INTRODUCTION
Many consumer species go through two or more life stages as they proceed from
birth to death. However, the majority of the models in the literature always
assumed that during the whole life histories, each individual admits the same
densitydependent rate as well as the identical ability to bear and compete
with other species, which clearly unrealistic. In many species, only the mature
predator food on the prey and its immature are too weak to food on the prey.
Therefore, it is practical to introduce the stagestructure into the competitive
or preypredator models. Some of the stage structure models can be found in
Aiello and Freedman (1990), Aiello et al. (1992), Freedman and Wu (1991),
Gambell (1985), Landahl and Hanson (1975), Wood et al. (1989) and the
references therein. A good overview on stagestructured models can be found
in the recent book by Murdoch et al. (2003). Recently, papers like Bosch
and Gabriel (1997), Kar (2003), Kar and Matsuda (2006), Zhang et al.
(2000) and Wikan (2004) study the stage structure of species with or without
time delays.
Harvesting has also a strong impact on a dynamic property of a population. Depending on the nature of applied harvesting strategy, the longrun stationary density of the population may be significantly smaller than the long run stationary density of a population in the absence of harvesting. In the absence of harvesting, a population can be free from extinction risk; however, harvesting can lead to the incorporation of a positive extinction probability and therefore, to potential extinction in a finite time. If a population is subject to a positive extinction rate then harvesting can drive the population density to a dangerously low level at which extinction becomes sure no matter how the harvester affects the population afterwards.
A fundamental issue in population biology is what are the minimal conditions
to ensure the long term survivorship for all of the interacting components.
When these conditions are meet the interacting populations are said to persist
or coexist. Permanence corresponds to the existence of a positive attractor
that attracts all positive population trajectories. Permanence implies positive
population trajectories can recover from large perturbations of the state variables.
This question is of particular interest to fishing managers. If it is known
that the system exhibits such a permanent behaviour, then ecological planning
based on a fixed eventual population can be carried out. Realizing the problem
we have obtained the conditions for permanence of the solutions of our system
considered.
Though the application of bioeconomics has usually limited to the determination of Total Allowable Catch (TAC), the sharing of the TAC between heterogeneous fisher groups is an important issue to be considered. The fact that different fisher groups harvest different cohorts within fish stocks and thereby have different effects upon both stocks growth and the economics of the fishery, is not taken into account. Political determination of harvest shares has bioeconomic effects, for instance, in the shape of reduced payoffs from the fishery or even extinction. This study is focused on the optimal steady state fish stock level and hence on the steady state Total Allowable Catch (TAC). This is done using a cooperative game theoretic approach to the issue of sharing the harvest. We assume that the harvest strategies of the two vessel groups are determined by their existing technologies and their respective fishing grounds. Thus, the overall optimal sharing of the resource can be determined, after deciding the weights that should be given to the two parties preferences.
THE MODEL
The ecosystem in our model consists of a preypredator system. We assume that the prey population (denoted by N_{1}) is subjected to a logistic growth condition. The predator population is divided into two stage groups: juvenile predators (denoted by N_{2}) and adult predators (denoted by N_{3}). Here we also assume that only adult predators are capable of preying on the prey species and that the juvenile predators live on their parents. For example, the Chinese Alligator can be regarded as a stagestructured species since the mature is more than 10 years old and can also be regarded as a predator because almost all aquatic animals are the chief food of the Chinese Alligator. Another key and somewhat novel feature of our model is to account for the universally prevalent intraspecific competition in the consumer growth dynamic (Kuang et al., 2003). This intraspecific competition is assumed to induce additional instantaneous deaths only to the adult population and the increased death rate is proportional to the square of the adult population. Holling type II functional response probably a better description of the actual predation seen in nature but for simplicity, we assume the simplest description.
With this assumptions, we have the following plausible two stage preypredator interaction model:
Here r_{1} is the specific growth rate of the prey and k is its carrying capacity. α is the predation parameter; m is the conversion factor; r_{3} is the death rate of mature predator species; γ is the proportionality constant of transformation of immature to mature predators; r_{2} = μ + γ, where μ is the death rate; β is the birth rate of the immature populations.
We can reduce the number of parameters by making the following transformation of variables:
Making these changes, system (1) assume the much simpler form
where
Now we assume that two groups of fishermen targeting two substocks consist of different stage groups of predators, with mature fish in one substock and immature fish in the other. Under this assumption system (2) becomes
Here q_{1}E_{1}x_{2} and q_{2}E_{2}x_{3} are based on the catchperuniteffort hypothesis (Clark, 1990), where q_{1}and q_{2} are catchability coefficients, E_{1} and E_{2} are harvesting efforts.
We are not making any case study but the NorthEast Atlantic cod fishery is a good example for it. The technology that the trawlers and coastal vessels utilize is different, as are the areas of fishing activity and therefore size of harvested fish. The trawlers catch fish of a smaller size than the coastal vessels, as the older fish tend to migrate in to the coast to spawn.
The Iberoatlantic Hake fishery is another example of multifleet fisheries. In this fishery, the species is caught using several fishing methods, particularly trawling, longlining and fixed gillnetting. Specially, trawling acts intensely on younger individuals, whereas the other fishing methods mainly affect more mature specimens.
EQUILIBRIA AND STABILITY ANALYSIS
System (3) has to be analyzed with the following initial conditions : x_{1}(0) > 0, x_{2}(0) > 0 and x_{3}(0) > 0. We observe that the righthand side of the system (3) is smooth function of the variables (x_{1}, x_{2}, x_{3}) and the parameters, as long as these quantities are nonnegative, so local existence and uniqueness properties hold in the positive octant. The state space for system (3) is in the positive octant, {(x_{1}, x_{2}, x_{3}): x_{1} > 0, x_{2} > 0 and x_{3} > 0}, which is clearly an invariant set, since the vector field on the boundary does not point to the exterior. Our next result concerns the existence of equilibrium points.
We find the steady states of system (3) by equating the derivatives on the left hand sides to zero and solving the resulting algebraic equations. This gives three possible steady states, namely, P_{0}(0, 0, 0), P_{1}(a, 0, 0) and P_{2}(x_{1}*, x_{2}*, x_{3}*) where
Here we want to remark that there exists another equilibrium in the absence
of prey if e>c. But it is not realistic since prey is the only source of
food for the predator. So throughout the paper we assume that
Therefore, P_{2} is feasible if
Particularly we are interested on the interior equilibrium point P_{2}(x*_{1}, x*_{2}, x*_{3}) for its usual importance.
Next we consider first the local stability of the equilibria. The variational matrix of the system (3) is given by
Now,
The diagonalization of the Jacobian matrix M (0, 0, 0), yields the following characteristic equation:
Which shows that P_{0}(0, 0, 0) is unstable.
Characteristic equation of M(a, 0, 0) is
∴P_{1}(a, 0, 0) is locally asymptotically stable for c>e + ad.
Now.
The characteristic equation of M_{}(x*_{1}, x*_{2},
x*_{3}) is λ^{3} + Aλ^{2} + Bλ + C =
0, where
Obviously, ABC>0. According to RouthHurwitz criteria, P_{2}(x*_{1},
x*_{2}, x*_{3}) is locally asymptotically stable if
Now we shall discuss the condition of global stability, permanence and extinction of system (3). At first, we give the following notations and definitions
Definition 1
An equilibrium point P_{2}(x_{1}, x_{2}, x_{3})
is said to be globally asymptotically stable in
if it is locally asymptotically stable and all trajectories in converges to
P(x_{1}, x_{2}, x_{3}).
Lemma 1
• 
If,
then the equilibrium P_{1}(a, 0, 0) is globally asymptotically
stable in . 
• 
If,
then the only interior equilibrium point P_{2}(x*_{1},
x*_{2}, x*_{3}) is globally asymptotically stable in . 
Proof
• 
We construct the following Lyapunov function 
where α_{i}, i = 1, 2, 3 are positive constants to be determined
in the subsequent steps.
Calculating the derivative of V_{1} along each solution of (3), we have
Let
and α_{3} = 1.
Then,
in Int. ,
for c≥e+ad.
Therefore, by LyapunovLaSalle (Hale, 1969), it follows that P_{1}
is locally asymptotically stable and all trajectories starting in Int.
approaches to P_{1} as t goes to infinity. This establishes the global
asymptotic stability (Fig. 1).
• 
Let us take the Lyapunov function 
where α_{i}, i = 1, 2, 3 are positive constants to be determined in the subsequent steps.
Calculating the derivative along each solution of (3), we have

Fig. 1: 
Phase space trajectories of system (3) beginning with different
initial states. It is seen that
is a global attractor, where a = 3, b = 4, c = 2.5, d = 3.0, e = 0.2,
f = 0.7, q_{1} = 0.3, q_{2} = 0.5, E_{1} = 1.5,
E_{2} = 2 
Let
and .
By similar arguments as we have used for P_{1}, we may state that P_{2}(x*_{1},
x*_{2}, x*_{3}) is globally asymptotically stable if
hold.
Definition 2
System (3) is said to be permanent if there are positive constants m and
M such that each positive solution x(t, x_{0}) of (3) with initial condition
satisfies
Definition 3
The ith species of system (3) is said to be extinctive if each positive
solution x(t, x_{0}) of (3) with initial condition
satisfies
Combining all these results we get the following theorem.
Theorem 1
• 
The predator species of system (3) is extinctive and the
prey species is not extinctive if and only if c+q_{2}E_{2}≥e+ad
hold. 
• 
Both prey and predator species are permanent if and only
if 
.Proof
By Definitions 1 and 2 and the Lemma 1, we can easily prove the Theorem.
THE BIOECONOMIC ENVIRONMENT
Once the process of harvesting the resource is started, the problem of management
of the fisheries can be viewed in terms of rent maximization, as is the case
of several theories of fisheries economics. In this section, the goal is to
find the efforts E_{1} and E_{2} to maximize a weighted average
of the objective functionals obtain from immature and mature predator fish,
respectively.
Let II_{1} = (p_{1}q_{1}x_{2}
c_{1})E_{1} and II_{2} = (p_{2}q_{2}x_{3}
c_{2})E_{2} 
represent the net revenues for the immature and mature species, respectively.
Thereby the present values of the two substocks are, respectively
where δ is the discount rate, p_{i} is the unit price of the resource,
c_{i} is the unit cost of harvesting the resource substock 1 and 2.
The aim of the social manager would be to select the efforts E_{i}
and stock level to maximize a weighted average of their objective functional,
PV. The weights β and (1β) indicate how much weight is given to the
objective functionals. For a given β∈0[0, 1], the management objective
functional translates into maximize
subject to the stock dynamics given by Eq. 3 and to the control
constraints 0 &le E_{i} &le E_{imax}. Given the structure of
our problem, we arrive at two equations that implicitly define the optimal (steady
state) equilibrium for the doubly singular solution (Appendix):
The solutions to Eq. 4 and 5 are pursued
numerically. From these solutions, we can determine the optimal equilibrium
harvest of the two substocks and efforts.
Remark
and

Fig. 2: 
Phase space trajectories of system (3) corresponding to
the optimal harvesting efforts. It is seen that the optimal equilibrium
point (0.18, 4.59, 1.62) is also a global attractor 
This implies that, for each species, the user cost of harvest per unit effort must equal the discounted value of the future marginal profit of effort at the steady state effort level.
For simulation let us take a = 3, b = 4, c = 2.5, d = 3.0, e = 0.2, f = 0.7, q_{1} = 0.3, q_{2} = 0.5, c_{1} = 7, c_{2} = 8, p_{1} = 90, p_{2} = 100, δ = 0.05 in appropriate units.
For the above values of the parameters, optimum harvest is attained for β=0.4.
This implies that the highest discounted total profit is achieved when about
40% of the harvest is taken from juvenile fish and 60% taken from the mature
fish. The optimum harvesting efforts are
and
Optimum equilibrium is (0.18, 4.59, 1.62) and discounted profit is 605.7 (Fig.
2).
CONCLUDING REMARKS
An important and one of the interesting questions in mathematical ecology is persistence or permanence, which ensures the survival of biological species and exclude extinction of species for all positive initial conditions. The question of permanence of biological species is of particular interest to fishery. If it known that a system exhibits such a permanent behaviour, then ecological planning based on a fixed eventual population can be carried out. Realizing the problem we have obtained the conditions for persistence of the solutions of our systems.
We have studied how the weights should be divided between two fleets to maximize the harvesting returns from the fishery. We are aware of the simplification involved in converting a highly complex situation into a theoretical model. However, the model that has been developed in the current paper would be a guidelines for the future research.
Before ending this article, we would like to mention that there is still tremendous
amount of work to do in this model. For example,
• 
One can consider the stagestructure of prey population. 
• 
Gestation period for predator is also an important characteristic
to be considered. We leave it for future considerations. 
• 
Uncertainties is another important point to be considered. 
APPENDIX
The Hamiltonian for the weighted average objective function is
where
and
are adjoint variables.
The control variables E_{1} and E_{2} appear linearly in the Hamiltonian function H. Therefore, optimal control will be a combination of bangbang control and singular control. The optimal control E_{i}(t) which maximizes H, must satisfy the following conditions:
For the singular control we have,
Now
and
We intend to derive here an optimal equilibrium solution of the problem. Since we are considering an equilibrium solution, x, y and E are to be treated as constants in the subsequent steps.
Now the adjoint equations are
and
Substituting
in (A.4) we get
Again substituting and
in (A.5) we get
Substituting
in (A.3) we get
ACKNOWLEDGMENT
Authors would like to thank Japan Society for the Promotion in Science (JSPS) for financial support of this research (P05109).