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An age-structured fishery model: Dynamics and optimal management with perfect elastic demand
Accepted Manuscript An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand S.M. Bouguima, S. Benzerdjeb PII: DOI: Reference:
S0307-904X(15)00362-5 http://dx.doi.org/10.1016/j.apm.2015.04.049 APM 10597
To appear in:
Appl. Math. Modelling
Please cite this article as: S.M. Bouguima, S. Benzerdjeb, An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm. 2015.04.049
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand. S.M .BOUGUIMA(1)
and
S.BENZERDJEB
Department of Mathematics, Faculty of sciences, University of Tlemcen, B. P.119, Tlemcen 13000Algeria (1)
Corresponding author: E-mail: [email protected] January 15, 2015
Abstract. We consider a juvenile adult model with spatial structure and selective harvesting e¤ort on adult population .The dynamic behavior of the model system is investigated.The results indicate that competition between adults and cannibalism on juveniles do not a¤ect the population survival.Bioeconomic equilibrium and optimal harvesting are discussed.Finally some numerical simulations are given to illustrate theoretical results. Key words: Adult-Juvenile Model, Optimal Control, Periodic Solutions, Fishing e¤ort.
1. Introduction Mathematical models describing the e¤ect of harvesting, are complex. There have been a need to know the harvesting strategies that result in maximum sustainable yields and not leading to instabilities or extinction of the species. Various dynamic models have been analyzed by considering economic and ecological factors, see for instance the works of (Chaudhuri, 1986; Clarck, 1979, 1990; Denis, 2008; Jelijer et al., 2004; Kar and Kumar, 2009; Murray, 1993) and the references therein. In this paper, we perform the analysis of a …shing model. To reduce complexity; some assumptions are made. Firstly, it is assumed that only mature population is susceptible to exploitation .This can be achieved with measures such as increasing the mesh size of the net. Secondly, we assume two times scales, a fast one associated to a quick movement of …sh in , and a slow one which corresponds to the demographic parameters of …sh population.The model is a reaction-di¤usion version of those based on classical stock-recruitment relation , see for instance Beverton-Holt Model for competition between juveniles, studied by Beverton and Holt (1993), and Ricker Model with cannibalism developed by Ricker (1954).We will show that cannibalism and competition have no direct e¤ect on the population survival. We will study the global existence and asymptotic behavior in time of solutions .The model is partially cooperative. This di¢ culty will be handled using the theory of fast -slow systems recently investigated by Sanchez et al. (2011) , the approach used by the authors allows one to focus the study on an ordinary di¤erential equations governing the total population density, obtained by integration over the spatial domain and slow time scale. We will illustrate the results by giving an application to an optimal harvesting problem. Optimal control on bioeconomic systems without age-dependence is largely investigated, we refer the reader to ( Clarck, 1990; Conrad, 1995) 1
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.2
and the references therein for a nice overview. Incorporating age-structure in modelling, is more realistic and we refer to(Chakrobory, 2012; Jerry and Raissi., 2005; Wu and Chen., 2009, Lenhart and Montero, 2001; Tahvonen, 2011, 2014; Wang and Wang, 2004).Belyakov and Velivio (2014) discussed a problem with age-dependence and selective harvesting. To our knowledge, the coupling between di¤usion and age-dependence has not been completely investigated in …shing models, and due to complexity they introduce, a few results are obtained for these models, see for instance the results obtained by Fu et al. (2014) , Lenhart and Montero (2001) where the authors establish existence of optimal control and characterize it by a system of partial di¤erential equation.This paper is organized as follows. The next section is devoted to the formulation of the model. In section 3, global existence and positiveness of solutions to system (1) are established. In section 4, we focus on system (2) and derive conditions of population survival. Then in section 5, we establish asymptotic behavior for system (1) .In section 6, we discuss optimal harvesting. Some simulations are given in section 7, to illustrate the results. In the last section, we discuss achievements of our paper with some open problems. 2.
Description of the model
We consider a time continuous stage structured model of a harvested marine population, living in a spatial region ; that is bounded an regular in Rn , n 2: The population is composed of two classes, adult and juveniles. Let u (t; x) and v (t; x) be the biomass of adults and juveniles that at time t are occupying the position x, respectively. We assume that the whole population is subject to a random dispersal with di¤usion constant d > 0: An application of balance law gives the following system @ @t U @ @t V
d d
U = (x)V V = b (x)U
(x)U 2 E (x)U f (x)V d (x)U V
in in
[0; T ]; [0; T ]:
(1)
The system is completed by the Neumann boundary condition @U @V = =0 @n @n where n is an outward pointing normal to @
on @
[0; T ]:
, and the initial conditions
U (0; x) = U0; V (0; x) = V0
in
:
The homogeneous Neumann boundary condition means that there is no migration of the species across the boundary of their habitat. The prescribed initial functions U0 and V0 are continuous and non negative. The functions ; ; E ; b ; f and d are positive continuous functions on . The rate of maturation of young population into adult population is given by : The birth rate by which young population is produced by adult population is the function b . The coe¢ cient measures competition between adults for food and space. The function E can be interpreted as harvesting a portion of adult populations. Function f measures all causes of mortality on young population, including natural mortality, predation by other species, and mortality caused by …shing activity. We assume that the adult prey on juveniles and the rate of cannibalism is d :
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.3
If d =
1 "
with " > 0 small enough, then by integrating (1) over
we obtain
Z Z Z U (t; x)dx = (x)V (t; x)dx (x)U (t; x)2 dx E (x)U (t; x)dx; Z Z Z Z > d V (t; x)dx = b (x)U (t; x)dx : f (x)V (t; x)dx d (x)V (t; x)U (t; x)dx: @t 8 > <
d @t
Z
Taking as new variables the total populations of adults and juveniles u(t) =
Z
U (t; x; )dx , v(t) =
Z
V (t; x)dx:
we obtain u0 (t)
=
v 0 (t)
=
Z Z
Z
(x)V (t; x)dx
Z
b (x)U (t; x)dx
(x)U (t; x)2 dx Z
f (x)V (t; x)dx
Z
E (x)U (t; x)dx;
d (x)V (t; x)U (t; x)dx:
The expression of the right hand side is given in terms of the variables U and V . We make the approximation, (see Sanchez et al. (2011)) u(t) ; V (t; x) vol ( )
U (t; x)
v(t) : vol ( )
therefore 8 <
du dt dv dt
= v u2 Eu; = bu f v duv; : u(0) = u0 , v(0) = v0 :
where
=
E
d
=
=
Z Z Z
(x)dx vol ( )
,
b=
Z
E (x)dx vol ( )
,
f=
(2)
b (x)dx ; vol ( ) Z f (x)dx vol ( )
;
d (x)dx 2
vol ( )
:
By theorem 4 in Sanchez et al.(2011), system (2) is a good approximation to system (1): The spatial structure has been taken into account in the parameters. The dynamic of system (1) depends on the dynamics of the aggregated system (2).
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.4
3.
Existence of Solutions to system (1)
All solutions of (1) starting with non negative functions U0 ; V0 0 are global in time and non negative for any t 0: Indeed, the local existence and the positiveness are consequence of theorem 3.1 in [24]. Global existence, that is the solutions are de…ned on the whole t 0 is established for positive solutions. Proposition 1. The solutions U (t; ); V (t; ) are de…ned on [0; +1) : Proof.
For T > 0, consider the cooperative system 8 @ + d U+ = V + > @t U > < @ + d V + = bU + @t V + U (0; x) = kU0; k1; V + (0; x) = kV0 k1 > > : @V + @U + on @ [0; T ] @n = @n = 0
in in
[0; +1[ [0; +1[ in
@
[0; +1[
Let U + (t) and V + (t) be a solution of the previous system that depends only on t. Then (U + ; V + ) satisfy the linear ordinary di¤erential equation 8 d + = V+ in [0; +1[ ; > dt U > < d + + V = bU in [0; +1[ ; dt + + U (0) = kU k V (0) = kV k in : > 0; 1; 0 1 > : From comparison principle, see theorem 3.4 in Smith (1995), we have 0
U (t; x)
U + (t) and 0
V (t; x)
V + (t):
We conclude that (U (t; ); V (t; )) are de…ned globally 3.1.
Steady state solutions for system (1). Consider the system 8 (x)U 2 E (x)U in ; < d U = (x)V d V = b (x)U f (x)V d (x)U V in ; : @U @V = = 0 on @ : @n @n
(S)
Similar arguments as in Brown and Zhang (2003) and Bouguima et al.(2007), show that a necessary and su¢ cient condition for the existence of at least one positive solution to (S), is that the trivial solution must be unstable. For g a continuous function on , we denote by g=inf g and g = sup g : Proposition 2. If Proof.
b > E f , then the trivial solution of system (1) is unstable.
Let L=
d 0
0 d
and A(x) =
E (x) b (x)
(x) f (x)
:
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.5
then the operator L A has a principal eigenvalue denoted by 1 (L A) : To show that (0; 0) is unstable, it su¢ ces to prove that 1 (L A) < 0: Let (Up ; Vp ) be the principal eigenfunction associated to 1 (L A) : Hence 8 (x)Vp + E (x)Up = 1 (L A) Up in ; < d Up d Vp b (x)Up + f (x)Vp = 1 (L A) Vp in ; : @Up @Vp on @ : @n = @n = 0 Integrating on
, we obtain Z
d
Up dx =
Z
d
Vp dx = 0:
This implies that
and
Z
(
(x)Vp
E (x)Up ) dx =
Z
1
(L
A) Up dx:
Z
(b (x)Up
f (x)Vp ) dx =
Z
1
(L
A) Vp dx:
Since Up and Vp are strictly positive, we conclude that 1 (L
A)
Z
Up dx
Z
Vp dx
and 1
(L
A)
Let
Z
=Z
b
Z
Vp dx
E
Z
Up dx:
Z
Up dx
f
Z
Vp dx:
Vp dx > 0: Up dx
We have 1
(L
A)
and 1
(L
A)
E : b
f :
We will study inf max >0
E ;
b
f
:
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.6
Consider the equation E =
b
A positive root of the previous equation is q 2 (E f )+ E = 2
f :
f
2
+4
b
:
Consequently max
E ;
>0
b
f
=
(
E if f if
b
; :
Hence inf max
E ;
>0
b
f
=
E =
b E f q (E + f ) + 2 (E f )2 + 4
It follows that (L
A)
f ;
4
=
1
b
4
b E f q (E + f ) + 2 (E f )2 + 4
: b
: b
Remark 3. See remark 5 below for an ecological interpretation of the condition given in the previous proposition.
4.
Equilibrium and dissipativness of System (2)
System (2) has to be analyzed with the initial conditions u(0) > 0; v(0) > 0: We observe that the right-hand side of (2) is a smooth function of the variables (u; v), hence the local existence and uniqueness properties are obtained for the corresponding Cauchy problem. The state space of(2) remains in the positive octant R2+ := f(u; v) : u > 0; v > 0g : Indeed ,the set R2+ is positively invariant since the vector …eld of equation (2) is inward 2 on the boundary @R+ :Now, we shall prove the dissipativness of system (2). Theorem 4. All solutions of the system(2) that start in R2+ , are in the compact K=
(u; v) 2 R2+ : u + v
b f
:
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.7
Proof.
We de…ne the function w(t) = dw dt
u(t)
+
v(t) f :
b u f
Therefore,
u2 :
From standard comparison principle, we get w
z:
where z is the solution of the logistic equation z0 =
b z f
z2:
Hence w
z
lim supz(t) t!+1
b f
:
So, we obtain the dissipativness properties of system (2) . Now, we investigate non-negative equilibria for system (2). The following proposition, gives the existence of equilibria . Proposition 5. System (2) have two equilibriums: i) The trivial equilibrium point P0 (0; 0): ii) The coexistence equilibrium P (u ; v ) exists if and only if E < condition , u and v are given by p f dE + ( f Ed)2 + 4 db u = ; 2 d 2 p ( f Ed)2 + 4 db f d2 E 2 v = : 4 d2
b f :
Under this
(3)
Proof. A direct way for …nding equilibriums is to consider the intersection of the nuclines. We should solve the two algebraic equations v bu
u2 fv
Eu = 0;
(4)
duv = 0:
(5)
The shape of the curves suggests that there is only the trivial point and an interior equilibrium provided that b E< : (6) f This equilibrium corresponds to the value p f dE + ( f Ed)2 + 4 db u = ; 2 d 2 p ( f Ed)2 + 4 db f d2 E 2 v = : 4 d2 The …gure2 shows the shape of the nuclines.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.8
Remark 6. Ecologically, condition(6) means that for small …shing e¤ort, the …sh population can be sustained at an appropriate equilibrium level.
Figure 1: Represent existence of the trivial equilibrum if E > fb
b ;then f there exists an unique equilibrium point P which must be globally stable. The arrows denote the direction of the trajectory, indeed any trajectory enters from the exterior to the interior Figure 2: The isoclines shows that if E <
Remark 7. Since d (x)
0 and not identical to zero, then
Z
d (x)dx > 0. This implies
that d > 0: We argue similarly for other coe¢ cients, except for E : In fact, E can be identically zero in the case of no harvesting.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.9
4.1.
Dynamic behavior. The local stability of an equilibrium can be studied by
analyzing the Jacobian matrix of the system (2 u + E) b dv
J (u; v) =
(f + du)
:
The stability of the trivial equilibrium P0 (0; 0) characterizes the ability of the species to survive. Theorem 8. i) If E > fb ;then the origin P0 is a stable node, ii) If E < fb ;then the origin P0 is a saddle point. Proof.
The Jacobian matrix calculated at the origin P0 is E b
J (0; 0) =
f
:
It is clear that the trace satis…es T rJ =
1
+
2
< 0:
and the determinant DetJ =
1 2
= Ef
b:
Since a necessary and su¢ cient condition for local stability in two dimensions is T rJ < 0 and DetJ > 0: Then the assertions i) and ii) are immediate. Remark 9. Although, …shing is restricted to adults; juveniles may disappear when …shing e¤ort exceeds a threshold value. Indeed, mature females cannot produce eggs when they are captured. Theorem 10. If E < Proof.
b f
is veri…ed, then the equilibrium P given by (3) is locally stable.
Observe that u =
E (2 and v = 2
with =
p
f 2d
Since u > 0; then necessarily >
E : 2
E) (2 4
> 0:
+ E)
;
(8)
(9)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.10
The Jacobian matrix calculated at the point (u ; v ) is " 2 J (u ; v ) = d b 4 (2 E) (2 + E) It is clear that T rJ =
(2
+
2 f +2 d 2
Ed
2 f +2 d 2
Ed
#
:
) < 0;
and
d (2 E) (2 + E) b: (10) 4 A simple analysis implies that det J > 0: Indeed, we shall consider this determinant as a second degree polynomial in , denoted by: det J = (2 f + 2 d
P( ) = 3 d
Ed) +
2
+ (2 f
E2d 4
Ed)
The polynomial P ( ) has two roots of di¤erent signs 2
=
(2 f
Ed)+ 6 d
p
> 0 where f
= (2 f Ed >
2
b: 1
=
2 f
p Ed+ 6 d
< 0 and
2 2
Ed) + 3E d + 12d b: Since; u > 0; we have r f Ed 2 db ( ) +4 ;
and 2 2
>
q
(
f
Ed 2 )
+4
db
p =
6d
3 2
q
(
2( b
f E)
f
+4
Ed 2 )
db
+
p
> 0:
(10)
We deduce that det J > 0: Therefore P = (u ; v ) ;is a positive equilibrium locally stable. In the next theorem, we shall show that the system (2) has no positive periodic solutions. Our method involves an application of the criterion of Bendixon Amann (1990) (see theorem.4.1, inVerhulst(1996)).The dissipativity of the system and Poincaré-Bendixon theorem Amann (1990) imply that the positive solution of system (2) tends either to the origin or to (u ; v ) : Theorem 11. The system given by (2) admits no periodic solution. In addition, If b < Ef then (u(t); v(t)) tends to (0; 0): If b > Ef then (u(t); v(t)) tends to (u ; v ): Proof.
and
Let
du = v dt dv = bu dt
then div(F; G) =
u2
fv
@F @G + = @u @v
Eu = F (u; v);
duv = G(u; v):
(2 u + e)
(f + du):
(11)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.11
For all u 0 and v 0, since all other parameters are strictly positive, the sign of div(F; G) in Eq(11) is strictly negative. Therefore, by applying Bendixon Criterion Amann (1990), there are no limit cycles within the interior of the positive octant in the state space (u; v). From Dulac principle and dissipativness of the system, it follows that (u(t); v(t)) tends either to (0; 0) or (u ; v ) : Remark 12. When the fecundity b and the maturity rate are low, or mortality f is high, there will not be enough juveniles that will support the population in longtime and the population goes to extinction. Corollary 13. i) If E > ii) If E < Proof.
b f ;then
b f ;then
the origin P0 is globally asymptotically stable.
the equilibrium P is globally asymptotically stable .
It is a direct consequence of theorem 8, 10 and 11.
It is mathematically interesting to see what happens if b = Ef , since in this case the equilibrium is not hyperbolic and the method of linearization is not conclusive. Proposition 14. If b = Ef , then (0; 0) is globally asymptotically stable in R2+ : Proof.
We construct the following Lyapunov function V =
u
+
v f
Calculating the derivative of V along solutions of (2), we obtain dV = dt
b f
in Int R2+ : In addition, ically stable in
E
u
1 duv f
u2 =
1 duv f
u2 < 0
lim V = +1: This implies that (0; 0) is globally asymptot-
u;v!+1
R2+ :
5. Fast slow dynamics In the planar case, the long time behavior of trajectories is simple when di¤usion is ignored. In this section, we will explore the asymptotic behavior in time when the di¤usion is supposed fast. Let " > 0 small enough. Proposition 15. a) If b > Ef; then system (1) has a compact attractor close to P b) If b < Ef; then system (1) has a compact attractor close to P0 : Proof.
The proof is a consequence of the theorem 4, in Sanchez et al .(2001).
The case of b = Ef is more complicate. Firstly, we will obtain some a priori estimates for system (1). For simplicity, we assume all the coe¢ cients of (1) are constant. The following theorem establishes the uniform boundedness of system (1).
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.12
Proposition 16. If b = Ef , then the continuous solutions to (1), corresponding to nonnegative continuous initial data are uniformly bounded on t and on " > 0: Proof.
Consider the cooperative system 8 @ + d U + = V + EU + > @t U > < @ + d V + = bU + f V + @t V + U (0; x) = kU0; k1; V + (0; x) = kV0 k1 > > : @U + @V + on @ [0; T ] @n = @n = 0
in in
[0; +1[ ; [0; +1[ ; in ;
@
[0; +1[ :
(C1)
Let U + (t) and V + (t) be a solution of (C1) that depends only on t. Then (U + ; V + ) satis…es the linear ordinary di¤erential equation 8 d + = V + EU + in [0; +1[ ; > dt U > < d + + + V = bU f V in [0; +1[ ; dt (C2) + + U (0) = kU k V (0) = kV k in : > 0; 0 1 1; > : From comparison principle, see theorem 3.4 in Smith (1995) we have 0
U + (t) and 0
U (t; x)
V + (t):
V (t; x)
The matrix E b has two eigenvalues 1 = 0 and are uniformly bounded on t:
2
f
;
< 0. It follows that the solutions U + (t) and V + (t)
Proposition 17. If b = Ef , then the continuous solutions to (1), corresponding to nonnegative continuous initial data, tend to (0; 0) when " goes to 0+ and t is large enough and …xed: Proof. From theorem 5 in [23], the solutions (U; V ) of system (1) veri…es for x 2 and t > 0 1 U u = + r" (x; t); V v V ol( ) where (u; v) is a solution of system (2) corresponding to the initial data u(0) = u0 =
Z
U (0; x; )dx, v(0) = v0 =
and jr" (x; t)j
a1 "eta2 + a3 ea4
Z
t "
V (0; x)dx;
:
uniformly on :Here ai are a positive constant. Since in this case (0; 0) is globally asymptotically stable for system (2), it follows that (u; v) is close to (0; 0) for t large enough. Hence(U; V ) tends to (0; 0) when " goes to 0+ and t, …xed and large enough.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.13
6.
Bioeconomic harvesting
We suppose that the selling price is constant. This corresponds to the case when the yield faces a perfectly elastic demand. For instance, the price is determined by the international market, see for instance (Knowler et al.2001). 6.1.
Open access …shery.
The optimization problem is to determine the optimal
…shing e¤ort to provide a maximum rent. We assume a constant price p per unit of harvest biomass. The sustainable yield is Y (E) = Eu ; where u is the adult equilibrium of system (2).The catch of resource multiplied by their price gives the total revenue T R = pY (E): The cost function is T C = c(u )E: The sustainable economic rent which provided by the …shing e¤ort E; is given by ER = T R
T C = pEu
c(u )E:
So, we have the bioeconomic e¤ort E1 when T R = T C: If E > E1 ; Then, the total cost of …shing is greater than the total revenue .Hence over…shing generates a fall in income for …shermen. If E < E1 ; The pro…t is positive. The economic objective is to maximize the pro…t which is the di¤erence between the income and the cost. (E) = E (pu
c(u )) :
(12)
We know that when …shable resource decreases, the e¤ort is more expensive. Hence c is assumed to decrease with stock size. We can choose a function cost such that c (u) = cu0 : with c0 a positive constant. Without lost of generality, we take c0 = 1: In this case, a simple calculation gives the bioeconomic e¤ort p p b p p E1 = p ; f p+d p
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.14
and E1 > 0 if
f+
p>
p
2f 2
+4 f b
2 b
!2
:
Under the condition b > Ef; the function (E) is continuous and (0) = 0: We assume that 0 (0) > 0: Hence, we obtain 0
(E) = (pu
c(u )) + E
d (pu dE
c(u )) ;
and 0
(0) = pu (0)
This implies that u (0) >
c(u (0)) > 0: r 2
1 : p
Remark 18. The stock of mature …sh should be large enough to allow harvesting. Now, our purpose is to obtain the best control of …shing e¤ort E , so it is necessary to have the following condition @ = 0 () pu @E
@u @c (u ) p = 0: @E @u h h The function (E) is continuous on the I = 0; fb interval and lim (E) = c (u ) + E
E!
b f
(13) 1: This
implies that the maximum of (E) is achieved at a point Eopt 2 I: @c Substituting c (u ) and its derivative @u (u ) into Eq (13) ;we get the optimal harvesting equation h i 2p b) 3 (1 ) E 4 + f (d + b(ppfb2 dd22d ) E 2 + pf 2 d2 E + p (14) d(1 ) f b E p(pfd2 bd2 ) = 0 : p(pf 2 d2 ) pf 2 d2 It is easy to calculate the algebraic solution Eop and is depending both on biological, environmental and economic parameters. If d is large enough, then the number of sign changes in the sequence of coe¢ cients in the previous polynomial is 4: From Descartes Principle, see for instance Murray (1993), p.704, there are exactly either 4 or 2 positive roots of equation (14). In other side, from the main theorem in Stefansson (2008), (see the appendix) any positive root Eop of (14) has an upper bound given by Eop It is clear that
2f (d d2
2f (d d2 2p b) pf 2
2p b) : pf 2 b f
Remark 19. Hence, if d is large enough, then the optimal …shing Eop goes to zero and this leads to a dissipation of economic rent.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.15
If d is small and is large enough, then the number of sign changes in the sequence of coe¢ cients in the previous polynomial is 1 and Descartes principle, there is exactly one positive root of equation (14): Remark 20. The previous uniqueness result is di¤erent from that obtained in [18], where the authors, under conditions on the smallness of the time interval, uniqueness of optimal control is established. 6.2. Optimal harvesting policy. Assume a constant discount rate : for simplicity, we suppose that the harvesting cost is constant. Let c be the constant …shing cost per unit e¤ort, and let p be the constant price per unit biomass of the mature. The net revenue of harvesting at any time is given by (E) = E (pu
c) :
In order to maximize the rent for all future, we consider the following optimization problem Z +1 supJ(E) = sup e t (E)dt; E
E
0
subjected to the state equation. We apply Pontryagin’s maximum principle. The pseudo-Hamiltonian is t
H=e
E (pu
c) +
u2
v
1
Eu +
2
(bu
fv
duv) :
(15)
where 1 (t), 2 (t) are the adjoint variables, E is the control variable satisfying the constraints 0 E Emax : Here Emax is the feasible upper limit of the …shing e¤ort. Since the pseudo-Hamiltonian H is linear in the control variable E with coe¢ cient e t (pu c) 1 u , the optimal control will be a combination of extreme controls and the singular control. The necessary condition for the maximization of H is @H e t (pu c) = 0 =) 1 = : @E u According to Pontryagin’s maximum principle, the adjoint equations are
Let
1
=
_1
=
_2
=
1e
t
@H = @u @H = @v
and
d dt d dt
2
1
=
1
=
2
=
=
t
Epe 1 t
2e
pu
c u
+
2
1
(2 u + E)
2
(b
dv) ;
(f + du) :
(17) (18)
;then, the equations (16),(17) and (18), becomes:
=p
Ep + 1
+
+
(16)
1
c ; u
(2 u + E + )
2
(19) 2
(b
dv) ;
(f + du + ) :
The optimal control E that maximizes H must satisfy the following conditions:
(20) (21)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.16
E
=
0;
when
E
= Emax ;
when
1 1
>p
c ; u c : u
Remark 21. The quantity 1 e t is called the shadow price and p uc is the net economic revenue on a unit harvest. One must …sh as much as possible when the shadow price is low enough 1 e t < p uc and no …sh at all when the shadow price is high 1 e t > p uc : When 1 equals the net economic revenue on a unit harvest p uc , then the pseudoHamiltonian H becomes independent of the control variable E and the maximum condition does not provide any information about E in this case. Using (19); the derivative of (19)and (20) we get cu0 = u2
pu
c
(2 u + E + )
u
Substituting u0 = v
u2 2
=
2 (b
dv)
Ep:
(22)
Eu into (22) , we obtain u(pu
c) ( u + ) + pu3 u2 (b dv)
c v
:
(23)
We restrict ourselves to optimization at equilibrium which is much easier to handle than the general case. At the positive equilibrium, we have u0 = v 0 = 0 and 0 consequently 2 = 0. A simple calculation gives in this case, the harvest rate E = h(u) = (pu
c)
(f + du + ) (2 u + ) + du2 c (f + du + ) du(pu c)
b
:
It refelects the direct change in …shing e¤ort due to a change in adult stock. Since E must satisfy E < fb ; then E is not feasible for every value of u.Since d dt 2 = 0, then 1 + 2 (f + du + ) = 0; and 1
=
2
(f + du + )
=
(f + du + ) u(pu
c) ( u + ) + pu3 u2 (b dv)
c v
:
and the necessary condition for maximization of the pseudo-Hamiltonian becomes c (f + du + ) u(pu c) ( u + ) + pu3 = u u2 (b dv)
p
c v
;
This implies that pu
c=
u2 (b
dv)
pu3 c v ; (f + du + ) u ( u + )
Hence (E) = E (pu is a decreasing function with respect to when goes to +1:
c) :
and the economic rent (E) tends to zero
Remark 22. When the time preference of the present generation is high, the net bene…ts is lower.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.17
7.
Numerical simulations
In this section, we give some numerical examples. Due to a lack of real world data, we have used hypothetical data to illustrate the analytical results. We assume that = 3:9, = 1:04, b = 2:3; f = 0:5, d = 0:2: Example 23. In this case, we choose the value of E = 1 which veri…es E < Considering system (2) with initial condition (6; 2);we have the following
b f
= 17:94:
This …gure represents the growth of the mature and immature populations in time. The intersection of both curves gives the non-negative equilibrium (u ; v ), which is u = 0:2112; and v = 0:0660:
Example 24. Let p = 1:5 be the selling price, using the data of the previous example, we …nd the control value which is represented by the intersection of the curve with the
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.18
x-axis.
Figure 4: Roots of equation (14) The …gure 04 shows the existence of 3 real roots of equation (14), which are represented in the …gure by the intersection the curve with the x-axes (y = 0): Here E = 6; 121:
8. Conclusion In this paper, a bioeconomic model attempting to relate the reproduction of a renewable resource with its exploitation is performed. We have analyzed the dynamics of …shery model with a stage structured population. It has been assumed the existence of competition between adults and cannibalism on juveniles populations. The exploitation of …sh population is assumed on adult population which has a commercial value.Jerry and Raissi (2005) studied a similar …shing model without intraspeci…c competition between species and cannibalism. Firstly, using stability theory of ordinary di¤erential equations, it has been proved that the interior equilibrium exists under certain conditions and it is globally asymptotically stable. Even when the …shing e¤ort is restricted to adult population, both adults and juveniles can become extinct.because the ecosystem is often altered by the human activities, we analyze the harvesting strategy that results in maximizing the pro…t and not leading to extinction. We obtain the optimal harvesting equation and deduce the e¤ect of cannibalism on economic rent. Our approach consists on the following idea. We take advantage from existence of two time scales (see Sanchez et al. 2011), a fast one for …sh di¤usion and a slow one for …sh growth and mortality to reduce our model to an ODE system. The major contributions of the present paper are: a) the persistence of the species depend on two factors:the …rst is biological and concerns the ability that a species reproduces itself, this is a function of birth rate, natural mortality of young population and the rate of maturity. For instance, if the growth rate is no longer su¢ cient to replace
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.19
loses, the stock collapses. The second is linked to the exploitation and mecanisms that reduce the …shing e¤ort. This latter can be reduced by limiting the time of …shing, reducing the capacity that a vessel can carry, or closing areas where the birth rate is high. b) Cannibalism and competition have no direct e¤ect on the population survival. c) Using a recent result of Stefansson (2008), we show that under suitable conditions, there will be a dissipation of economic rent in an open access …shery. d) In the case of total discounted net bene…t, the …shing e¤ort is given as a function of the mature stock. This result is di¢ cult to obtain without the method of fast-slow system, see for instance Fu et al. (2014) , Lenhart and Montero (2001). There are still many interesting and challenging mathematical questions need to be studied for system (1). For example, the model ignore seasonal or stochastic variation, reserve area (see for instance Lv et al .(2013) and the references therein). A further research may incorporate these aspects in the model. It would happen that …shers do not exclude the …shing of the juvenile and it is interesting to consider exploitation taking into account juvenile stage. The selling price is assumed constant, it would be interesting to consider a variable price of …sh. For related questions with other methodologies using the EOQ model we refer to S.K.De and S.S.Sana (2014) and the references therein).
Acknowledgements The authors are grateful to the reviewers for their critical evaluations and suggestions. A.
Appendix:
Let us recall the following result on upper bounds of positive roots of a polynomial given in [28] Theorem 25. Let p(x) = xd
b1 xd
m1
bk xd
mk
+
X
j6=m1 ;
with b1 ; :::; bk are strictly positive and aj B1 (p) = max
0 for all j 2 fm1 ; 1
kb1 ) m1 ; ::::(kbk
aj xd
j
;mk
; mk g :The number
1 mk
is an upper bound for the positive roots of p: References [1] Amann, H(1990). Ordinary Di¤ erential Equations: An Introduction to Nonlinear Analysis, Walter de Gruyter, Berlin. New York : [2] Beverton, R.J.H &Holt, S.J(1993). Recruitment and Egg-production.In: On the Dynamics of Exploited Fish Population , Vol.6, Chapman &Hall, London, 44-67. [3] Belyakov, A.O & Veliov, V.M.(2014).Constant versus periodic …shing:age structured optimal control approach, Math.Model. Nat.Phenom .9, 20-37.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.20
[4] Bouguima, S.M., Fekih, S.& Hennaoui, W.(2007).Spatial structure in a juvenile-adult model, Nonlinear Anal.Real .World.Appl. 1184-1201. [5] Brown, K.J &Zhang, Y (2003).On a system of reaction-di¤usion equations describing a population with two-age groups, J.Math.Anal.Appl., 282(2) , 444-452. [6] Clark, C.W(1979) .Mathematical models in the economics of renewable resources., SIAM Rev.21, 81-99. [7] Clarck, C.W(1990). Mathematical Bioeconomics:The optimal Management of Renewable Resource, Wiley, New York, 10-69. [8] Conrad, J.M(1995). Bioeconomic models of the …shery.In: Bromley (Ed). Handbook of Environmental Economics. Blackwells, Oxford & Cambridge, 405-432. [9] Chakrabory, K ., Jana, S, & Kar,T.K (2012), Global dynamics and bifurcation in a stage-structured prey-predator …shery model with harvesting, Appl.Math.Comput., 218, 9271-9290. [10] De, S.K., & Sana, S.S(2014).An alternative fuzzy EOQ model with backlogging for selling price and promotional e¤ort sensitive demand., Int.J.Appl.Comput.Math.(to appear). [11] L. Denis , Notions de controle optimal appliquées à la gestion d’une ressource renouvelable, Collège militaire royal de Saint-Jean, Bulletin AMQ, Vol. XLVIII, No4; décembre 2008: [12] Chaudhuri, K.S(1986). A bioeconomic model of harvesting, a multispecies …shery, Ecol.Model., 32(1986), 267-279. [13] Fu, J., Wu, X. & Zhu, H(2014) .Optimal harvesting control for agedependent competing population with di¤usion, Bound.Value Probl, 147, 1-11. [14] Wu, H. & Chen, F(2009). Harvesting of a single-species system incorporating stage structure and toxicity, Discrete. Dyn.Nat.Soc., 1-16. [15] Jelijer, H., Jeroen, C.J.M & Vanden, B(2004).Harvesting and conservation in a predator–prey system, J.Econom.Dynam.Control., 29, 1097-1120. [16] Kar, T. K.& Kumar, C. S(2009).Bioeconomic modelling: An application to the North-East- Atlantic Cod Fishery, Journal of Mathematics research., vol.1, No.2, 164 178: [17] Knowler, D., Barbier, E.B., & Strand, I (2001).An open access-Model of …sheries and Nutrient Enrichment in the Black sea, Marine Resource Economics., 16(3) , 195-217. [18] Lenhart , S & Montero, J.A(2001) .Optimal control of harvesting in a parabolic system modeling two subpopulations, Math.Methods Appl.Sci.,11, 1129-1141. [19] Murray, J.D(1993). Mathematical Biology, Second Edition, Springer, Berlin .
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.21
[20] Jerry, M & Raissi, N (2005). Optimal strategy for structured model of …shing problem, C.R.Biologies.? 328, 351-356. [21] Lv, Y, Yuan, R& Pei, Y(2013). A prey-predator model with harvesting for …shery resource with reserve area, Appl.Math.Modell., 37, 3048-3062. [22] Ricker,W.E. (1954). Stock and recruitment, J.Fish.Res.Board.Canada.11 , 559-623. [23] Sanchez, E, Auger, P & Poggiale, J.C (2011). Two -time scales in spatially structured models of population dynamics: Asemigroup approach, J. Math.Anal.and Appl., 375, 149-165. [24] Smith, H.L (1995) .Monotone Dynamical Systems: An introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, AMS, Vol.41, Providence. [25] Smoller, J (1994). Shock Waves and Reaction-Di¤ usion Equations, Springer-Verlag, NewYork . [26] Tahvonen, O. (2011) .Age structured optimization models in …sheries bioeconomics: a survey ’Optimal Control of Age-structured population in Economy, Demography, and the Environment’. In: Bouccekine, R, Hritonenko, N& Y.Yartensko (Eds). Environmental Economics, Routledge ,Taylor and Francis, UK, 140-173. [27] Tahvonen, O., Kumpuler, J. &Pekkarinen, A..J(2014). Optimal harvesting of an age-structured, two sex herbivore-plant systems, Ecol.Model., 272, 348-361. [28] Stefansson, D(2008). Computation of Dominant Reals Roots of Polynomilas, Prog.and Compt.Software, 34, 69-74 . [29] Verhulst, F. (1996) Nonlinear Di¤ erential Equations and Dynamical Systems, Springer, Berlin. [30] Wang , J & Wang, K.E. (2004) Optimal Harvesting Policy for Single Population with Stage Structure, Comput.Math.Appl.,48, 943-950.
S0307-904X(15)00362-5 http://dx.doi.org/10.1016/j.apm.2015.04.049 APM 10597
To appear in:
Appl. Math. Modelling
Please cite this article as: S.M. Bouguima, S. Benzerdjeb, An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm. 2015.04.049
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand. S.M .BOUGUIMA(1)
and
S.BENZERDJEB
Department of Mathematics, Faculty of sciences, University of Tlemcen, B. P.119, Tlemcen 13000Algeria (1)
Corresponding author: E-mail: [email protected] January 15, 2015
Abstract. We consider a juvenile adult model with spatial structure and selective harvesting e¤ort on adult population .The dynamic behavior of the model system is investigated.The results indicate that competition between adults and cannibalism on juveniles do not a¤ect the population survival.Bioeconomic equilibrium and optimal harvesting are discussed.Finally some numerical simulations are given to illustrate theoretical results. Key words: Adult-Juvenile Model, Optimal Control, Periodic Solutions, Fishing e¤ort.
1. Introduction Mathematical models describing the e¤ect of harvesting, are complex. There have been a need to know the harvesting strategies that result in maximum sustainable yields and not leading to instabilities or extinction of the species. Various dynamic models have been analyzed by considering economic and ecological factors, see for instance the works of (Chaudhuri, 1986; Clarck, 1979, 1990; Denis, 2008; Jelijer et al., 2004; Kar and Kumar, 2009; Murray, 1993) and the references therein. In this paper, we perform the analysis of a …shing model. To reduce complexity; some assumptions are made. Firstly, it is assumed that only mature population is susceptible to exploitation .This can be achieved with measures such as increasing the mesh size of the net. Secondly, we assume two times scales, a fast one associated to a quick movement of …sh in , and a slow one which corresponds to the demographic parameters of …sh population.The model is a reaction-di¤usion version of those based on classical stock-recruitment relation , see for instance Beverton-Holt Model for competition between juveniles, studied by Beverton and Holt (1993), and Ricker Model with cannibalism developed by Ricker (1954).We will show that cannibalism and competition have no direct e¤ect on the population survival. We will study the global existence and asymptotic behavior in time of solutions .The model is partially cooperative. This di¢ culty will be handled using the theory of fast -slow systems recently investigated by Sanchez et al. (2011) , the approach used by the authors allows one to focus the study on an ordinary di¤erential equations governing the total population density, obtained by integration over the spatial domain and slow time scale. We will illustrate the results by giving an application to an optimal harvesting problem. Optimal control on bioeconomic systems without age-dependence is largely investigated, we refer the reader to ( Clarck, 1990; Conrad, 1995) 1
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.2
and the references therein for a nice overview. Incorporating age-structure in modelling, is more realistic and we refer to(Chakrobory, 2012; Jerry and Raissi., 2005; Wu and Chen., 2009, Lenhart and Montero, 2001; Tahvonen, 2011, 2014; Wang and Wang, 2004).Belyakov and Velivio (2014) discussed a problem with age-dependence and selective harvesting. To our knowledge, the coupling between di¤usion and age-dependence has not been completely investigated in …shing models, and due to complexity they introduce, a few results are obtained for these models, see for instance the results obtained by Fu et al. (2014) , Lenhart and Montero (2001) where the authors establish existence of optimal control and characterize it by a system of partial di¤erential equation.This paper is organized as follows. The next section is devoted to the formulation of the model. In section 3, global existence and positiveness of solutions to system (1) are established. In section 4, we focus on system (2) and derive conditions of population survival. Then in section 5, we establish asymptotic behavior for system (1) .In section 6, we discuss optimal harvesting. Some simulations are given in section 7, to illustrate the results. In the last section, we discuss achievements of our paper with some open problems. 2.
Description of the model
We consider a time continuous stage structured model of a harvested marine population, living in a spatial region ; that is bounded an regular in Rn , n 2: The population is composed of two classes, adult and juveniles. Let u (t; x) and v (t; x) be the biomass of adults and juveniles that at time t are occupying the position x, respectively. We assume that the whole population is subject to a random dispersal with di¤usion constant d > 0: An application of balance law gives the following system @ @t U @ @t V
d d
U = (x)V V = b (x)U
(x)U 2 E (x)U f (x)V d (x)U V
in in
[0; T ]; [0; T ]:
(1)
The system is completed by the Neumann boundary condition @U @V = =0 @n @n where n is an outward pointing normal to @
on @
[0; T ]:
, and the initial conditions
U (0; x) = U0; V (0; x) = V0
in
:
The homogeneous Neumann boundary condition means that there is no migration of the species across the boundary of their habitat. The prescribed initial functions U0 and V0 are continuous and non negative. The functions ; ; E ; b ; f and d are positive continuous functions on . The rate of maturation of young population into adult population is given by : The birth rate by which young population is produced by adult population is the function b . The coe¢ cient measures competition between adults for food and space. The function E can be interpreted as harvesting a portion of adult populations. Function f measures all causes of mortality on young population, including natural mortality, predation by other species, and mortality caused by …shing activity. We assume that the adult prey on juveniles and the rate of cannibalism is d :
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.3
If d =
1 "
with " > 0 small enough, then by integrating (1) over
we obtain
Z Z Z U (t; x)dx = (x)V (t; x)dx (x)U (t; x)2 dx E (x)U (t; x)dx; Z Z Z Z > d V (t; x)dx = b (x)U (t; x)dx : f (x)V (t; x)dx d (x)V (t; x)U (t; x)dx: @t 8 > <
d @t
Z
Taking as new variables the total populations of adults and juveniles u(t) =
Z
U (t; x; )dx , v(t) =
Z
V (t; x)dx:
we obtain u0 (t)
=
v 0 (t)
=
Z Z
Z
(x)V (t; x)dx
Z
b (x)U (t; x)dx
(x)U (t; x)2 dx Z
f (x)V (t; x)dx
Z
E (x)U (t; x)dx;
d (x)V (t; x)U (t; x)dx:
The expression of the right hand side is given in terms of the variables U and V . We make the approximation, (see Sanchez et al. (2011)) u(t) ; V (t; x) vol ( )
U (t; x)
v(t) : vol ( )
therefore 8 <
du dt dv dt
= v u2 Eu; = bu f v duv; : u(0) = u0 , v(0) = v0 :
where
=
E
d
=
=
Z Z Z
(x)dx vol ( )
,
b=
Z
E (x)dx vol ( )
,
f=
(2)
b (x)dx ; vol ( ) Z f (x)dx vol ( )
;
d (x)dx 2
vol ( )
:
By theorem 4 in Sanchez et al.(2011), system (2) is a good approximation to system (1): The spatial structure has been taken into account in the parameters. The dynamic of system (1) depends on the dynamics of the aggregated system (2).
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.4
3.
Existence of Solutions to system (1)
All solutions of (1) starting with non negative functions U0 ; V0 0 are global in time and non negative for any t 0: Indeed, the local existence and the positiveness are consequence of theorem 3.1 in [24]. Global existence, that is the solutions are de…ned on the whole t 0 is established for positive solutions. Proposition 1. The solutions U (t; ); V (t; ) are de…ned on [0; +1) : Proof.
For T > 0, consider the cooperative system 8 @ + d U+ = V + > @t U > < @ + d V + = bU + @t V + U (0; x) = kU0; k1; V + (0; x) = kV0 k1 > > : @V + @U + on @ [0; T ] @n = @n = 0
in in
[0; +1[ [0; +1[ in
@
[0; +1[
Let U + (t) and V + (t) be a solution of the previous system that depends only on t. Then (U + ; V + ) satisfy the linear ordinary di¤erential equation 8 d + = V+ in [0; +1[ ; > dt U > < d + + V = bU in [0; +1[ ; dt + + U (0) = kU k V (0) = kV k in : > 0; 1; 0 1 > : From comparison principle, see theorem 3.4 in Smith (1995), we have 0
U (t; x)
U + (t) and 0
V (t; x)
V + (t):
We conclude that (U (t; ); V (t; )) are de…ned globally 3.1.
Steady state solutions for system (1). Consider the system 8 (x)U 2 E (x)U in ; < d U = (x)V d V = b (x)U f (x)V d (x)U V in ; : @U @V = = 0 on @ : @n @n
(S)
Similar arguments as in Brown and Zhang (2003) and Bouguima et al.(2007), show that a necessary and su¢ cient condition for the existence of at least one positive solution to (S), is that the trivial solution must be unstable. For g a continuous function on , we denote by g=inf g and g = sup g : Proposition 2. If Proof.
b > E f , then the trivial solution of system (1) is unstable.
Let L=
d 0
0 d
and A(x) =
E (x) b (x)
(x) f (x)
:
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.5
then the operator L A has a principal eigenvalue denoted by 1 (L A) : To show that (0; 0) is unstable, it su¢ ces to prove that 1 (L A) < 0: Let (Up ; Vp ) be the principal eigenfunction associated to 1 (L A) : Hence 8 (x)Vp + E (x)Up = 1 (L A) Up in ; < d Up d Vp b (x)Up + f (x)Vp = 1 (L A) Vp in ; : @Up @Vp on @ : @n = @n = 0 Integrating on
, we obtain Z
d
Up dx =
Z
d
Vp dx = 0:
This implies that
and
Z
(
(x)Vp
E (x)Up ) dx =
Z
1
(L
A) Up dx:
Z
(b (x)Up
f (x)Vp ) dx =
Z
1
(L
A) Vp dx:
Since Up and Vp are strictly positive, we conclude that 1 (L
A)
Z
Up dx
Z
Vp dx
and 1
(L
A)
Let
Z
=Z
b
Z
Vp dx
E
Z
Up dx:
Z
Up dx
f
Z
Vp dx:
Vp dx > 0: Up dx
We have 1
(L
A)
and 1
(L
A)
E : b
f :
We will study inf max >0
E ;
b
f
:
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.6
Consider the equation E =
b
A positive root of the previous equation is q 2 (E f )+ E = 2
f :
f
2
+4
b
:
Consequently max
E ;
>0
b
f
=
(
E if f if
b
; :
Hence inf max
E ;
>0
b
f
=
E =
b E f q (E + f ) + 2 (E f )2 + 4
It follows that (L
A)
f ;
4
=
1
b
4
b E f q (E + f ) + 2 (E f )2 + 4
: b
: b
Remark 3. See remark 5 below for an ecological interpretation of the condition given in the previous proposition.
4.
Equilibrium and dissipativness of System (2)
System (2) has to be analyzed with the initial conditions u(0) > 0; v(0) > 0: We observe that the right-hand side of (2) is a smooth function of the variables (u; v), hence the local existence and uniqueness properties are obtained for the corresponding Cauchy problem. The state space of(2) remains in the positive octant R2+ := f(u; v) : u > 0; v > 0g : Indeed ,the set R2+ is positively invariant since the vector …eld of equation (2) is inward 2 on the boundary @R+ :Now, we shall prove the dissipativness of system (2). Theorem 4. All solutions of the system(2) that start in R2+ , are in the compact K=
(u; v) 2 R2+ : u + v
b f
:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.7
Proof.
We de…ne the function w(t) = dw dt
u(t)
+
v(t) f :
b u f
Therefore,
u2 :
From standard comparison principle, we get w
z:
where z is the solution of the logistic equation z0 =
b z f
z2:
Hence w
z
lim supz(t) t!+1
b f
:
So, we obtain the dissipativness properties of system (2) . Now, we investigate non-negative equilibria for system (2). The following proposition, gives the existence of equilibria . Proposition 5. System (2) have two equilibriums: i) The trivial equilibrium point P0 (0; 0): ii) The coexistence equilibrium P (u ; v ) exists if and only if E < condition , u and v are given by p f dE + ( f Ed)2 + 4 db u = ; 2 d 2 p ( f Ed)2 + 4 db f d2 E 2 v = : 4 d2
b f :
Under this
(3)
Proof. A direct way for …nding equilibriums is to consider the intersection of the nuclines. We should solve the two algebraic equations v bu
u2 fv
Eu = 0;
(4)
duv = 0:
(5)
The shape of the curves suggests that there is only the trivial point and an interior equilibrium provided that b E< : (6) f This equilibrium corresponds to the value p f dE + ( f Ed)2 + 4 db u = ; 2 d 2 p ( f Ed)2 + 4 db f d2 E 2 v = : 4 d2 The …gure2 shows the shape of the nuclines.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.8
Remark 6. Ecologically, condition(6) means that for small …shing e¤ort, the …sh population can be sustained at an appropriate equilibrium level.
Figure 1: Represent existence of the trivial equilibrum if E > fb
b ;then f there exists an unique equilibrium point P which must be globally stable. The arrows denote the direction of the trajectory, indeed any trajectory enters from the exterior to the interior Figure 2: The isoclines shows that if E <
Remark 7. Since d (x)
0 and not identical to zero, then
Z
d (x)dx > 0. This implies
that d > 0: We argue similarly for other coe¢ cients, except for E : In fact, E can be identically zero in the case of no harvesting.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.9
4.1.
Dynamic behavior. The local stability of an equilibrium can be studied by
analyzing the Jacobian matrix of the system (2 u + E) b dv
J (u; v) =
(f + du)
:
The stability of the trivial equilibrium P0 (0; 0) characterizes the ability of the species to survive. Theorem 8. i) If E > fb ;then the origin P0 is a stable node, ii) If E < fb ;then the origin P0 is a saddle point. Proof.
The Jacobian matrix calculated at the origin P0 is E b
J (0; 0) =
f
:
It is clear that the trace satis…es T rJ =
1
+
2
< 0:
and the determinant DetJ =
1 2
= Ef
b:
Since a necessary and su¢ cient condition for local stability in two dimensions is T rJ < 0 and DetJ > 0: Then the assertions i) and ii) are immediate. Remark 9. Although, …shing is restricted to adults; juveniles may disappear when …shing e¤ort exceeds a threshold value. Indeed, mature females cannot produce eggs when they are captured. Theorem 10. If E < Proof.
b f
is veri…ed, then the equilibrium P given by (3) is locally stable.
Observe that u =
E (2 and v = 2
with =
p
f 2d
Since u > 0; then necessarily >
E : 2
E) (2 4
> 0:
+ E)
;
(8)
(9)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.10
The Jacobian matrix calculated at the point (u ; v ) is " 2 J (u ; v ) = d b 4 (2 E) (2 + E) It is clear that T rJ =
(2
+
2 f +2 d 2
Ed
2 f +2 d 2
Ed
#
:
) < 0;
and
d (2 E) (2 + E) b: (10) 4 A simple analysis implies that det J > 0: Indeed, we shall consider this determinant as a second degree polynomial in , denoted by: det J = (2 f + 2 d
P( ) = 3 d
Ed) +
2
+ (2 f
E2d 4
Ed)
The polynomial P ( ) has two roots of di¤erent signs 2
=
(2 f
Ed)+ 6 d
p
> 0 where f
= (2 f Ed >
2
b: 1
=
2 f
p Ed+ 6 d
< 0 and
2 2
Ed) + 3E d + 12d b: Since; u > 0; we have r f Ed 2 db ( ) +4 ;
and 2 2
>
q
(
f
Ed 2 )
+4
db
p =
6d
3 2
q
(
2( b
f E)
f
+4
Ed 2 )
db
+
p
> 0:
(10)
We deduce that det J > 0: Therefore P = (u ; v ) ;is a positive equilibrium locally stable. In the next theorem, we shall show that the system (2) has no positive periodic solutions. Our method involves an application of the criterion of Bendixon Amann (1990) (see theorem.4.1, inVerhulst(1996)).The dissipativity of the system and Poincaré-Bendixon theorem Amann (1990) imply that the positive solution of system (2) tends either to the origin or to (u ; v ) : Theorem 11. The system given by (2) admits no periodic solution. In addition, If b < Ef then (u(t); v(t)) tends to (0; 0): If b > Ef then (u(t); v(t)) tends to (u ; v ): Proof.
and
Let
du = v dt dv = bu dt
then div(F; G) =
u2
fv
@F @G + = @u @v
Eu = F (u; v);
duv = G(u; v):
(2 u + e)
(f + du):
(11)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.11
For all u 0 and v 0, since all other parameters are strictly positive, the sign of div(F; G) in Eq(11) is strictly negative. Therefore, by applying Bendixon Criterion Amann (1990), there are no limit cycles within the interior of the positive octant in the state space (u; v). From Dulac principle and dissipativness of the system, it follows that (u(t); v(t)) tends either to (0; 0) or (u ; v ) : Remark 12. When the fecundity b and the maturity rate are low, or mortality f is high, there will not be enough juveniles that will support the population in longtime and the population goes to extinction. Corollary 13. i) If E > ii) If E < Proof.
b f ;then
b f ;then
the origin P0 is globally asymptotically stable.
the equilibrium P is globally asymptotically stable .
It is a direct consequence of theorem 8, 10 and 11.
It is mathematically interesting to see what happens if b = Ef , since in this case the equilibrium is not hyperbolic and the method of linearization is not conclusive. Proposition 14. If b = Ef , then (0; 0) is globally asymptotically stable in R2+ : Proof.
We construct the following Lyapunov function V =
u
+
v f
Calculating the derivative of V along solutions of (2), we obtain dV = dt
b f
in Int R2+ : In addition, ically stable in
E
u
1 duv f
u2 =
1 duv f
u2 < 0
lim V = +1: This implies that (0; 0) is globally asymptot-
u;v!+1
R2+ :
5. Fast slow dynamics In the planar case, the long time behavior of trajectories is simple when di¤usion is ignored. In this section, we will explore the asymptotic behavior in time when the di¤usion is supposed fast. Let " > 0 small enough. Proposition 15. a) If b > Ef; then system (1) has a compact attractor close to P b) If b < Ef; then system (1) has a compact attractor close to P0 : Proof.
The proof is a consequence of the theorem 4, in Sanchez et al .(2001).
The case of b = Ef is more complicate. Firstly, we will obtain some a priori estimates for system (1). For simplicity, we assume all the coe¢ cients of (1) are constant. The following theorem establishes the uniform boundedness of system (1).
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.12
Proposition 16. If b = Ef , then the continuous solutions to (1), corresponding to nonnegative continuous initial data are uniformly bounded on t and on " > 0: Proof.
Consider the cooperative system 8 @ + d U + = V + EU + > @t U > < @ + d V + = bU + f V + @t V + U (0; x) = kU0; k1; V + (0; x) = kV0 k1 > > : @U + @V + on @ [0; T ] @n = @n = 0
in in
[0; +1[ ; [0; +1[ ; in ;
@
[0; +1[ :
(C1)
Let U + (t) and V + (t) be a solution of (C1) that depends only on t. Then (U + ; V + ) satis…es the linear ordinary di¤erential equation 8 d + = V + EU + in [0; +1[ ; > dt U > < d + + + V = bU f V in [0; +1[ ; dt (C2) + + U (0) = kU k V (0) = kV k in : > 0; 0 1 1; > : From comparison principle, see theorem 3.4 in Smith (1995) we have 0
U + (t) and 0
U (t; x)
V + (t):
V (t; x)
The matrix E b has two eigenvalues 1 = 0 and are uniformly bounded on t:
2
f
;
< 0. It follows that the solutions U + (t) and V + (t)
Proposition 17. If b = Ef , then the continuous solutions to (1), corresponding to nonnegative continuous initial data, tend to (0; 0) when " goes to 0+ and t is large enough and …xed: Proof. From theorem 5 in [23], the solutions (U; V ) of system (1) veri…es for x 2 and t > 0 1 U u = + r" (x; t); V v V ol( ) where (u; v) is a solution of system (2) corresponding to the initial data u(0) = u0 =
Z
U (0; x; )dx, v(0) = v0 =
and jr" (x; t)j
a1 "eta2 + a3 ea4
Z
t "
V (0; x)dx;
:
uniformly on :Here ai are a positive constant. Since in this case (0; 0) is globally asymptotically stable for system (2), it follows that (u; v) is close to (0; 0) for t large enough. Hence(U; V ) tends to (0; 0) when " goes to 0+ and t, …xed and large enough.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.13
6.
Bioeconomic harvesting
We suppose that the selling price is constant. This corresponds to the case when the yield faces a perfectly elastic demand. For instance, the price is determined by the international market, see for instance (Knowler et al.2001). 6.1.
Open access …shery.
The optimization problem is to determine the optimal
…shing e¤ort to provide a maximum rent. We assume a constant price p per unit of harvest biomass. The sustainable yield is Y (E) = Eu ; where u is the adult equilibrium of system (2).The catch of resource multiplied by their price gives the total revenue T R = pY (E): The cost function is T C = c(u )E: The sustainable economic rent which provided by the …shing e¤ort E; is given by ER = T R
T C = pEu
c(u )E:
So, we have the bioeconomic e¤ort E1 when T R = T C: If E > E1 ; Then, the total cost of …shing is greater than the total revenue .Hence over…shing generates a fall in income for …shermen. If E < E1 ; The pro…t is positive. The economic objective is to maximize the pro…t which is the di¤erence between the income and the cost. (E) = E (pu
c(u )) :
(12)
We know that when …shable resource decreases, the e¤ort is more expensive. Hence c is assumed to decrease with stock size. We can choose a function cost such that c (u) = cu0 : with c0 a positive constant. Without lost of generality, we take c0 = 1: In this case, a simple calculation gives the bioeconomic e¤ort p p b p p E1 = p ; f p+d p
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.14
and E1 > 0 if
f+
p>
p
2f 2
+4 f b
2 b
!2
:
Under the condition b > Ef; the function (E) is continuous and (0) = 0: We assume that 0 (0) > 0: Hence, we obtain 0
(E) = (pu
c(u )) + E
d (pu dE
c(u )) ;
and 0
(0) = pu (0)
This implies that u (0) >
c(u (0)) > 0: r 2
1 : p
Remark 18. The stock of mature …sh should be large enough to allow harvesting. Now, our purpose is to obtain the best control of …shing e¤ort E , so it is necessary to have the following condition @ = 0 () pu @E
@u @c (u ) p = 0: @E @u h h The function (E) is continuous on the I = 0; fb interval and lim (E) = c (u ) + E
E!
b f
(13) 1: This
implies that the maximum of (E) is achieved at a point Eopt 2 I: @c Substituting c (u ) and its derivative @u (u ) into Eq (13) ;we get the optimal harvesting equation h i 2p b) 3 (1 ) E 4 + f (d + b(ppfb2 dd22d ) E 2 + pf 2 d2 E + p (14) d(1 ) f b E p(pfd2 bd2 ) = 0 : p(pf 2 d2 ) pf 2 d2 It is easy to calculate the algebraic solution Eop and is depending both on biological, environmental and economic parameters. If d is large enough, then the number of sign changes in the sequence of coe¢ cients in the previous polynomial is 4: From Descartes Principle, see for instance Murray (1993), p.704, there are exactly either 4 or 2 positive roots of equation (14). In other side, from the main theorem in Stefansson (2008), (see the appendix) any positive root Eop of (14) has an upper bound given by Eop It is clear that
2f (d d2
2f (d d2 2p b) pf 2
2p b) : pf 2 b f
Remark 19. Hence, if d is large enough, then the optimal …shing Eop goes to zero and this leads to a dissipation of economic rent.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.15
If d is small and is large enough, then the number of sign changes in the sequence of coe¢ cients in the previous polynomial is 1 and Descartes principle, there is exactly one positive root of equation (14): Remark 20. The previous uniqueness result is di¤erent from that obtained in [18], where the authors, under conditions on the smallness of the time interval, uniqueness of optimal control is established. 6.2. Optimal harvesting policy. Assume a constant discount rate : for simplicity, we suppose that the harvesting cost is constant. Let c be the constant …shing cost per unit e¤ort, and let p be the constant price per unit biomass of the mature. The net revenue of harvesting at any time is given by (E) = E (pu
c) :
In order to maximize the rent for all future, we consider the following optimization problem Z +1 supJ(E) = sup e t (E)dt; E
E
0
subjected to the state equation. We apply Pontryagin’s maximum principle. The pseudo-Hamiltonian is t
H=e
E (pu
c) +
u2
v
1
Eu +
2
(bu
fv
duv) :
(15)
where 1 (t), 2 (t) are the adjoint variables, E is the control variable satisfying the constraints 0 E Emax : Here Emax is the feasible upper limit of the …shing e¤ort. Since the pseudo-Hamiltonian H is linear in the control variable E with coe¢ cient e t (pu c) 1 u , the optimal control will be a combination of extreme controls and the singular control. The necessary condition for the maximization of H is @H e t (pu c) = 0 =) 1 = : @E u According to Pontryagin’s maximum principle, the adjoint equations are
Let
1
=
_1
=
_2
=
1e
t
@H = @u @H = @v
and
d dt d dt
2
1
=
1
=
2
=
=
t
Epe 1 t
2e
pu
c u
+
2
1
(2 u + E)
2
(b
dv) ;
(f + du) :
(17) (18)
;then, the equations (16),(17) and (18), becomes:
=p
Ep + 1
+
+
(16)
1
c ; u
(2 u + E + )
2
(19) 2
(b
dv) ;
(f + du + ) :
The optimal control E that maximizes H must satisfy the following conditions:
(20) (21)
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.16
E
=
0;
when
E
= Emax ;
when
1 1
>p
c ; u c : u
Remark 21. The quantity 1 e t is called the shadow price and p uc is the net economic revenue on a unit harvest. One must …sh as much as possible when the shadow price is low enough 1 e t < p uc and no …sh at all when the shadow price is high 1 e t > p uc : When 1 equals the net economic revenue on a unit harvest p uc , then the pseudoHamiltonian H becomes independent of the control variable E and the maximum condition does not provide any information about E in this case. Using (19); the derivative of (19)and (20) we get cu0 = u2
pu
c
(2 u + E + )
u
Substituting u0 = v
u2 2
=
2 (b
dv)
Ep:
(22)
Eu into (22) , we obtain u(pu
c) ( u + ) + pu3 u2 (b dv)
c v
:
(23)
We restrict ourselves to optimization at equilibrium which is much easier to handle than the general case. At the positive equilibrium, we have u0 = v 0 = 0 and 0 consequently 2 = 0. A simple calculation gives in this case, the harvest rate E = h(u) = (pu
c)
(f + du + ) (2 u + ) + du2 c (f + du + ) du(pu c)
b
:
It refelects the direct change in …shing e¤ort due to a change in adult stock. Since E must satisfy E < fb ; then E is not feasible for every value of u.Since d dt 2 = 0, then 1 + 2 (f + du + ) = 0; and 1
=
2
(f + du + )
=
(f + du + ) u(pu
c) ( u + ) + pu3 u2 (b dv)
c v
:
and the necessary condition for maximization of the pseudo-Hamiltonian becomes c (f + du + ) u(pu c) ( u + ) + pu3 = u u2 (b dv)
p
c v
;
This implies that pu
c=
u2 (b
dv)
pu3 c v ; (f + du + ) u ( u + )
Hence (E) = E (pu is a decreasing function with respect to when goes to +1:
c) :
and the economic rent (E) tends to zero
Remark 22. When the time preference of the present generation is high, the net bene…ts is lower.
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.17
7.
Numerical simulations
In this section, we give some numerical examples. Due to a lack of real world data, we have used hypothetical data to illustrate the analytical results. We assume that = 3:9, = 1:04, b = 2:3; f = 0:5, d = 0:2: Example 23. In this case, we choose the value of E = 1 which veri…es E < Considering system (2) with initial condition (6; 2);we have the following
b f
= 17:94:
This …gure represents the growth of the mature and immature populations in time. The intersection of both curves gives the non-negative equilibrium (u ; v ), which is u = 0:2112; and v = 0:0660:
Example 24. Let p = 1:5 be the selling price, using the data of the previous example, we …nd the control value which is represented by the intersection of the curve with the
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.18
x-axis.
Figure 4: Roots of equation (14) The …gure 04 shows the existence of 3 real roots of equation (14), which are represented in the …gure by the intersection the curve with the x-axes (y = 0): Here E = 6; 121:
8. Conclusion In this paper, a bioeconomic model attempting to relate the reproduction of a renewable resource with its exploitation is performed. We have analyzed the dynamics of …shery model with a stage structured population. It has been assumed the existence of competition between adults and cannibalism on juveniles populations. The exploitation of …sh population is assumed on adult population which has a commercial value.Jerry and Raissi (2005) studied a similar …shing model without intraspeci…c competition between species and cannibalism. Firstly, using stability theory of ordinary di¤erential equations, it has been proved that the interior equilibrium exists under certain conditions and it is globally asymptotically stable. Even when the …shing e¤ort is restricted to adult population, both adults and juveniles can become extinct.because the ecosystem is often altered by the human activities, we analyze the harvesting strategy that results in maximizing the pro…t and not leading to extinction. We obtain the optimal harvesting equation and deduce the e¤ect of cannibalism on economic rent. Our approach consists on the following idea. We take advantage from existence of two time scales (see Sanchez et al. 2011), a fast one for …sh di¤usion and a slow one for …sh growth and mortality to reduce our model to an ODE system. The major contributions of the present paper are: a) the persistence of the species depend on two factors:the …rst is biological and concerns the ability that a species reproduces itself, this is a function of birth rate, natural mortality of young population and the rate of maturity. For instance, if the growth rate is no longer su¢ cient to replace
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An age-Structured Fishery Model:Dynamics and Optimal Management with Perfect Elastic Demand.19
loses, the stock collapses. The second is linked to the exploitation and mecanisms that reduce the …shing e¤ort. This latter can be reduced by limiting the time of …shing, reducing the capacity that a vessel can carry, or closing areas where the birth rate is high. b) Cannibalism and competition have no direct e¤ect on the population survival. c) Using a recent result of Stefansson (2008), we show that under suitable conditions, there will be a dissipation of economic rent in an open access …shery. d) In the case of total discounted net bene…t, the …shing e¤ort is given as a function of the mature stock. This result is di¢ cult to obtain without the method of fast-slow system, see for instance Fu et al. (2014) , Lenhart and Montero (2001). There are still many interesting and challenging mathematical questions need to be studied for system (1). For example, the model ignore seasonal or stochastic variation, reserve area (see for instance Lv et al .(2013) and the references therein). A further research may incorporate these aspects in the model. It would happen that …shers do not exclude the …shing of the juvenile and it is interesting to consider exploitation taking into account juvenile stage. The selling price is assumed constant, it would be interesting to consider a variable price of …sh. For related questions with other methodologies using the EOQ model we refer to S.K.De and S.S.Sana (2014) and the references therein).
Acknowledgements The authors are grateful to the reviewers for their critical evaluations and suggestions. A.
Appendix:
Let us recall the following result on upper bounds of positive roots of a polynomial given in [28] Theorem 25. Let p(x) = xd
b1 xd
m1
bk xd
mk
+
X
j6=m1 ;
with b1 ; :::; bk are strictly positive and aj B1 (p) = max
0 for all j 2 fm1 ; 1
kb1 ) m1 ; ::::(kbk
aj xd
j
;mk
; mk g :The number
1 mk
is an upper bound for the positive roots of p: References [1] Amann, H(1990). Ordinary Di¤ erential Equations: An Introduction to Nonlinear Analysis, Walter de Gruyter, Berlin. New York : [2] Beverton, R.J.H &Holt, S.J(1993). Recruitment and Egg-production.In: On the Dynamics of Exploited Fish Population , Vol.6, Chapman &Hall, London, 44-67. [3] Belyakov, A.O & Veliov, V.M.(2014).Constant versus periodic …shing:age structured optimal control approach, Math.Model. Nat.Phenom .9, 20-37.
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