Abstract
This paper presents a singular prey-predator fishery model, where maturation delay for prey and gestation delay for predator are considered. Fishing efforts are introduced to harvest prey and predator population, which are developed as control instruments to investigate optimal utilization of fishery resource. By analyzing associated characteristic equation, local stability analysis is studied due to combined variations of double time delays. Furthermore, Pontryagin’s maximum principle is utilized to characterize optimal harvest control, and the optimality system is numerically solved based on an iterative method.
1. Introduction
According to statistics from the Food and Agriculture Organization of United Nations [1], approximate fifty-three percent of fish stock under observation has experienced overexploitation or depletion, which reiterate the fact that fishery needs to be managed with an effective and carefully defined objective to prevent overexploitation and replenish depleted stock [1]. In order to ensure sustainable fishery, it requires precision in stock assessment and reliability in fishery modelling [2, 3]. In recent decades, there are growing research interests in formulating bioeconomic mathematical models to obtain more comprehensive indications of feedback effect between exploitation activity and fishery resources [4–13].
In [5], authors establish a dynamic model where both prey and predator fishery resource population are exploitable, which is as follows:where and represent population density of prey and predator population with time , respectively. is the intrinsic growth rate of prey, stands for maximum environmental capacity, denotes predation rate of predator, represents biomass transmission rate from predation activity, and is death rate of predator.
It is well known that commercial harvesting fluctuates with dynamical variation of economic interest [14, 15]. Although the harvested biological models have received great attention from both mathematical and theoretical biologists, little work has been done on dynamic effect of economic interest on harvested prey-predator fishery model [14, 15]. In 1954, Gordon proposes common-property resource economic theory [14], which investigates dynamic effect of commercial harvesting on ecosystem from an economic perspective [14, 15]. In [14], an algebraic equation is proposed to investigate economic interest of commercial harvesting:
In order to show dynamical variation of unit price of harvested population due to change of amount of harvested population, is assumed to be unit price of commercially harvested prey, and is assumed to be unit price of commercially harvested predator. Hence, TR in (2) takes the following form , where are all positive constants. Let represent cost of commercial harvesting. Hence, TC in (2) takes the following form .
Recently, many theoreticians and experimentalists have investigated complex dynamics of prey-predator fishery system [16, 17]; it reveals that time delay may cause the loss of stability and other complicated dynamical behavior such as the periodic structure and bifurcation phenomenon [15, 18, 19].
Keeping these aspects in view, we will extend work in [5] by incorporating maturation delay for prey and gestation delay for predator into system (1). In this paper, prey is considered to be delayed by maturation delay due to population crowding [15, 16], and predator is assumed to be delayed by gestation delay [16]. Based on (1) and (2), a singular prey-predator fishery model is established as follows:where represents cost of commercial harvesting, is assumed to be economic interest of commercial harvesting, represents commercial harvesting amount with respect to time , and interpretations for other parameters and state variables share the same interpretations introduced in system (1) and (2). Furthermore, system (3) is investigated with initial conditions
System (3) is rewritten as follows:where the third equation in system (3) does not contain any differentiated variables; hence the third row in leading matrix has a zero row.
The rest of the sections of this paper are organized as follows. Positivity of any solution of system (3) and uniform persistence of system (3) are investigated in the second section. Local stability analysis of system (3) around interior equilibrium is discussed in the third section. The optimal system is derived and solved numerically based on an iterative method with Runge-Kutta fourth order scheme. Optimal harvest control problems associated with maximizing total discounted net revenues from the fishery and minimizing harvest cost are discussed in the fourth section. Numerical simulations are carried out to support theoretical analysis in the fifth section. Finally, this paper ends with a conclusion.
2. Positivity and Uniform Persistence
Theorem 1. Any solutions of system (3) with initial conditions (4) are positive.
Proof. Due to lemma in [20] and Theorem A.4 in [21], any solution of system (3) with initial conditions (4) exists uniquely and each component of solution remains within interval for some . Standard and simple arguments show that any solution of system (3) always exists and stays positive.
Theorem 2. If , and are bounded, and the following inequalities (6) hold, and then system (3) with initial conditions (4) is uniformly persistent:
Proof. By using Taylor series expansion [2], for and we haveConsequently, it can be derived thatBy using (8) and the first equation of system (3), it gives that which derives thatIf is bounded, then it is easy to show that and is bounded. According to (10), it can be derived that there exists such that holds for .
Based on the first equation of system (3) and practical interpretations of prey survival, it can be derived that there exists such that , andSimilarly, based on the second equation of system (3) and practical interpretations of predator survival, it is shown that there exists such thatBy using (8), if is bounded then it follows that , andIf , then it can be derived that and is bounded. According to (13), it is shown that there exists such that holds for .
By virtue of the third equation of system (3), when , it can be derived that there exists such that which implies thatIf , then it is easy to show that and is bounded, and then there exists such that holds for .
By virtue of the third equation of system (3), when , it can be derived that there exists such that According to (10), (11), and (13), it can be obtained that If the following two inequalities hold then and and is bounded, and there exists such that and hold for .
Based on the above analysis, it can be concluded that system (3) with initial conditions (4) is uniformly persistent.
3. Local Stability Analysis
Firstly, interior equilibrium of system (3) in the case of is computed as follows: , and satisfy the following equation:where () are defined as follows:
Furthermore, it follows from the practical interpretations of interior equilibrium that exists provided that and , which derives that
It follows from Routh-Hurwitz criterion [21], simple sufficient conditions for existence of of system (3) are as follows:
Based on Jacobian matrix of system (3) evaluated around and defined in (5), characteristic equation of system (3) around is as follows:where , , are defined as follows:
3.1. Case I: ,
When and , (23) can be rewritten as follows:
By substituting () into (25) and separating real and imaginary parts, it gives thatwhich derives that
According to Routh-Hurwitz criterion [21], a simple sufficient condition guarantees existence of positive root of (27). Hence, (25) has a pair of purely imaginary roots of the form .
By eliminating from (26), it can be calculated that corresponding to is as follows:where .
It follows from Butler’s lemma [22] that system (3) is locally stable around when .
Based on the above analysis, local stability analysis of system (3) around is concluded in Theorem 3.
Theorem 3. If , then system (3) is locally stable around when , and is defined in (28).
3.2. Case II: ,
When , , by using similar arguments in Section 3.1, local stability analysis of system (3) around can be concluded in the following Theorem 4.
Theorem 4. If , then system (3) is locally stable around when , , and is defined as follows:where and is a positive root of the following equation:
3.3. Case III: , ,
In this subsection, is considered to be bifurcation parameter and is regarded as fixed value , where is determined in Theorem 4.
Let () be the root of (23), and two transcendental equations are obtained as follows:where , , are defined in (23).
By eliminating from (31), it can be derived thatwhere , , are defined as follows:
Based on , , defined in (32), we define
It is difficult to discuss root properties of transcendental equation (34) due to its complicated form. Without investigating the properties of roots of (34) in detail, it follows from simple computations that (23) has a pair of purely imaginary roots when (34) has at least two roots .
By denoting and which is considered to be bifurcation parameter, we can obtain the corresponding critical value :where satisfies the following equations:
Based on the above analysis, local stability analysis around due to dynamical variation of bifurcation parameter can be summarized as follows.
Theorem 5. If (34) has at least two roots , and there exists defined in (35), then system (3) is locally stable around when .
3.4. Case IV: , ,
In this subsection, is considered to bifurcation parameter and is regarded as fixed value , where is determined in Theorem 3. By using similar arguments in Theorem 5 of this paper, local stability analysis around can be concluded in Theorem 6.
Firstly, we define following equation:where , , are defined as follows:
Theorem 6. If (37) has at least two roots , there exists a critical delay , and then system (3) is locally stable around when , where and satisfies the following equations:
4. Optimal Control Problem
In this section, and are considered to be a fixed value and , respectively; , , and have been determined in Theorems 5 and 4, respectively. It follows from Theorem 5 that system (3) with and is locally stable around , and we will investigate optimal control with maximizing total discounted net revenues and minimizing commercial harvesting cost during . The optimal harvest control problem is formulated as follows:where is instantaneous annual discount rate [7], and optimal control problem (40) is subject to the first and second delayed differential equations of system (3) with and . Furthermore, convexity of with respect to , linearity of the delayed differential equations with , and range values compactness of state variables can be uniformly investigated when discussing existence of optimal harvest control . By assuming as an optimal harvest control with corresponding optimal state variables and of system (3) with and , and satisfies the following condition:where represents control set that is defined as follows:
The Hamiltonian function associated with system (3) with and iswhere , , and are adjoint functions.
Theorem 7. There exists an optimal harvest which is determined by (44); furthermore, there exist adjoint functions satisfying system (47) with .
Proof. By differentiating , the adjoint system can be obtained as follows:whereBy using the optimal conditions, (45) can be rewritten as follows:In optimal control problem (40), there is not terminal cost and final state is free. Consequently, transversality conditions (bounded conditions) for adjoint functions are as follows: .
It is easy to obtain characterization of optimal harvest control , which derives that . Consequently, it follows from simple computations that there exists an optimal harvest determined bywhere satisfies system (47) with .
Remark 8. It follows from similar arguments in Theorem 7 that symmetric optimal control problem on cases in Section 3.4 can be also obtained, which are omitted in this paper. and can be set to be a fixed value and , respectively; , , and are determined in Theorems 3 and 6, respectively.
In the following part, the optimal system is numerically solved based on an iterative method with Runge-Kutta fourth order scheme. Firstly, by assuming that there exist a step size and integers with , and . Secondly, we initialize optimal harvest control with a given initial value . By using forward difference approximation, we solve the first and second differential equation of system (3) with initial conditions , . Thirdly, by using backward difference approximation, we solve the adjoint functions (47) with transversality conditions. Finally, the optimal harvest control is updated by values of the state and adjoint variables, and the updated optimal harvest control is replaced with the initial value given in the second step. This process will be repeated until successive iterates of harvest control values become sufficiently close.
By utilizing combinations of forward and backward difference approximations, it can be derived that for
In order to show effectiveness of optimal harvest control designed in Theorem 7, parameters values of system (3) are partially from [5], , , , , , , , , , , , , , and with appropriate units. By using given values of parameters and simple computations, interior equilibrium exists provided that . In the following part, is utilized in the numerical simulations, which is arbitrarily selected within and is enough to merit theoretical analysis obtained in this paper. It follows from Theorem 3, Theorem 7, and simple computations that system (3) is locally stable around when and . By using similar arguments and Theorems 4 and 7, it is revealed that system (3) is locally stable around when and . As analyzed in Section 3.3, is arbitrarily selected within , that is, , and it follows from Theorems 5 and 7 that system (3) is locally stable around when and . As analyzed in Section 3.4, is arbitrarily selected within , that is, , and it follows from Theorems 6 and 7 that system (3) is locally stable around when and . Numerical simulations are made to investigate dynamic effects of double time delays on variation of optimal prey, predator biomass, and optimal harvest control, which can be found in Figures 1 and 2, respectively. The initial values of simulation work made in Figure 1 are as follows: , and initial values of simulation work made in Figure 2 are as follows: .
5. Conclusion
By incorporating maturation delay for prey population and gestation delay for predator population, we extend work done in [5]. In this paper, a singular prey-predator fishery model with double time delays is established. Fishing efforts are introduced to commercially harvest prey population and predator population, which are developed as control instruments to investigate optimal utilization of prey-predator fishery resource. Positivity of solutions and uniform persistence of system are discussed in Theorems 1 and 2. By analyzing associated characteristic transcendental equation, local stability around interior equilibrium is discussed due to combined variations of double time delays, which can be found in Theorem 3 to Theorem 6. Furthermore, an optimal harvest control is designed in Theorem 7. By using the proposed mathematical model and dynamical analysis, authors aim to obtain some results which are theoretically beneficial to discussing optimal harvesting strategies and sustainability mechanism of harvested prey-predator fishery system with double time delays. Furthermore, the theoretical results may be potentially constructive for administrative agencies to formulate regulatory policies to preserve economic optimality through ensuring global sustainability.
Competing Interests
All authors of this article declare that there is no conflict of interests regarding the publication of this article. We have no proprietary, financial, professional, or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this article.
Acknowledgments
This work is supported by National Natural Science Foundation of China, Grant No. 61104003 and Grant No. 61673099, Research Program for Liaoning Excellent Talents in University, Grant No. LJQ2014027, and Hebei Province Natural Science Foundation, Grant No. F2015501047. This work is supported by Hong Kong Admission Scheme for Mainland Talents and Professionals, Hong Kong Special Administrative Region.