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 [1ā€“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 [6ā€“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 [15ā€“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, 20ā€“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:

š‘„ī…ž1(š‘”)=š‘Žš‘„2āˆ’š‘‘1š‘„1āˆ’š‘‘2š‘„21āˆ’š›½š‘„1āˆ’š‘Ÿ1š‘„31,š‘„ī…ž2(š‘”)=š›½š‘„1āˆ’š‘1š‘„2āˆ’š‘2šøš‘„2,(1.1) where š‘„1(š‘”),š‘„2(š‘”) represent the population density of the immature and the mature at time š‘”, respectively, š‘Ÿ1š‘„31 is the effects of toxicant on the immature, šø is the harvesting effort, š‘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 šøī…ž is proportion to the economic revenue [23], that is,

šøī…žī€·š‘(š‘”)=š‘˜šø2š‘2š‘„2ī€øāˆ’š‘.(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 šø0(0,0), (ii)the equilibrium šøāˆ—(š‘„āˆ—1,š‘„āˆ—2), where

š‘Žš‘„āˆ—2āˆ’š‘‘1š‘„āˆ—1āˆ’š‘‘2š‘„1āˆ—2āˆ’š›½š‘„āˆ—1āˆ’š‘Ÿ1š‘„1āˆ—3=0,š›½š‘„āˆ—1āˆ’š‘1š‘„āˆ—2āˆ’š‘2šøš‘„āˆ—2=0.(2.1) By simple calculation we have

š‘„āˆ—1=āˆ’š‘‘2+ī”š‘‘22+4š‘Ÿ1ī€·ī€·š‘š‘Žš›½/1+š‘2šøī€øāˆ’š‘‘1ī€øāˆ’š›½2š‘Ÿ1,š‘„āˆ—2=š›½š‘1+š‘2šøš‘„āˆ—1.(2.2) To ensure the positivity of the equilibrium šøāˆ—(š‘„āˆ—1,š‘„āˆ—2), we assume that

ī€·š‘š‘Žš›½>1+š‘2šøš‘‘ī€øī€·1ī€ø+š›½(2.3) holds. We can see that š‘„āˆ—1,š‘„āˆ—2 decrease as š‘Ÿ1 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

īƒ¬āˆ’š‘‘1āˆ’š›½āˆ’2š‘‘2š‘„1āˆ’3š‘Ÿ1š‘„21š‘Žš›½āˆ’š‘1āˆ’š‘2šøīƒ­.(2.4) For šø0(0,0), the characteristic equation is šœ†2+ī€·š‘‘1+š›½+š‘1+š‘2šøī€øī€·š‘‘šœ†+1š‘+š›½ī€øī€·1+š‘2šøī€øāˆ’š‘Žš›½=0.(2.5) It is not hard to see that when š‘Žš›½<(š‘‘1+š›½)(š‘1+š‘2šø), (2.5) has two negative roots or two complex roots with negative real parts; thus šø0(0,0) is locally asymptotically stable; when š‘Žš›½>(š‘‘1+š›½)(š‘1+š‘2šø), šø0(0,0) is a saddle point.

For šøāˆ—(š‘„āˆ—1,š‘„āˆ—2), the characteristic equation is

šœ†2+ī€·š‘‘1+š›½+š‘1+š‘2šø+2š‘‘2š‘„āˆ—1+3š‘Ÿ1š‘„1āˆ—2ī€øī€·š‘‘šœ†+1+š›½+2š‘‘2š‘„āˆ—1+3š‘Ÿ1š‘„1āˆ—2š‘ī€øī€·1+š‘2šøī€øāˆ’š‘Žš›½=0.(2.6) By applying (2.1), we have

ī€·š‘‘1+š›½+2š‘‘2š‘„āˆ—1+3š‘Ÿ1š‘„1āˆ—2š‘ī€øī€·1+š‘2šøī€øī€·š‘āˆ’š‘Žš›½=1+š‘2šøš‘‘ī€øī€·2š‘„āˆ—1+2š‘Ÿ1š‘„1āˆ—2ī€ø>0.(2.7) Therefore, the characteristic equation of šøāˆ—(š‘„āˆ—1,š‘„āˆ—2) has two negative roots or two complex roots with negative real parts; thus šøāˆ—(š‘„āˆ—1,š‘„āˆ—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 šøāˆ—(š‘„āˆ—1,š‘„āˆ—2), it exists if š‘Žš›½>(š‘1+š‘2šø)(š‘‘1+š›½); in this case šø0(0,0) is a saddle point; thus, šøāˆ—(š‘„āˆ—1,š‘„āˆ—2) is the unique stable equilibrium in the first quadrant if it exists. Let š“šµ be the line segment of šæ1āˆ¶š‘„1=š‘ and šµš¶ the line segment of šæ2āˆ¶š‘„2=š‘ž, where š“(š‘,0),šµ(š‘,š‘ž),š¶(0,š‘ž), and š‘,š‘ž are positive constants which satisfy š‘>š‘„āˆ—1, and

š›½š‘š‘1+š‘2šøš‘ī€·š‘‘<š‘ž<1+š›½+š‘‘2š‘+š‘Ÿ1š‘2ī€øš‘Ž.(2.8) By simple calculation, we have

Ģ‡š‘„1āˆ£š“šµ=š‘Žš‘„2āˆ’š‘‘1š‘āˆ’š‘‘2š‘2āˆ’š›½š‘āˆ’š‘Ÿ1š‘3āˆ£0ā‰¤š‘„2ā‰¤š‘ž<0,Ģ‡š‘„2āˆ£šµš¶=š›½š‘„1āˆ’ī€·š‘1+š‘2šøī€øš‘žāˆ£0ā‰¤š‘„1ā‰¤š‘<0.(2.9) Thus š“šµ,šµš¶ are the transversals of system (1.1). It is no hard to check that š‘‚š“,š‘‚š¶ are the transversals of system (1.1), and any trajectory enters region š‘‚š“šµš¶š‘‚ from its exterior to interior (see Figure 1).


š‘„ī…ž1(š‘”)=š‘Žš‘„2āˆ’š‘‘1š‘„1āˆ’š‘‘2š‘„21āˆ’š›½š‘„1āˆ’š‘Ÿ1š‘„31āˆ’š‘1šøš‘„1ī€·š‘„=š‘ƒ1,š‘„2ī€ø,š‘„ī…ž2(š‘”)=š›½š‘„1āˆ’š‘1š‘„2āˆ’š‘2šøš‘„2ī€·š‘„=š‘„1,š‘„2ī€ø.(2.10) It is easy to see that

šœ•š‘ƒšœ•š‘„1+šœ•š‘„šœ•š‘„2=āˆ’š‘‘1āˆ’š›½āˆ’2š‘‘2š‘„1āˆ’3š‘Ÿ1š‘„21āˆ’š‘1āˆ’š‘2šø<0.(2.11) By Poincare-Bendixson theorem, there are no limit cycles in the first quadrant; thus šøāˆ—(š‘„āˆ—1,š‘„āˆ—2) is globally asymptotically stable if it exists.

For šø0(0,0), it is a unique equilibrium which is locally asymptotical stable if š‘Žš›½<(š‘1+š‘2šø)(š‘‘1+š›½). Similarly to the above analysis we can show that šø0(0,0) is globally asymptotically stable if š‘Žš›½<(š‘1+š‘2šø)(š‘‘1+š›½) holds.

Therefore, we have the following.

(i)If š‘Žš›½<(š‘‘1+š›½)(š‘1+š‘2šø), the trivial equilibrium šø0(0,0) is globally asymptotically stable. (ii)If š‘Žš›½>(š‘‘1+š›½)(š‘1+š‘2šø), the positive equilibrium šøāˆ—(š‘„āˆ—1,š‘„āˆ—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 š‘ be the constant fishing cost per unit effort, and let š‘2 be the constant price per unit biomass of the mature. The net revenue of harvesting at any time is given by:

š‘ƒī€·š‘„1,š‘„2ī€ø,šø=š‘2š‘2šøš‘„2āˆ’š‘šø.(3.1) A bionomic equilibrium is both a biological equilibrium and a economic equilibrium, the biological equilibrium is given by š‘„ī…ž1(š‘”)=š‘„ī…ž2(š‘”)=0, and the economic equilibrium occurs when the economic rent is š‘ƒ=0, thus the bionomic equilibrium šø(š‘„1āˆž,š‘„2āˆž,šøāˆž) satisfying

š‘Žš‘„2āˆžāˆ’š‘‘1š‘„1āˆžāˆ’š‘‘2š‘„21āˆžāˆ’š›½š‘„1āˆžāˆ’š‘Ÿ1š‘„31āˆž=0,(3.2)š›½š‘„1āˆžāˆ’š‘1š‘„2āˆžāˆ’š‘2š‘„2āˆžšøāˆžš‘=0,(3.3)2š‘2š‘„2āˆžāˆ’š‘=0.(3.4) From (3.4) we get š‘„2āˆž=š‘/š‘2š‘2. Combining (3.4) and (3.2) we can obtain that š‘„1āˆž is one of the roots of the following equation:

š‘Ÿ1š‘„31+š‘‘2š‘„21+ī€·š‘‘1ī€øš‘„+š›½1āˆ’š‘Žš‘š‘2š‘2=0.(3.5) Denoting š‘“(š‘„)=š‘Ÿ1š‘„3+š‘‘2š‘„2+(š‘‘1+š›½)š‘„āˆ’š‘Žš‘š‘2š‘2, we have

š‘“(0)=āˆ’š‘Žš‘š‘2š‘2[<0,š‘“(+āˆž)=+āˆž,š‘“ā€²(š‘„)>0(š‘„āˆˆ0,āˆž)).(3.6) Hence, by the continuity of š‘“(š‘„), there exists exactly one root in (0,+āˆž). From (3.3) and (3.4), to ensure the positivity of šøāˆž, one needs

š‘„1āˆž>š‘1š‘š›½š‘2š‘2,(3.7) Thus we need to find a solution of š‘“(š‘„) in (š‘1š‘/š›½š‘2š‘2,+āˆž). Since (3.6) always holds, we only need

š‘“ī‚µš‘1š‘š›½š‘2š‘2ī‚¶<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 šø>šøāˆž cannot be maintained all the time, it will decrease because the total cost of harvesting exceed the total revenues; (ii) šø<šøāˆž 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 šøāˆž. However, there is no answer about whether it will become stable or not because of the complex changing of šø.

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 šøī…ž(š‘”)=š‘˜šø(š‘2š‘2š‘„2āˆ’š‘); then system (1.1) becomes

š‘„ī…ž1(š‘”)=š‘Žš‘„2āˆ’š‘‘1š‘„1āˆ’š‘‘2š‘„21āˆ’š›½š‘„1āˆ’š‘Ÿ1š‘„31,š‘„ī…ž2(š‘”)=š›½š‘„1āˆ’š‘1š‘„2āˆ’š‘2šøš‘„2,šøī…žī€·š‘(š‘”)=š‘˜šø2š‘2š‘„2ī€ø.āˆ’š‘(4.1) System (4.1) has three possible equilibrium:

(i)the trivial equilibriumš‘‰0(0,0,0),(ii)equilibrium in the absence of harvesting š‘‰1(Ģƒš‘„1,Ģƒš‘„2,0), where Ģƒš‘„1=āˆ’š‘‘2+ī”š‘‘22+4š‘Ÿ1ī€·š›½š‘Ž/š‘1āˆ’š‘‘1ī€øāˆ’š›½2š‘Ÿ1,Ģƒš‘„2=š›½š‘1Ģƒš‘„1,(4.2) and for the positiveness of Ģƒš‘„1,Ģƒš‘„2, we need


(iii)the interior equilibrium šø(š‘„1āˆž,š‘„2āˆž,šøāˆž), which is the bionomic equilibrium in Section 3; it exists if (3.8) holds.

For š‘‰0(0,0,0), the characteristic equation is given by

(šœ†+š‘˜š‘)ī€·ī€·šœ†+š‘‘1+š›½ī€øī€·šœ†+š‘1ī€øī€øāˆ’š›½š‘Ž=0.(4.4) It is easy to see that all of the roots of (4.4) are negative if š›½š‘Ž<š‘1(š‘‘1+š›½) holds; thus š‘‰0(0,0,0) is locally asymptotically stable if š›½š‘Ž<š‘1(š‘‘1+š›½), and unstable if š›½š‘Ž>š‘1(š‘‘1+š›½).

For š‘‰1(Ģƒš‘„1,Ģƒš‘„2,0), the characteristic equation is given by

ī€·ī€·š‘šœ†āˆ’š‘˜2š‘2Ģƒš‘„2āˆ’š‘ī€øī€øī€·ī€·šœ†+š‘‘1+š›½+2š‘‘2Ģƒš‘„1+3š‘Ÿ1Ģƒš‘„21ī€øī€·šœ†+š‘1ī€øī€øāˆ’š‘Žš›½=0.(4.5) It is no hard to see that š‘‰1(Ģƒš‘„1,Ģƒš‘„2,0) is locally asymptotically stable if š‘2š‘2Ģƒš‘„2āˆ’š‘<0, and unstable if š‘2š‘2Ģƒš‘„2āˆ’š‘>0.

From the condition for the stability of š‘‰0,š‘‰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 šø(š‘„1āˆž,š‘„2āˆž,šøāˆž), the characteristic equation is

šœ†3+š‘ˆšœ†2+š‘‰šœ†+šæ=0,(4.6) where

š‘ˆ=š‘1+š‘2šøāˆž+š‘‘1+š›½+2š‘‘2š‘„1āˆž+3š‘Ÿ1š‘„21āˆžī€·š‘>0,š‘‰=1+š‘2šøāˆžš‘‘ī€øī€·1+š›½+2š‘‘2š‘„1āˆž+3š‘Ÿ1š‘„21āˆžī€ø+š‘22š‘˜š‘2š‘„2āˆžšøāˆž=ī€·š‘āˆ’š‘Žš›½1+š‘2šøāˆžš‘‘ī€øī€·2š‘„1āˆž+2š‘Ÿ1š‘„21āˆžī€ø+š‘22š‘˜š‘2š‘„2āˆžšøāˆž>0,šæ=š‘22š‘˜š‘2š‘„2āˆžšøāˆžī€·š‘‘1+š›½+2š‘‘2š‘„1āˆž+3š‘Ÿ1š‘„21āˆžī€ø>0.(4.7) By Routh-Hurwitz criterion, all roots of (4.6) have negative real parts if and only if

š‘ˆ>0,šæ>0,š‘ˆš‘‰>šæ.(4.8) By simple calculation, we know that condition (4.8) holds always, Thus, šø(š‘„1āˆž,š‘„2āˆž,šøāˆž) is locally asymptotically stable.

For the global stability of šø(š‘„1āˆž,š‘„2āˆž,šøāˆž), we construct the following Lyapunov function:

š‘‰=š‘„1āˆ’š‘„1āˆžāˆ’š‘„1āˆžš‘„ln1š‘„1āˆž+ī‚µš‘„2āˆ’š‘„2āˆžāˆ’š‘„2āˆžš‘„ln2š‘„2āˆžī‚¶ī‚µ+š‘›šøāˆ’šøāˆžšøāˆ’lnšøāˆžī‚¶.(4.9) The time derivative of š‘‰ along the positive solution of system (4.1) is

Ģ‡š‘„š‘‰=1āˆ’š‘„1āˆžš‘„1š‘„ī…ž1š‘„(š‘”)+2āˆ’š‘„2āˆžš‘„2š‘„ī…ž2(š‘”)+š‘›šøāˆ’šøāˆžšø=š‘„šøā€²(š‘”)1āˆ’š‘„1āˆžš‘„1ī€½š‘Žī€·š‘„2āˆ’š‘„2āˆžī€øāˆ’ī€·š‘‘1š‘„+š›½ī€øī€·1āˆ’š‘„1āˆžī€øāˆ’š‘‘2ī€·š‘„21āˆ’š‘„21āˆžī€øāˆ’š‘Ÿ1ī€·š‘„31āˆ’š‘„31āˆž+š‘„ī€øī€¾2āˆ’š‘„2āˆžš‘„2ī€½š›½ī€·š‘„1āˆ’š‘„1āˆžī€øāˆ’š‘1ī€·š‘„2āˆ’š‘„2āˆžī€øāˆ’š‘2ī€·šøš‘„2āˆ’šøāˆžš‘„2āˆžī€øī€¾+š‘›š‘˜šøāˆ’šøāˆžšøšøī€½š‘2š‘2ī€·š‘„2āˆ’š‘„2āˆžī€·š‘„ī€øī€¾=āˆ’1āˆ’š‘„1āˆžī€ø2š‘„1ī€½š‘‘1+š›½+š‘‘2ī€·š‘„1+š‘„1āˆžī€ø+š‘Ÿ1ī€·š‘„21+š‘„1š‘„1āˆž+š‘„21āˆžāˆ’ī€·š‘„ī€øī€¾2āˆ’š‘„2āˆžī€ø2š‘„2ī€·š‘1+š‘2šøāˆžī€ø+ī‚µš‘Žš‘„1+š›½š‘„2ī‚¶ī€·š‘„1āˆ’š‘„1āˆžš‘„ī€øī€·2āˆ’š‘„2āˆžī€ø+ī€·āˆ’š‘2+š‘›š‘˜š‘2š‘2š‘„ī€øī€·2āˆ’š‘„2āˆžī€øī€·šøāˆ’šøāˆžī€ø.(4.10) Let š‘›š‘˜š‘1=1, then we have

Ģ‡ī€·š‘„š‘‰=āˆ’1āˆ’š‘„1āˆžī€ø2š‘„1ī€½š‘‘1+š›½+š‘‘2ī€·š‘„1+š‘„1āˆžī€ø+š‘Ÿ1ī€·š‘„21+š‘„1š‘„1āˆž+š‘„21āˆžāˆ’ī€·š‘„ī€øī€¾2āˆ’š‘„2āˆžī€ø2š‘„2ī€·š‘1+š‘2šøāˆžī€ø+ī‚µš‘Žš‘„1+š›½š‘„2ī‚¶ī€·š‘„1āˆ’š‘„1āˆžš‘„ī€øī€·2āˆ’š‘„2āˆžī€ø.(4.11) If inequality

1š‘„1š‘„2ī€·š‘‘1+š›½+š‘‘2ī€·š‘„1+š‘„1āˆžī€ø+š‘Ÿ1ī€·š‘„21+š‘„1š‘„1āˆž+š‘„21āˆžš‘ī€øī€øī€·1+š‘2šøī€ø>14ī‚µš‘Žš‘„1+š›½š‘„2ī‚¶2(4.12) holds, then Ģ‡š‘‰(š‘”)<0 in set Ī©={š‘„1>0,š‘„2>0}. Set

š‘”ī€·š‘„1,š‘„2ī€ø=š‘„1š‘„2ī€·š‘‘1+š›½+š‘‘2ī€·š‘„1+š‘„1āˆžī€ø+š‘Ÿ1ī€·š‘„21+š‘„1š‘„1āˆž+š‘„21āˆžš‘ī€øī€øī€·1+š‘2šøī€øāˆ’14ī€·š‘Žš‘„2+š›½š‘„1ī€ø2,(4.13) then (4.12) holds in set Ī© if š‘”(š‘„1,š‘„2)>0. By applying (3.2) and (3.3), we have

š‘”ī€·š‘„1,š‘„2ī€ø=12š‘Žš›½š‘„1š‘„2+š‘„1š‘„2ī€·š‘‘1š‘„1+š‘Ÿ1š‘„21+š‘Ÿ1š‘„1š‘„1āˆžš‘ī€øī€·1+š‘2šøī€øāˆ’14š‘Ž2š‘„22āˆ’14š›½2š‘„21.(4.14) If š‘„1ā‰„š‘„2, then

š‘”ī€·š‘„1,š‘„2ī€øā‰„12š‘Žš›½š‘„22+š‘„22ī€·š‘‘1š‘„2+š‘Ÿ1š‘„22+š‘Ÿ1š‘„2š‘„1āˆžš‘ī€øī€·1+š‘2šøī€øāˆ’14ī€·š‘Ž2+š›½2ī€øš‘„21.(4.15) Thus, we can get that if

š‘„2ā‰¤š‘„1<ā„Ž2ī€·š‘„2ī€ø(4.16) holds, then š‘”(š‘„1,š‘„2)>0, where

ā„Ž2ī€·š‘„2ī€ø=š‘„2īƒŽī€·š‘‘2š‘Žš›½+41š‘„2+š‘Ÿ1š‘„22+š‘Ÿ1š‘„2š‘„1āˆžš‘ī€øī€·1+š‘2šøāˆžī€øš‘Ž2+š›½2.(4.17) If š‘„1<š‘„2, by the same way above, we can get the other sufficient condition for š‘”(š‘„1,š‘„2)>0, that is,

š‘„1<š‘„2<ā„Ž1ī€·š‘„1ī€ø,(4.18) where

ā„Ž1ī€·š‘„1ī€ø=š‘„1īƒŽī€·š‘‘2š‘Žš›½+41š‘„1+š‘Ÿ1š‘„21+š‘Ÿ1š‘„1š‘„1āˆžš‘ī€øī€·1+š‘2šøāˆžī€øš‘Ž2+š›½2.(4.19) Therefore, if (4.16) or (4.18) holds, then Ģ‡š‘‰(š‘”)<0 and the bionomic equilibrium is globally asymptotically stable.

The globally asymptotically stability of the bionomic equilibrium means that harvesting effect šø which changes along (1.2) will make system (4.1) drive to the ā€œbionomic equilibriumā€ 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 š½ of a continuous time-stream:

ī€œš½=āˆž0š‘ƒī€·š‘„1,š‘„2ī€øš‘’,šø,š‘”āˆ’š›æš‘”š‘‘š‘”,(5.1) where š‘ƒ is the net revenue given by š‘ƒ(š‘„1,š‘„2,šø,š‘”)=š‘2š‘2šøš‘„2āˆ’š‘šø, and š›æ denotes the instantaneous annual rate of discount; the aim of this section is to maximize š½ subjected to state equation (1.1). Firstly we construct the following Hamiltonian function:

ī€·š‘š»=2š‘2š‘„2ī€øāˆ’š‘šøš‘’āˆ’š›æš‘”+šœ†1ī€·š‘Žš‘„2āˆ’š‘‘1š‘„1āˆ’š‘‘2š‘„21āˆ’š›½š‘„1āˆ’š‘Ÿ1š‘„31ī€ø+šœ†2ī€·š›½š‘„1āˆ’š‘1š‘„2āˆ’š‘2šøš‘„2ī€ø,(5.2) where šœ†1(š‘”),šœ†2(š‘”) are the adjoint variables, šø is the control variable satisfying the constraints 0ā‰¤šøā‰¤šømax, and šœ™(š‘”)=š‘’āˆ’š›æš‘”(š‘2š‘2š‘„2āˆ’š‘)āˆ’šœ†2š‘2š‘„2 is called the switching function [23]. We aim to find an optimal equilibrium (š‘„1š›æ,š‘„2š›æ,šøš›æ) to maximize Hamiltonian š»; since Hamiltonian š» is linear in the control variable šø, the optimal control can be the extreme controls or the singular controls; thus, we have

šø=šømax,whenšœ™(š‘”)>0,thatis,whenšœ†2(š‘”)š‘’š›æš‘”<š‘2āˆ’š‘š‘2š‘„2;šø=0,whenšœ™(š‘”)<0,thatis,whenšœ†2(š‘”)š‘’š›æš‘”>š‘2āˆ’š‘š‘2š‘„2.(5.3) When šœ™(š‘”)=0, that is,

šœ†2(š‘”)š‘’š›æš‘”=š‘2āˆ’š‘š‘2š‘„2,oršœ•š»šœ•šø=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 š». By Pontrayaginā€™s maximal principle, the adjoint equations are

š‘‘šœ†1š‘‘š‘”=āˆ’šœ•š»šœ•š‘„1=šœ†1ī€·š‘‘1+2š‘‘2š‘„1+š›½+3š‘Ÿ1š‘„21ī€øāˆ’šœ†2š›½,š‘‘šœ†2š‘‘š‘”=āˆ’šœ•š»šœ•š‘„2=āˆ’š‘2š‘2šøš‘’āˆ’š›æš‘”+šœ†2ī€·š‘1+š‘2šøī€øāˆ’šœ†1š‘Ž.(5.5) From (5.4) and (5.5), we have

š‘‘šœ†1š‘‘š‘”āˆ’šµšœ†1=š“š‘’āˆ’š›æš‘”,(5.6) where šµ=š‘‘1+2š‘‘2š‘„1+š›½+3š‘Ÿ1š‘„21,š“=š›½(š‘/š‘2š‘„2āˆ’š‘2). We can calculate that

šœ†1š“=āˆ’š‘’šµ+š›æāˆ’š›æš‘”.(5.7) Substituting (5.7) into the second equation of (5.5), we get

š‘‘šœ†2š‘‘š‘”āˆ’šŗšœ†2=š·š‘’āˆ’š›æš‘”,(5.8) where šŗ=š‘1+š‘2šø,š·=āˆ’š‘2š‘2šø+š“/(šµ+š›æ). Therefore, we have

šœ†2š·=āˆ’š‘’šŗ+š›æāˆ’š›æš‘”.(5.9) It is obviously that šœ†1(š‘”),šœ†2(š‘”) are bounded as š‘”ā†’āˆž.

Substituting (5.9) into (5.4), we obtain

š‘2āˆ’š‘š‘2š‘„2š·=āˆ’.šŗ+š›æ(5.10) Our purpose is to find an optimal equilibrium solution; so we have

š‘„1š›æ=š‘„āˆ—1=āˆ’š‘‘2+ī”š‘‘22+4š‘Ÿ1ī€·ī€·š‘š‘Žš›½/1+š‘2šøī€øāˆ’š‘‘1ī€øāˆ’š›½2š‘Ÿ1,š‘„2š›æ=š‘„āˆ—2=š›½š‘1+š‘2šøš‘„āˆ—1.(5.11) By (5.10) and (5.11), we can get š‘„1š›æ,š‘„2š›æ, andšøš›æ. Thus, the optimal policy is

āŽ§āŽŖāŽŖāŽŖāŽØāŽŖāŽŖāŽŖāŽ©šøšø=max,whenšœ†2(š‘”)š‘’š›æš‘”<š‘2āˆ’š‘š‘2š‘„2,šøš›æ,whenšœ†2(š‘”)š‘’š›æš‘”=š‘2āˆ’š‘š‘2š‘„2,0,whenšœ†2(š‘”)š‘’š›æš‘”>š‘2āˆ’š‘š‘2š‘„2.(5.12) Again, from (5.10) we have

ī€·š‘š‘ƒ=2š‘2š‘„2ī€øāˆ’š‘šø=āˆ’š·š‘2š‘„2šŗ+š›æšø.(5.13) When š›æā†’āˆž, š‘ƒāˆ¼š‘œ(š›æāˆ’1). Therefore, š›æ=0 leads to the maximization of š‘ƒ.

6. Number Simulations

In the following examples, we take the parameters values as š‘Ž=2,š‘‘1=0.1,š‘‘2=0.1,š‘2=0.2,š‘1=0.1,andš›½=0.2. We will see how the system behavior is while the toxicant effect changes.

Example 6.1. šø=1; in this case, š‘Žš›½=0.4>0.09=(š‘‘1+š›½)(š‘1+š‘2šø). From the results in Section 2, we know that for a given š‘Ÿ1, 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)š‘Ÿ1=0, šøāˆ—(10.33,6.89) is global stable;(ii)š‘Ÿ1=0.01, šøāˆ—(6.33,4.22) is global stable (Figure 2);(iii)š‘Ÿ1=1, šøāˆ—(0.97,0.65) is global stable (see Figure 3);(iv)š‘Ÿ1=100, šøāˆ—(0.01,0.07) is global stable (Figure 4).

Example 6.2. š‘˜=0.1,š‘2=2,š‘=0.2,š›æ=0.01, and šøā€²(š‘”)=0.1šø(0.4š‘„2āˆ’0.2). Considering system (4.1) with initial condition (2,3,3),(4,5,6), and (1,1,1), we have the following.(i)š‘Ÿ1=0; the bionomic equilibrium šø(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)š‘Ÿ1=1; the bionomic equilibrium šø(0.87,0.5,1.24) is globally stable (Figure 6). The optimal equilibrium is (1.26,1.28,0.49).(iii)š‘Ÿ1=10; the bionomic equilibrium šø(0.44,0.5,0.38) is globally stable (Figure 7). The optimal equilibrium is (0.51,0.74,0.18).(iv)š‘Ÿ1=100; both the bionomic equilibrium šø(0.2,0.5,āˆ’0.08) and the optimal equilibrium (0.20,0.44,āˆ’0.046) are unfeasible.

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 š‘„1āˆž and the harvesting effect šø decrease while the mature population š‘„2āˆž 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.


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