular control, there arose four subcases, each possessing optimal periodic solution with distinct behaviour. The relative position of the extremal control to the extremal values of the singular control and inter distance between them seem to be a factor that decides the Subcase. The distinct behaviour of the optimal periodic solution is identified by the cyclic sequence of the singular and extremal controls that appear in the construction of the optimal periodic solution. These cyclic sequences are Es − Emin − Es − Emax − Es , Es − Emin − Emax − Es , Es − Emax − Emin − Es and Emin − Emax − Emin (or Emax − Emin − Emax ). In this work we considered the simplest T- periodic function Es(t) which has at most one local minimum and one local maximum on any interval of length T. This choice is made only for the sake of clarity and the results are also the same for a periodic effort function which has two or more local minim(and/or local maximum) on each period. The bounds Emin and Emax in the considered problem are assumed to be constants. Similar results hold for more general cases in which Emin and Emax are T- periodic functions. The parameters such as the discount rate ρ, the price p and the cost c are considered to be constants in this work. Similar results can be developed for more general case where ρ, p and c are T- periodic functions.

clearly seen. For the said subcases, the present value of the total net revenues (PV) is computed along the optimal path xo(t) as well as the suboptimal path xso(t), and the results are presented in Table 2. The computed results clearly exhibit the supremacy of optimal solution over the suboptimal solution. Lemma 2.1 is demonstrated through numerical simulation (for subcases i and ii), and the results are presented in Fig. 6. Similar results for subcases iii and iv can be found in online appendix associated with this article. 6. Discussion In this work we studied the bio-economics associated with a renewable resource in a seasonally varying environment involving binding constraints. From a real world perspective, binding constraints on the control variable such as fishing effort play a vital role in conservation and optimal management of resources. The intention to overcome the over exploitation of resources is one of the major factors to implement restrictions on the fishing effort such as number of vessels. The other factors include proper utilization of the available fishing effort and ensuring resource sustainability. This highlights the need to study control problems associated with resource management wherein the singular control is influenced by the constraints on the control variable. In the present study, it is assumed that the resource grows as per logistic equation with involved coefficients being periodic of the same period. Here, the periodic coefficients accommodate seasonality in the environment. The work involves solving an optimal linear control problem associated with binding constraints on the control variable. Although, the considered optimal control problem admits a periodic singular control, it fails to be optimal in the current study due to the singular control being restricted by the binding constraints. This leads to the interesting situation where periodic occurrence of premature switching is observed in the optimal control. Taking into consideration the periodic influence of binding constraints on the singular control, an optimal periodic solution has been formulated for the considered control problem involving binding constraints. Thus, we obtain an optimal periodic harvest policy that is a combination of singular and extremal controls. In this study we considered six distinct cases involving binding constraints on the control variable and derived respective solutions. Though the environmental periodicity drives the problem to admit an optimal periodic solution, any mismatch with the initial state of the stock with that of the periodic solution calls for an initial adjustment wherein the stock path is driven to the optimal periodic solution optimally. The considered problem being linear in the control variable, this is achieved by most rapid approach path. Thus we also derive an optimal harvest policy along with optimal periodic harvest policy. In any case we find the optimal control to be a combination of bang-

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