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

Abstract

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.

1. 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 [15–19] 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:

𝑥𝑥^1_2(𝑡)=𝑎𝑥₂−𝑑₁𝑥₁−𝑑₂𝑥²¹−𝛽𝑥₁−𝑟₁𝑥³¹,(𝑡)=𝛽𝑥₁−𝑏₁𝑥₂−𝑐₂𝐸𝑥₂,(1.1) where 𝑥₁(𝑡),𝑥₂(𝑡) represent the population density of the immature and the mature at time 𝑡, respectively, 𝑟₁𝑥³¹ is the effects of toxicant on the immature, 𝐸 is the harvesting effort, 𝑐₂ 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,

𝐸^(𝑡)=𝑘𝐸𝑝₂𝑐₂𝑥₂−𝑐.(1.2) 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.

2. The Steady States and Stability

It can be calculated that system (1.1) has two possible equilibriums:

(i)the trivial Equilibrium 𝐸₀(0,0), (ii)the equilibrium 𝐸_∗(𝑥^∗1,𝑥^∗2), where

𝑎𝑥^∗2−𝑑𝛽₁𝑥𝑥^∗1_∗1−𝑑₂𝑥^1∗2−𝛽𝑥^∗1−𝑟₁𝑥^1∗3=0,−𝑏₁𝑥^∗2−𝑐₂𝐸𝑥^∗2=0.(2.1) By simple calculation we have

𝑥^∗1=−𝑑²+𝑑²²+4𝑟₁𝑎𝛽/𝑏₁+𝑐₂𝐸−𝑑₁−𝛽2𝑟₁,𝑥^∗2=𝛽𝑏₁+𝑐₂𝐸𝑥^∗1.(2.2) To ensure the positivity of the equilibrium 𝐸_∗(𝑥^∗1,𝑥^∗2), we assume that

𝑎𝛽>𝑏₁+𝑐₂𝐸𝑑₁+𝛽(2.3) holds. We can see that 𝑥^∗1,𝑥^∗2 decrease as 𝑟₁ 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

−𝑑¹−𝛽−2𝛽𝑑₂𝑥₁−3𝑟₁𝑥²¹−𝑏₁𝑎−𝑐₂𝐸.(2.4) For 𝐸₀(0,0), the characteristic equation is 𝜆²+𝑑₁+𝛽+𝑏₁+𝑐₂𝐸𝜆+𝑑₁+𝛽𝑏₁+𝑐₂𝐸−𝑎𝛽=0.(2.5) It is not hard to see that when 𝑎𝛽<(𝑑₁+𝛽)(𝑏₁+𝑐₂𝐸), (2.5) has two negative roots or two complex roots with negative real parts; thus 𝐸₀(0,0) is locally asymptotically stable; when 𝑎𝛽>(𝑑₁+𝛽)(𝑏₁+𝑐₂𝐸), 𝐸₀(0,0) is a saddle point.

For 𝐸_∗(𝑥^∗1,𝑥^∗2), the characteristic equation is

𝜆²+𝑑₁+𝛽+𝑏₁+𝑐₂𝐸+2𝑑₂𝑥^∗1+3𝑟₁𝑥^1∗2𝜆+𝑑₁+𝛽+2𝑑₂𝑥^∗1+3𝑟₁𝑥^1∗2𝑏₁+𝑐₂𝐸−𝑎𝛽=0.(2.6) By applying (2.1), we have

𝑑₁+𝛽+2𝑑₂𝑥^∗1+3𝑟₁𝑥^1∗2𝑏₁+𝑐₂𝐸−𝑎𝛽=𝑏₁+𝑐₂𝐸𝑑₂𝑥^∗1+2𝑟₁𝑥^1∗2>0.(2.7) 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 𝐸_∗(𝑥^∗1,𝑥^∗2), it exists if 𝑎𝛽>(𝑏₁+𝑐₂𝐸)(𝑑₁+𝛽); in this case 𝐸₀(0,0) is a saddle point; thus, 𝐸_∗(𝑥^∗1,𝑥^∗2) is the unique stable equilibrium in the first quadrant if it exists. Let 𝐴𝐵 be the line segment of 𝐿₁∶𝑥₁=𝑝 and 𝐵𝐶 the line segment of 𝐿₂∶𝑥₂=𝑞, where 𝐴(𝑝,0),𝐵(𝑝,𝑞),𝐶(0,𝑞), and 𝑝,𝑞 are positive constants which satisfy 𝑝>𝑥^∗1, and

𝛽𝑝𝑏₁+𝑐₂𝐸<𝑞<𝑝𝑑¹+𝛽+𝑑₂𝑝+𝑟₁𝑝²𝑎.(2.8) By simple calculation, we have

𝑥𝑥̇̇₁₂∣_𝐴𝐵=𝑎𝑥₂−𝑑₁𝑝−𝑑₂𝑝²−𝛽𝑝−𝑟₁𝑝³∣_{0≤𝑥2≤𝑞}<0,∣_𝐵𝐶=𝛽𝑥₁−𝑏₁+𝑐₂𝐸𝑞∣_{0≤𝑥1≤𝑝}<0.(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).

Figure 1 

Trajectories enter rectangle 𝑂𝐴𝐵𝐶𝑂 from exterior to interior.

Denote

𝑥𝑥^1_2(𝑡)=𝑎𝑥₂−𝑑₁𝑥₁−𝑑₂𝑥²¹−𝛽𝑥₁−𝑟₁𝑥³¹−𝑐₁𝐸𝑥₁=𝑃𝑥₁,𝑥₂,(𝑡)=𝛽𝑥₁−𝑏₁𝑥₂−𝑐₂𝐸𝑥₂=𝑄𝑥₁,𝑥₂.(2.10) It is easy to see that

𝜕𝑃𝜕𝑥₁+𝜕𝑄𝜕𝑥₂=−𝑑₁−𝛽−2𝑑₂𝑥₁−3𝑟₁𝑥²¹−𝑏₁−𝑐₂𝐸<0.(2.11) By Poincare-Bendixson theorem, there are no limit cycles in the first quadrant; thus 𝐸_∗(𝑥^∗1,𝑥^∗2) is globally asymptotically stable if it exists.

For 𝐸₀(0,0), it is a unique equilibrium which is locally asymptotical stable if 𝑎𝛽<(𝑏₁+𝑐₂𝐸)(𝑑₁+𝛽). Similarly to the above analysis we can show that 𝐸₀(0,0) is globally asymptotically stable if 𝑎𝛽<(𝑏₁+𝑐₂𝐸)(𝑑₁+𝛽) holds.

Therefore, we have the following.

(i)If 𝑎𝛽<(𝑑₁+𝛽)(𝑏₁+𝑐₂𝐸), the trivial equilibrium 𝐸₀(0,0) is globally asymptotically stable. (ii)If 𝑎𝛽>(𝑑₁+𝛽)(𝑏₁+𝑐₂𝐸), the positive equilibrium 𝐸_∗(𝑥^∗1,𝑥^∗2) 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.

3. 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 𝑝₂ be the constant price per unit biomass of the mature. The net revenue of harvesting at any time is given by:

𝑃𝑥₁,𝑥₂,𝐸=𝑝₂𝑐₂𝐸𝑥₂−𝑐𝐸.(3.1) 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 𝐸(𝑥_1∞,𝑥_2∞,𝐸_∞) satisfying

𝑎𝑥_2∞−𝛽𝑑𝑥_11∞𝑥_1∞−𝑑₂𝑥^21∞−𝛽𝑥_1∞−𝑟₁𝑥^31∞=0,(3.2)−𝑝𝑏₁𝑥_2∞−𝑐₂𝑥_2∞𝐸_∞=0,(3.3)₂𝑐₂𝑥_2∞−𝑐=0.(3.4) From (3.4) we get 𝑥_2∞=𝑐/𝑝₂𝑐₂. Combining (3.4) and (3.2) we can obtain that 𝑥_1∞ is one of the roots of the following equation:

𝑟₁𝑥³¹+𝑑₂𝑥²¹+𝑑₁+𝛽𝑥₁−𝑎𝑐𝑝₂𝑐₂=0.(3.5) Denoting 𝑓(𝑥)=𝑟₁𝑥³+𝑑₂𝑥²+(𝑑₁+𝛽)𝑥−𝑎𝑐𝑝₂𝑐₂, we have

𝑓(0)=−𝑎𝑐𝑝₂𝑐₂<0,𝑓(+∞)=+∞,𝑓′(𝑥)>0(𝑥∈[0,∞)).(3.6) 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

𝑥_1∞>𝑏¹𝑐𝛽𝑝₂𝑐₂,(3.7) Thus we need to find a solution of 𝑓(𝑥) in (𝑏₁𝑐/𝛽𝑝₂𝑐₂,+∞). Since (3.6) always holds, we only need

𝑓𝑏¹𝑐𝛽𝑝₂𝑐₂<0.(3.8) 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.

4. 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(𝑝₂𝑐₂𝑥₂−𝑐); then system (1.1) becomes

𝑥𝑥^1_2(𝑡)=𝑎𝑥₂−𝑑₁𝑥₁−𝑑₂𝑥²¹−𝛽𝑥₁−𝑟₁𝑥³¹,𝐸_(𝑡)=𝛽𝑥₁−𝑏₁𝑥₂−𝑐₂𝐸𝑥₂,(𝑡)=𝑘𝐸𝑝₂𝑐₂𝑥₂−𝑐.(4.1) System (4.1) has three possible equilibrium:

(i)the trivial equilibrium𝑉₀(0,0,0),(ii)equilibrium in the absence of harvesting 𝑉₁(𝑥̃₁,𝑥̃₂,0), where 𝑥̃₁=−𝑑²+𝑑²²+4𝑟₁𝛽𝑎/𝑏₁−𝑑₁−𝛽2𝑟₁,𝑥̃₂=𝛽𝑏₁𝑥̃₁,(4.2) and for the positiveness of 𝑥̃₁,𝑥̃₂, we need

𝛽𝑎>𝑑₁+𝛽𝑏₁,(4.3)

(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 𝑉₀(0,0,0), the characteristic equation is given by

(𝜆+𝑘𝑐)𝜆+𝑑₁+𝛽𝜆+𝑏₁−𝛽𝑎=0.(4.4) It is easy to see that all of the roots of (4.4) are negative if 𝛽𝑎<𝑏₁(𝑑₁+𝛽) holds; thus 𝑉₀(0,0,0) is locally asymptotically stable if 𝛽𝑎<𝑏₁(𝑑₁+𝛽), and unstable if 𝛽𝑎>𝑏₁(𝑑₁+𝛽).

For 𝑉₁(𝑥̃₁,𝑥̃₂,0), the characteristic equation is given by

𝜆−𝑘𝑝₂𝑐₂𝑥̃₂−𝑐𝜆+𝑑₁+𝛽+2𝑑₂𝑥̃₁+3𝑟₁𝑥̃²¹𝜆+𝑏₁−𝑎𝛽=0.(4.5) It is no hard to see that 𝑉₁(𝑥̃₁,𝑥̃₂,0) is locally asymptotically stable if 𝑝₂𝑐₂𝑥̃₂−𝑐<0, and unstable if 𝑝₂𝑐₂𝑥̃₂−𝑐>0.

From the condition for the stability of 𝑉₀,𝑉₁, 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

𝜆³+𝑈𝜆²+𝑉𝜆+𝐿=0,(4.6) where

𝑈𝑉==𝑏𝑏₁₁+𝑐₂𝐸_∞+𝑑₁+𝛽+2𝑑₂𝑥_1∞+3𝑟₁𝑥^21∞>0,=𝑏+𝑐₂𝐸_∞𝑑₁+𝛽+2𝑑₂𝑥_1∞+3𝑟₁𝑥^21∞+𝑐²²𝑘𝑝₂𝑥_2∞𝐸_∞−𝑎𝛽₁𝐿=𝑐₂₂+𝑐₂𝐸_∞𝑑₂𝑥_1∞+2𝑟₁𝑥^21∞+𝑐²²𝑘𝑝₂𝑥_2∞𝐸_∞>0,𝑘𝑝₂𝑥_2∞𝐸_∞𝑑₁+𝛽+2𝑑₂𝑥_1∞+3𝑟₁𝑥^21∞>0.(4.7) By Routh-Hurwitz criterion, all roots of (4.6) have negative real parts if and only if

𝑈>0,𝐿>0,𝑈𝑉>𝐿.(4.8) 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:

𝑉=𝑥₁−𝑥_1∞−𝑥_1∞ln𝑥¹𝑥_1∞+𝑥₂−𝑥_2∞−𝑥_2∞ln𝑥²𝑥_2∞+𝑛𝐸−𝐸^∞−ln𝐸𝐸_∞.(4.9) The time derivative of 𝑉 along the positive solution of system (4.1) is

𝑉̇=𝑥¹−𝑥_1∞𝑥₁𝑥^1(𝑡)+𝑥²−𝑥_2∞𝑥₂𝑥^2(𝑡)+𝑛𝐸−𝐸^∞=𝑥𝐸𝐸′(𝑡)₁−𝑥_1∞𝑥₁𝑎𝑥₂+𝑥−𝑥_2∞−𝑑₁+𝛽𝑥₁−𝑥_1∞−𝑑₂𝑥²¹−𝑥^21∞−𝑟₁𝑥³¹−𝑥^31∞₂−𝑥_2∞𝑥₂𝛽𝑥₁+𝑛𝑘𝐸−𝐸_∞−𝑥_1∞−𝑏₁𝑥₂−𝑥_2∞−𝑐₂𝐸𝑥₂−𝐸_∞𝑥_2∞𝐸𝐸𝑝²=−𝑥₁𝑐₂𝑥₂−𝑥_2∞−𝑥_1∞²𝑥₁𝑑₁−𝑥+𝛽+𝑑₂𝑥₁+𝑥_1∞+𝑟₁𝑥²¹+𝑥₁𝑥_1∞+𝑥^21∞₂−𝑥_2∞²𝑥₂𝑏₁+𝑐₂𝐸_∞+𝑎𝑥₁+𝛽𝑥₂𝑥¹+−𝑐₂−𝑥_1∞𝑥₂−𝑥_2∞+𝑛𝑘𝑝₂𝑐₂𝑥₂−𝑥_2∞𝐸−𝐸_∞.(4.10) Let 𝑛𝑘𝑝₁=1, then we have

𝑉̇=−𝑥¹−𝑥_1∞²𝑥₁𝑑₁−𝑥+𝛽+𝑑₂𝑥₁+𝑥_1∞+𝑟₁𝑥²¹+𝑥₁𝑥_1∞+𝑥^21∞₂−𝑥_2∞²𝑥₂𝑏₁+𝑐₂𝐸_∞+𝑎𝑥₁+𝛽𝑥₂𝑥¹−𝑥_1∞𝑥₂−𝑥_2∞.(4.11) If inequality

1𝑥₁𝑥₂𝑑₁+𝛽+𝑑₂𝑥₁+𝑥_1∞+𝑟₁𝑥²¹+𝑥₁𝑥_1∞+𝑥^21∞𝑏₁+𝑐₂𝐸>14𝑥𝑎₁+𝛽𝑥₂²(4.12) holds, then 𝑉̇(𝑡)<0 in set Ω={𝑥₁>0,𝑥₂>0}. Set

𝑔𝑥₁,𝑥₂=−𝑥₁1𝑥₂𝑑₁+𝛽+𝑑₂𝑥₁+𝑥_1∞+𝑟₁𝑥²¹+𝑥₁𝑥_1∞+𝑥^21∞𝑏₁+𝑐₂𝐸4𝑎𝑥²+𝛽𝑥₁²,(4.13) then (4.12) holds in set Ω if 𝑔(𝑥₁,𝑥₂)>0. By applying (3.2) and (3.3), we have

𝑔𝑥₁,𝑥₂=12𝑎𝛽𝑥¹𝑥₂+𝑥₁𝑥₂𝑑₁𝑥₁+𝑟₁𝑥²¹+𝑟₁𝑥₁𝑥_1∞𝑏₁+𝑐₂𝐸−14𝑎²𝑥²²−14𝛽²𝑥²¹.(4.14) If 𝑥₁≥𝑥₂, then

𝑔𝑥₁,𝑥₂≥12𝑎𝛽𝑥²²+𝑥²²𝑑₁𝑥₂+𝑟₁𝑥²²+𝑟₁𝑥₂𝑥_1∞𝑏₁+𝑐₂𝐸−14𝑎²+𝛽²𝑥²¹.(4.15) Thus, we can get that if

𝑥₂≤𝑥₁<ℎ₂𝑥₂(4.16) holds, then 𝑔(𝑥₁,𝑥₂)>0, where

ℎ₂𝑥₂=𝑥₂2𝑎𝛽+4𝑑₁𝑥₂+𝑟₁𝑥²²+𝑟₁𝑥₂𝑥_1∞𝑏₁+𝑐₂𝐸_∞𝑎²+𝛽².(4.17) If 𝑥₁<𝑥₂, by the same way above, we can get the other sufficient condition for 𝑔(𝑥₁,𝑥₂)>0, that is,

𝑥₁<𝑥₂<ℎ₁𝑥₁,(4.18) where

ℎ₁𝑥₁=𝑥₁2𝑎𝛽+4𝑑₁𝑥₁+𝑟₁𝑥²¹+𝑟₁𝑥₁𝑥_1∞𝑏₁+𝑐₂𝐸_∞𝑎²+𝛽².(4.19) Therefore, if (4.16) or (4.18) holds, then 𝑉̇(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.

5. 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:

𝐽=^∞0𝑃𝑥₁,𝑥₂,𝐸,𝑡𝑒^−𝛿𝑡𝑑𝑡,(5.1) where 𝑃 is the net revenue given by 𝑃(𝑥₁,𝑥₂,𝐸,𝑡)=𝑝₂𝑐₂𝐸𝑥₂−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:

𝐻=𝑝₂𝑐₂𝑥₂−𝑐𝐸𝑒^−𝛿𝑡+𝜆₁𝑎𝑥₂−𝑑₁𝑥₁−𝑑₂𝑥²¹−𝛽𝑥₁−𝑟₁𝑥³¹+𝜆₂𝛽𝑥₁−𝑏₁𝑥₂−𝑐₂𝐸𝑥₂,(5.2) where 𝜆₁(𝑡),𝜆₂(𝑡) are the adjoint variables, 𝐸 is the control variable satisfying the constraints 0≤𝐸≤𝐸_max, and 𝜙(𝑡)=𝑒^−𝛿𝑡(𝑝₂𝑐₂𝑥₂−𝑐)−𝜆₂𝑐₂𝑥₂ 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,when𝜙(𝑡)>0,thatis,when𝜆₂(𝑡)𝑒^𝛿𝑡<𝑝₂−𝑐𝑐₂𝑥₂𝐸=0,when𝜙(𝑡)<0,thatis,when𝜆₂;(𝑡)𝑒^𝛿𝑡>𝑝₂−𝑐𝑐₂𝑥₂.(5.3) When 𝜙(𝑡)=0, that is,

𝜆₂(𝑡)𝑒^𝛿𝑡=𝑝₂−𝑐𝑐₂𝑥₂,or𝜕𝐻𝜕𝐸=0.(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

𝑑𝜆₁𝑑𝑡=−𝜕𝜕𝐻𝑥₁𝑑𝜆₂=𝜆₁𝑑₁+2𝑑₂𝑥₁+𝛽+3𝑟₁𝑥²¹−𝜆₂𝛽,𝑑𝑡=−𝜕𝜕𝐻𝑥₂=−𝑝₂𝑐₂𝐸𝑒^−𝛿𝑡+𝜆₂𝑏₁+𝑐₂𝐸−𝜆₁𝑎.(5.5) From (5.4) and (5.5), we have

𝑑𝜆₁𝑑𝑡−𝐵𝜆¹=𝐴𝑒^−𝛿𝑡,(5.6) where 𝐵=𝑑₁+2𝑑₂𝑥₁+𝛽+3𝑟₁𝑥²¹,𝐴=𝛽(𝑐/𝑐₂𝑥₂−𝑝₂). We can calculate that

𝜆₁=−𝐴𝐵+𝛿𝑒^−𝛿𝑡.(5.7) Substituting (5.7) into the second equation of (5.5), we get

𝑑𝜆₂𝑑𝑡−𝐺𝜆²=𝐷𝑒^−𝛿𝑡,(5.8) where 𝐺=𝑏₁+𝑐₂𝐸,𝐷=−𝑝₂𝑐₂𝐸+𝐴/(𝐵+𝛿). Therefore, we have

𝜆₂=−𝐷𝐺+𝛿𝑒^−𝛿𝑡.(5.9) It is obviously that 𝜆₁(𝑡),𝜆₂(𝑡) are bounded as 𝑡→∞.

Substituting (5.9) into (5.4), we obtain

𝑝₂−𝑐𝑐₂𝑥₂=−𝐷𝐺+𝛿.(5.10) Our purpose is to find an optimal equilibrium solution; so we have

𝑥_1𝛿=𝑥^∗1=−𝑑²+𝑑²²+4𝑟₁𝑎𝛽/𝑏₁+𝑐₂𝐸−𝑑₁−𝛽2𝑟₁,𝑥_2𝛿=𝑥^∗2=𝛽𝑏₁+𝑐₂𝐸𝑥^∗1.(5.11) By (5.10) and (5.11), we can get 𝑥_1𝛿,𝑥_2𝛿, and𝐸_𝛿. Thus, the optimal policy is

𝐸=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐸^max,when𝜆₂(𝑡)𝑒^𝛿𝑡<𝑝₂−𝑐𝑐₂𝑥₂𝐸_𝛿,,when𝜆₂(𝑡)𝑒^𝛿𝑡=𝑝₂−𝑐𝑐₂𝑥₂0,when𝜆₂,(𝑡)𝑒^𝛿𝑡>𝑝₂−𝑐𝑐₂𝑥₂.(5.12) Again, from (5.10) we have

𝑃=𝑝₂𝑐₂𝑥₂−𝑐𝐸=−𝐷𝑐²𝑥₂𝐺+𝛿𝐸.(5.13) When 𝛿→∞, 𝑃∼𝑜(𝛿⁻¹). Therefore, 𝛿=0 leads to the maximization of 𝑃.

6. Number Simulations

In the following examples, we take the parameters values as 𝑎=2,𝑑₁=0.1,𝑑₂=0.1,𝑐₂=0.2,𝑏₁=0.1,and𝛽=0.2. We will see how the system behavior is while the toxicant effect changes.

Example 6.1. 𝐸=1; in this case, 𝑎𝛽=0.4>0.09=(𝑑₁+𝛽)(𝑏₁+𝑐₂𝐸). From the results in Section 2, we know that for a given 𝑟₁, the system admits a unique global stable positive equilibrium. Indeed, considering system (1.1) and the initial conditions (6,2),(5,10), and(1,5), respectively, we can see that(i)𝑟₁=0, 𝐸^∗(10.33,6.89) is global stable;(ii)𝑟₁=0.01, 𝐸^∗(6.33,4.22) is global stable (Figure 2);(iii)𝑟₁=1, 𝐸^∗(0.97,0.65) is global stable (see Figure 3);(iv)𝑟₁=100, 𝐸^∗(0.01,0.07) is global stable (Figure 4).

Figure 2 

Solution curves of system (1.1) with the parameters given by Example 6.1 when 𝑟1=0.01.

Figure 3 

Solution curves of system (1.1) with the parameters given by Example 6.1 when 𝑟1=1.

Figure 4 

Solution curves of system (1.1) with the parameters given by Example 6.1 when 𝑟1=100.

Example 6.2. 𝑘=0.1,𝑝₂=2,𝑐=0.2,𝛿=0.01, and 𝐸′(𝑡)=0.1𝐸(0.4𝑥₂−0.2). Considering system (4.1) with initial condition (2,3,3),(4,5,6), and (1,1,1), we have the following.(i)𝑟₁=0; the bionomic equilibrium 𝐸(2,0.5,3.5) is globally stable (Figure 5). The optimal equilibrium (10.32,6.87,1) is far away from the bionomic equilibrium. (ii)𝑟₁=1; the bionomic equilibrium 𝐸(0.87,0.5,1.24) is globally stable (Figure 6). The optimal equilibrium is (1.26,1.28,0.49).(iii)𝑟₁=10; the bionomic equilibrium 𝐸(0.44,0.5,0.38) is globally stable (Figure 7). The optimal equilibrium is (0.51,0.74,0.18).(iv)𝑟₁=100; both the bionomic equilibrium 𝐸(0.2,0.5,−0.08) and the optimal equilibrium (0.20,0.44,−0.046) are unfeasible.

Figure 5 

Solution curves of system (4.1) with the parameters given by Example 6.2 when 𝑟1=0.

Figure 6 

Solution curves of system (4.1) with the parameters given by Example 6.2 when 𝑟1=1.

Figure 7 

Solution curves of system (4.1) with the parameters given by Example 6.2 when 𝑟1=10.

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.

7. 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.

Acknowledgment

This work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).