Harvesting of a Single-Species System Incorporating Stage Structure and Toxicity

A single species stage-structured model incorporating both toxicant and harvesting is proposed and studied. It is shown that toxicant has no influence on the persistent property of the system. The existence of the bionomic equilibrium is also studied. After that, we consider the system with variable harvest effect; sufficient conditions are obtained for the global stability of bionomic equilibrium by constructing a suitable Lyapunov function. The optimal policy is also investigated by using Pontryagin’s maximal principle. Some numeric simulations are carried out to illustrate the feasibility of the main results. We end this paper by a brief discussion.


Introduction
As the development of industry, the influence of toxicant becomes more and more serious; toxicant which was produced by water pollution, air pollution, heavy metal pollution and organisms themselves, and so on, has great effects on the ecological communities.
Mathematical models which concerned with the influence of toxicant were first studied by Hallam and his colleagues 1-3 .After that, Freedman and Shukla 4 studied the single-species and predator-prey model; Chattopadhyay 5 and many scholars paid attention to the competition model 6-10 ; Ma et al. 11 , Das et al. 12 , and Saha and Bandyopadhyay 13 laid emphasis on the predator-prey models.However, seldom did scholars investigated the stage-structured models with toxicant effects; to the best of authors' knowledge, only Xiao and Chen 14 explored a single-species model with stage-structured and toxicant substance.It is well known that many species in the natural world have a lifetime going through many stages, and in different stages, they have different reactions to the environment.For example, the immature may be more susceptible to the toxicant than the mature.Although there are many works on the stage-structured model see [15][16][17][18][19] and the references cited therein , seldom did scholars consider the influence of the toxicant substance on the immature species.
In this paper, we study the single-species model with simplified toxicant effect, and we also take the commercially exploit into account.Since many species can be resources as human food, harvesting has a great influence both on the species population and on the economic revenue.There are many papers that deal with the effects of harvesting 10, 12, 20-22 ; such topics as the optimal harvesting policy and the bionomic equilibrium are well studied by them.However, only recently scholars considered the ecosystem with both harvesting and toxicant effects see 10, 12 , while no scholar investigated the stage structure population dynamics with both harvesting and toxicant effect.
We will study the following singe species stage structure ecosystem with both toxicant effect and harvesting: where x 1 t , x 2 t represent the population density of the immature and the mature at time t, respectively, r 1 x 3 1 is the effects of toxicant on the immature, E is the harvesting effort, c 2 is the catchability coefficient.We assume that the immature is density restriction, toxicant affects the immature population and only harvesting the mature species.
The paper is arranged as follows The stability property of equilibria is studied in the next section, and the existence of the bionomic equilibrium is explored in Section 3. In order to investigate the stability of the bionomic equilibrium and discuss how the population will be changed according to the the variable harvest effects, we assume that the E is proportion to the economic revenue 23 , that is, Sufficient condition which ensures the global stability of bionomic equilibrium is then investigated in Section 4. The optimal harvesting policy is studied in Section 5 and some numeric simulations are carried out in Section 6 to illustrate the feasibility of the main results.We end this paper by a briefly discussion.

The Steady States and Stability
It can be calculated that system 1.1 has two possible equilibriums: where

2.1
By simple calculation we have To ensure the positivity of the equilibrium E * x * 1 , x * 2 , we assume that holds.We can see that x * 1 , x * 2 decrease as r 1 increases.Next, we use the Jacobian matrix to determine the locally stability of the equilibriums.By simple calculation, we see that the Jacobian matrix of system 1.1 is For E 0 0, 0 , the characteristic equation is It is not hard to see that when aβ < d 1 β b 1 c 2 E , 2.5 has two negative roots or two complex roots with negative real parts; thus E 0 0, 0 is locally asymptotically stable; when

2.6
By applying 2.1 , we have Therefore, the characteristic equation of E * x * 1 , x * 2 has two negative roots or two complex roots with negative real parts; thus E * x * 1 , x * 2 is locally asymptotically stable.Following we will take the idea and method of Xiao and Chen 14 to investigate the globally asymptotically stability property of the equilibriums, and we need to determine the existence or nonexistence of the limit cycle in the first quadrant. For 2 is the unique stable equilibrium in the first quadrant if it exists.Let AB be the line segment of L 1 : x 1 p and BC the line segment of L 2 : x 2 q, where A p, 0 , B p, q , C 0, q , and p, q are positive constants which satisfy p > x * 1 , and Trajectories enter rectangle OABCO from exterior to interior.
By simple calculation, we have ẋ1 2.9 Thus AB, BC are the transversals of system 1.1 .It is no hard to check that OA, OC are the transversals of system 1.1 , and any trajectory enters region OABCO from its exterior to interior see Figure 1 .Denote

2.10
It is easy to see that By Poincare-Bendixson theorem, there are no limit cycles in the first quadrant; thus is globally asymptotically stable if it exists.For E 0 0, 0 , it is a unique equilibrium which is locally asymptotical stable if aβ < b 1 c 2 E d 1 β .Similarly to the above analysis we can show that E 0 0, 0 is globally Therefore, we have the following.
is globally asymptotically stable.
We mention here that since condition 2.3 is independent of the toxicant of the system, thus, the globally asymptotically stability of the systems is independent of the intensities of toxicant, but from the expression of positive equilibrium we know that the density of both the immature and the mature species decreases while the toxicant increases; specially, the density of species will tend to indefinitely small if the toxicant substance is large enough.

Bionomic Equilibrium
For simplicity, we assume that the harvesting cost is a constant.Let c be the constant fishing cost per unit effort, and let p 2 be the constant price per unit biomass of the mature.The net revenue of harvesting at any time is given by: A bionomic equilibrium is both a biological equilibrium and a economic equilibrium, the biological equilibrium is given by x 1 t x 2 t 0, and the economic equilibrium occurs when the economic rent is P 0, thus the bionomic equilibrium From 3.4 we get x 2∞ c/p 2 c 2 .Combining 3.4 and 3.2 we can obtain that x 1∞ is one of the roots of the following equation: Hence, by the continuity of f x , there exists exactly one root in 0, ∞ .From 3.3 and 3.4 , to ensure the positivity of E ∞ , one needs Thus we need to find a solution of f x in b 1 c/βp 2 c 2 , ∞ .Since 3.6 always holds, we only need Thus, there exists a unique bionomic equilibrium if inequality 3.8 holds.
The existence of the bionomic equilibrium means that i Harvesting efforts E > E ∞ cannot be maintained all the time, it will decrease because the total cost of harvesting exceed the total revenues; ii E < E ∞ cannot be maintained indefinitely, harvesting is profitable in this occasion, and it will make the harvesting effort increases.Hence, the harvesting effort is always oscillating around E ∞ .However, there is no answer about whether it will become stable or not because of the complex changing of E.

Globally Stability of the Bionomic Equilibrium
In this section, we study system 1.1 with variable harvest effects; sufficient condition for the globally asymptotically stability of the bionomic equilibrium will be derived.We assume that E t kE p 2 c 2 x 2 − c ; then system 1.1 becomes

4.1
System 4.1 has three possible equilibrium: i the trivial equilibrium V 0 0, 0, 0 , ii equilibrium in the absence of harvesting V 1 x 1 , x 2 , 0 , where and for the positiveness of x 1 , x 2 , we need iii the interior equilibrium E x 1∞ , x 2∞ , E ∞ , which is the bionomic equilibrium in Section 3; it exists if 3.8 holds.
For V 0 0, 0, 0 , the characteristic equation is given by It is easy to see that all of the roots of 4.4 are negative if βa < b 1 d 1 β holds; thus V 0 0, 0, 0 is locally asymptotically stable if βa < b 1 d 1 β , and unstable if βa > b 1 d 1 β .For V 1 x 1 , x 2 , 0 , the characteristic equation is given by From the condition for the stability of V 0 , V 1 , we can see that low birth rate can make the population be driven to extinction, high harvesting cost has negative effect on fishing effort, and it can make the harvesting effect approach zero.
For E x 1∞ , x 2∞ , E ∞ , the characteristic equation is where

4.7
By Routh-Hurwitz criterion, all roots of 4.6 have negative real parts if and only if By simple calculation, we know that condition 4.8 holds always, Thus, E x 1∞ , x 2∞ , E ∞ is locally asymptotically stable.
For the global stability of E x 1∞ , x 2∞ , E ∞ , we construct the following Lyapunov function: The time derivative of V along the positive solution of system 4.1 is

4.11
If inequality 4.13 then 4.12 holds in set Ω if g x 1 , x 2 > 0. By applying 3.2 and 3.3 , we have

4.15
Thus, we can get that if holds, then g x 1 , x 2 > 0, where 4.17 If x 1 < x 2 , by the same way above, we can get the other sufficient condition for g x 1 , x 2 > 0, that is, where Therefore, if 4.16 or 4.18 holds, then V t < 0 and the bionomic equilibrium is globally asymptotically stable.
The globally asymptotically stability of the bionomic equilibrium means that harvesting effect E which changes along 1.2 will make system 4.1 drive to the "bionomic equilibrium" and keep stable in the bionomic equilibrium.

Optimal Harvesting Policy
In this section, we study the optimal harvesting policy of system 1.1 , and we consider the following present value J of a continuous time-stream: where P is the net revenue given by P x 1 , x 2 , E, t p 2 c 2 Ex 2 − cE, and δ denotes the instantaneous annual rate of discount; the aim of this section is to maximize J subjected to state equation 1.1 .Firstly we construct the following Hamiltonian function:

5.2
where λ 1 t , λ 2 t are the adjoint variables, E is the control variable satisfying the constraints 0 ≤ E ≤ E max , and φ t e −δt p 2 c 2 x 2 − c − λ 2 c 2 x 2 is called the switching function 23 .We aim to find an optimal equilibrium x 1δ , x 2δ , E δ to maximize Hamiltonian H; since Hamiltonian H is linear in the control variable E, the optimal control can be the extreme controls or the singular controls; thus, we have E E max , when φ t > 0, that is, when λ 2 t e δt < p 2 − c c 2 x 2 ; E 0, when φ t < 0, that is, when λ 2 t e δt > p 2 − c c 2 x 2 .

5.4
In this case, the optimal control is called the singular control 23 , and 5.4 is the necessary condition for the maximization of Hamiltonian H.By Pontrayagin's maximal principle, the adjoint equations are

5.5
From 5.4 and 5.5 , we have where We can calculate that It is obviously that λ 1 t , λ 2 t are bounded as t → ∞.
Substituting 5.9 into 5.4 , we obtain Our purpose is to find an optimal equilibrium solution; so we have x * 1 .

5.11
By 5.10 and 5.11 , we can get x 1δ , x 2δ , and E δ .Thus, the optimal policy is

5.12
Again, from 5.10 we have

Number Simulations
In the following examples, we take the parameters values as a 2, d 1 0.1, d 2 0.1, c 2 0.2, b 1 0.1, and β 0.2.We will see how the system behavior is while the toxicant effect changes.
From the above examples we can found the following phenomena: i Increasing of toxicant will make the population of both mature and immature decrease.ii The bionomic equilibrium exists and globally stable both in the absence of toxicant and in the present of toxicant; however, with the increase of toxicant, the immature population x 1∞ and the harvesting effect E decrease while the mature population x 2∞ remains as the same.
iii The bionomic equilibrium and the optimal equilibrium will become unfeasible if the toxicant is large enough.
iv The immature, mature populations, and the harvesting effect in the optimal equilibrium are decreasing as the toxicant is increasing.
v The optimal equilibrium becomes more and more close to the bionomic equilibrium as the toxicant effect increases.

Discussion
In this paper, we consider the single-species stage structure model incorporating both toxicant and harvesting, and we assume that only the immature affected by the toxicant.Firstly, we explore the local and global stability properties of the equilibria of the system.Next, we investigate the existence and stability properties of the bionomic equilibrium.Finally, the optimal harvesting is studied, and it is found that there exists two optimal equilibria when the toxicant varies in a certain set.Some numeric examples to illustrate how the equilibrium include bionomic equilibrium and optimal equilibrium changes with the toxicant are also given.
Nevertheless, as we know, the immature needs a certain time to develop to mature stage, the model incorporating time delay may be more reasonable and worth further study, and we leave this for future study.

Figure 2 :
Figure 2: Solution curves of system 1.1 with the parameters given by Example 6.1 when r 1 0.01.

Figure 3 :
Figure 3: Solution curves of system 1.1 with the parameters given by Example 6.1 when r 1 1.

Figure 4 :Figure 5 :
Figure 4: Solution curves of system 1.1 with the parameters given by Example 6.1 when r 1 100.

Figure 6 :Figure 7 :
Figure 6: Solution curves of system 4.1 with the parameters given by Example 6.2 when r 1 1.