Abstract
A double delayed hybrid stochastic prey-predator bioeconomic system with Lévy jumps is established and analyzed, where commercial harvesting on prey and environmental stochasticity on population dynamics are considered. Two discrete time delays are utilized to represent the maturation delay of prey and gestation delay of predator, respectively. For a deterministic system, positivity of solutions and uniform persistence of system are discussed. Some sufficient conditions associated with double time delays are derived to discuss asymptotic stability of interior equilibrium. For a stochastic system, existence and uniqueness of a global positive solution are studied. By using the invariant measure theory and singular boundary theory of diffusion process, existence of stochastic Hopf bifurcation and stochastic stability are investigated. By constructing appropriate Lyapunov functions, asymptotic dynamic behavior of the proposed hybrid stochastic system with double time delays and Lévy jumps is discussed. Numerical simulations are provided to show consistency with theoretical analysis.
1. Introduction
It is well known that a harvest effort has a strong dynamical impact on the prey-predator system [1–5], which plays a significant role in bioeconomics management among various species in a harvested prey-predator system [6–8]. Furthermore, it is more realistic to investigate the coexistence and interaction mechanism of the harvested prey-predator system by introducing time delays into a model system, such as maturation delay and gestation delay of the population [9–12]. In recent years, combined dynamic effects of the harvest effort and time delay on the population dynamics of prey-predator systems have been widely investigated in [13–18] and references therein, where the asymptotic behavior of the model system around equilibrium is analyzed and stability of bifurcated periodic solutions is studied. Yuan et al. [18] incorporated gestation delay as a negative feedback of predator population density in the harvested prey-predator system, which is as follows: where and represent the population density of the prey and predator population, respectively. and stand for the birth rate of the prey and predator population, respectively. represents the intracompetition rate for the prey population, is the maximum value of the per capita reduction rate of the prey population due to predation, and has a similar interpretation to that of . measures the extent to which the surrounding environment provides protection to each population. denotes the catchability coefficient, is a constant parameter representing the harvest effort on the prey population, and represents the gestation delay of the predator population. By analyzing the dynamical behavior of interior equilibrium and properties of bifurcation phenomena, it reveals that sustainable development of the harvested prey-predator system may be guaranteed by adopting an appropriate harvest effort.
It should be noted that the dynamical behavior of the commercially harvested bioeconomic system can be precisely predicted by using stochastic mathematical models [19–23], which can provide an additional degree of realistic reflection in the real world compared to its corresponding deterministic counterpart. Many scholars have incorporated stochastic perturbations into deterministic mathematical models to discuss dynamic effects of environmental noises on population dynamics of the harvested bioeconomic system [19–21], which show that persistence and extinction of population are relevant to time delay and stochastic fluctuations. Combined dynamic effects of time delay and Gaussian white noises on population dynamics of the harvested bioeconomic system as well as optimal harvest control problems are studied in [22, 23]. Recently, it is proved that Lévy jumps can efficiently depict sudden and severe environmental perturbations arising in the real world [24, 25], while these phenomena cannot be described better by Brownian motion.
Based on the above analysis, some assumptions are proposed as follows.
Assumption 1. In this paper, we will extend the work in [18] by incorporating commercial harvesting on prey into system (1). represents the commercial harvesting effort on prey at time , represents the harvesting reward coefficients, represents the cost per unit harvesting effort for the unit weight of prey, and is the economic interest of commercial harvesting on prey. Based on the economic theory proposed in [26], an algebraic equation is constructed to study the economic interest of commercial harvesting: Based on system (1), TR and TC in (2), it is easy to show that and .
Assumption 2. In this paper, the maturity of the prey population is assumed to be mediated by discrete time delay . Furthermore, the reproduction of the predator population after predating the prey population is not instantaneous but will be mediated by some time lag required for gestation of the predator population. represents the gestation delay of the predator population. Hence, we will extend the work in [18] by incorporating two different discrete time delays into system (1), and .
Assumption 3. In this paper, the population growth of prey and predator populations affected by environmental stochastic fluctuations is assumed to be a stochastic process. Gaussian white noises and Lévy jumps will be incorporated into system (1) to describe stochastic surrounding environmental factors. are nonnegative constants, and denote multiplicative stochastic excitation and external stochastic excitation related to surrounding environment, respectively. represents independent Gaussian white noise such that . For any time, , , and , where denotes the Dirac delta function. represents the left limit of and . denotes a Poisson counting measure with characteristic measure on a measurable subset with and is assumed to be a Lévy measure such that and denotes a measurable subset of . Throughout this paper, and are assumed to be independent.
In this paper, keeping all these Assumptions 1–3 in mind, a double delayed hybrid stochastic prey-predator bioeconomic system with Lévy jumps is established as follows: where the initial conditions for system (3) take the following form:
System (3) can be rewritten in the matrix form as follows:
Remark 1. Since the algebraic equation in (3) includes no differentiated variables, the third row in matrix has a corresponding zero row.
Recently, some hybrid bioeconomic systems with time delay and stochastic fluctuations are established in [27–30] to investigate the combined dynamic effects of stochastic fluctuation and commercial harvesting on population dynamics. These proposed systems in [27–30] are also constructed by several differential equations with stochastic fluctuations and an algebraic equation. Compared with the previously bioeconomic systems proposed in [19–23, 31] and the references therein, such delayed bioeconomic systems [27–30] can not only discuss coexistence and interaction mechanism of a delayed bioeconomic system under stochastic environmental fluctuations but also investigate population dynamics due to variations of economic interest of commercial harvesting. However, biological characteristics among interacting populations are not considered in [27–29]; time delays such as gestation delay and maturation delay for interacting populations in [27–29] are assumed to be the same discrete value, which contradicts to reality in the real world. Asymptotical stability of interior equilibrium and dynamic effects of Lévy jumps on population dynamics are not discussed in [27–29]. It is proved that Lévy jumps can efficiently depict sudden and severe environmental perturbations arising in the real world [24, 25], while these phenomena can not be described better by Brownian motion. Although the dynamic effects of double time delays have been investigated in [30], the combined dynamic effects of multiple time delays and Lévy jumps on population dynamics have not been investigated in [30]. Dynamic effects of multiple time delays on the hybrid bioeconomic prey-predator system are investigated in [15, 32]. However, the dynamic effect of Lévy jumps and asymptotical stability of solutions of the stochastic system are not studied in [15, 32].
To the authors’ best knowledge, population dynamics of the hybrid bioeconomic system with double time delays and Lévy jumps have not been investigated. The rest sections of this paper are organized as follows: in the second section, qualitative analysis of the deterministic system is investigated. Positivity of solutions and uniform persistence of the deterministic system are studied. Asymptotic stability of interior equilibrium of the deterministic system is discussed due to variations of double time delays. In the third section, qualitative analysis of the stochastic system is investigated. Existence and uniqueness of global positive solution are studied. In the absence of double time delays and Lévy jumps, the existence of stochastic Hopf bifurcation and stochastic stability is investigated. In the presence of double time delays and Lévy jumps, by constructing appropriate Lyapunov functions, the asymptotic dynamic behavior of the proposed hybrid stochastic system with double time delays and Lévy jumps is discussed. In the fourth section, numerical simulations are provided to support theoretical findings. Finally, this paper ends with a conclusion.
2. Qualitative Analysis of Deterministic System
In the absence of stochastic fluctuations, positivity of solutions and uniform persistence of system (3) with initial conditions (4) will be studied in Lemmas 1 and 2.
Lemma 1. When and , all solutions of system (3) with initial conditions (4) are positive for all .
Proof 1. When and , it is easy to show that is locally Lipschitz and satisfy the condition, , where can be found in (5).
Due to lemma in [33] and Theorem A.4 in [34], all solutions of system (3) with initial conditions (4) exist uniquely and each component of the solution remains within the interval for some . Standard and simple arguments show that any solution of system (3) with initial conditions (4) always exist and stay positive.
Lemma 2. When and , if and are bounded and satisfy the following inequality: then all solutions of system (3) with initial conditions (4) are uniformly persistent.
Proof 2. When and , by using Taylor series expansion [35], for , , , and , it is easy to show that Hence, it follows that there exists such that
According to Lemma 1, (9), and the first equation of system (3), it follows from simple computations that which derives that holds for .
For , it follows from (10) and the first equation of system (3) that which gives that
If is bounded, then is bounded for .
Based on Lemma 1 and the second equation of system (3), there exists and it follows from simple computations that holds for . By using standard comparison arguments, it gives that
According to biological interpretations, (12) and the first equation of system (3), it derives that there exists and holds for . Furthermore, it follows from Lemma 1 that and
If is bounded, then is bounded for .
By using the first equation of system (3) and (14), it can be obtained that there exists such that holds for and
Furthermore, by using simple computations, it follows from (12) and (16) and the third algebraic equation of system (3) that
If , then it is easy to show that and is bounded.
If , then it is easy to show that and are bounded.
Based on the above analysis, if and are bounded and satisfy the following inequality then all solutions of system (3) with initial conditions (4) are uniformly persistent.
Remark 2. From the practical and physical perspective of viewpoints, positivity of the solution of the proposed hybrid delayed stochastic bioeconomic system refers to each population survival for a long duration under combined dynamic effects of commercial harvesting, time delay, and stochastic fluctuations. Since natural resources for each population survival are relatively limited within a closed environment, permanence of the hybrid delayed stochastic bioeconomic system interprets that there exist positive finite upper bounds and lower bounds for each population density, which may avoid overpopulation and extinction of each interacting populations. Furthermore, in order to maintain sustainable development of commercially harvested population, commercial harvesting should be constrained within a certain range. It practically interprets there exist positive finite upper bounds and lower bounds for commercial harvesting amount on the predator population.
When economic interest , , and , the interior equilibrium of system (3) can be obtained as follows: and satisfies the following equation: where are defined as follows:
According to Routh-Hurwitz criterion [35], a sufficient condition for (19) has at least one positive root which is . Furthermore, exists provided that and . Based on the above analysis, if , then there exists at least one positive root for (19), and is defined as follows:
In the following part, some sufficient conditions associated with double time delays are derived to investigate the local asymptotic stability of the system (3) around . By using the third algebraic equation , the system (3) can be rewritten as follows:
For mathematical convenience, some transformations and are made and system (22) is rewritten as follows:
By utilizing the following mathematical relations: system (23) can be rewritten as follows:
Some sufficient conditions for local asymptotical stability of system (3) around the interior equilibrium can be concluded as follows.
Theorem 1. When and , if , then system (3) is asymptotically stable around interior equilibrium , where is defined in (A.10).
Proof 3. The proof of Theorem 1 can be found in Appendix A of this paper.
3. Qualitative Analysis of Stochastic System
If , then it follows from the third algebraic equation of system (3) that . Consequently, system (3) is transformed as follows:
In the following part, existence and uniqueness of the global positive solution of system (26) are discussed in Theorem 2.
Theorem 2. If and are positive constants and is sufficiently small, then system (26) has a unique global positive solution for all and the solution for almost surely.
Proof 4. The proof of Theorem 2 can be found in Appendix B of this paper.
3.1. Case I: System (3) without Double Time Delays and Lévy Jumps
In this subsection, when and , it follows from simple computations that the linearized form of system (26) around is as follows: where , , , and .
By using Khasminskii transformations, , , and system (27) are rewritten as follows:
According to Khasminskii limit theorem [36, 37], if , , and are sufficiently small, then weakly converges to the two-dimensional Markov diffusion process. Based on the stochastic averaging method, the Itô stochastic differential equation can be obtained as follows: where and represent standard and independent Wiener processes,
Furthermore, some parameter transformations are provided as follows:
According to and (31), system (29) can be rewritten as follows:
It follows from the above analysis and that averaging amplitude refers to the one-dimensional Markov diffusion process, which derives that
Theorem 3. When and , stochastic stability of system (26) is discussed due to the variation of stability of averaging amplitude in terms of probability. (i) is unstable, and system (26) is unstable around in terms of probability.(ii)If , then is attractively natural and system (26) is unstable around in terms of probability; system (26) does not undergo stochastic Hopf bifurcation around .(iii)If , then is strictly natural.(iv)If , then is exclusively natural and system (26) is unstable around in terms of probability.
Proof 5. It follows from the formulation of that and , which derives that is a regular boundary of system (29) (reachable) and is unstable. Hence, system (26) is unstable around , which is not relevant to the local stability of the deterministic system around . On the other hand, if , then , which derives that is the second singular boundary of system (29). Based on the singular boundary theory [36, 37], it is easy to show that diffusion exponent , drifting exponent , and characteristic value of boundary are computed as follows:
If , then is attractively natural and system (26) is unstable around ; system (26) does not undergo stochastic Hopf bifurcation around .
If , then is strictly natural, which is just the line of stochastic stability demarcation.
If , then is exclusively natural and system (26) is unstable around in terms of probability.
In the following part, existence of stochastic Hopf bifurcation will be investigated as is exclusively natural. By using Itô stochastic equation of amplitude of , Fokker-Planck equation can be obtained as follows: with the initial value condition where denotes the transition probability density of the diffusion process .
By virtue of (35), it follows from simple computations that invariant measure of is the steady-state probability density , which is as follows: where and .
Theorem 4. When and , if and , then there exists a maximum value for . Occurrence positions and probabilities of stochastic Hopf bifurcation vary due to values of .
Proof 6. By using Namachivaya’s theory [36, 37], it is easy to show that characteristic information of a steady state can be revealed based on the invariant measure. When intension of Gaussian white noises tends to zero, the dynamical behavior of the corresponding deterministic system can be approximated by extreme values of . Based on Oseledec ergodic’s theory [36, 37], is considered to be the time measurement in the neighbourhood of , which derives that , , , and .
If there exists a maximum value of around , then it is easy to show that is asymptotically stable in terms of probability. On the other hand, if there exists a minimum value of (zero), then the opposite case also holds. It is assumed that there exists a maximum value as for ; further computations show that and solutions of the above equation are computed as follows: or as . Furthermore, it can be computed that
It is easy to show that when . Based on the singular boundary theory [36, 37], it shows that system (33) is unstable around in terms of probability. Hence, when and , system (26) undergoes stochastic Hopf bifurcation around , which derives that
Remark 3. From practical and physical perspective of viewpoints, if the proposed hybrid delayed stochastic bioeconomic system is unstable around the interior equilibrium in terms of probability, then it practically implies that the population density of the prey and predator population will not eventually reach an ecological balance state with a high probability, which is not beneficial to sustainable survival of each population under a commercially harvested ecological environment. If stochastic Hopf bifurcation does not occur and boundary is attractively natural, then it practically interprets that either the prey or the predator population density will sharply increase with a high probability within a short duration, which may result in the corresponding shortage of survival space and resource and beyond environment carrying capacity. The ecological balance state will be destroyed in a high probability.
3.2. Case II: System (3) with Double Time Delays and Lévy Jumps
It should be noted that is not the equilibrium of the stochastic system (26); we will show that solution of system (26) is going around under certain conditions.
Theorem 5. For , when and are positive constants and is sufficiently small, then where and are defined as follows:
Proof 7. The proof of Theorem 5 can be found in Appendix C of this paper.
4. Numerical Simulation
Numerical simulations are carried out to show combined dynamic effects of double time delays and Lévy jumps on population dynamics. Parameters are partially taken from numerical simulations in [18], , , , , , , , , , , and with appropriate units.
4.1. Numerical Simulation for Deterministic System
By utilizing the above parameter values, it follows from (21) that there exists an interior equilibrium when . In order to facilitate the following analysis, is arbitrarily selected within which is enough to merit the corresponding analysis in this paper, and an interior equilibrium can be obtained as . Based on Theorem 1 and (A.10), if and , then it is easy to show that interior equilibrium of system (22) is asymptotically stable. Based on the above analysis, some numerical simulations are supported. System (22) with and is locally asymptotically stable around , whose dynamical responses are plotted in Figure 1(a). System (22) with and is unstable around , whose dynamical responses are plotted in Figure 1(b). Furthermore, system (22) with and is locally asymptotically stable around , whose dynamical responses are plotted in Figure 2(a). System (22) with and is unstable around , whose dynamical responses are plotted in Figure 2(b).
(a)
(b)
(a)
(b)
4.2. Numerical Simulation for Stochastic System
In the absence of double time delays and Lévy jumps, by taking and as fixed values, it follows from simple computations that , , and , which derive and . Based on Theorems 3 and 4 of this paper, when , probabilities and occurrence positions of stochastic Hopf bifurcation of system (27) with and and different values of are computed in Table 1. Corresponding to the computation results in Table 1, the probability density curve and occurrence positions of stochastic Hopf bifurcation of system (27) due to variations of are plotted in Figure 3. According to (40), the corresponding phase portraits of probability densities with respect to and are plotted in Figure 4.
(a)
(b)
(c)
(d)
In the presence of double time delays and Lévy jumps, if and , then it follows from Theorem 5 that and , where and have been defined in Theorem 5. It follows from the mathematical form of and that and affect asymptotic behaviors of system (3) around interior equilibrium as well as fluctuation intensity of population densities of and . By assuming , , , , and , dynamical responses of system (3) with and and different values of and are obtained based on Euler-Maruyama method and plotted in Figure 5, where and . Similarly, by assuming , , , , and , dynamical responses of system (3) with and and different values of and are obtained based on Euler-Maruyama method and plotted in Figure 6, where and .
(a)
(b)
(a)
(b)
Based on the numerical simulations in Figures 1 and 2, it follows that local asymptotic stability of system (22) around interior equilibrium switches due to variations of time delays. System (22) shows unstable dynamical responses when time delays cross critical values, which can be computed based on Theorem 1. It follows from the mathematical formulation of and numerical computational results in Table 1 that the value of increases as intensities of external excitations and values increase. Furthermore, it follows from Figure 3 the occurrence positions where system (27) undergoes stochastic Hopf bifurcation switch higher as the intensities of external excitations and values increase. On the other hand, the corresponding stochastic Hopf bifurcation will decrease in a higher probability, which is observed in Figure 4. Based on the numerical simulations in Figures 5 and 6, it can be concluded that the magnitude of environmental external stochastic excitation and values and Lévy jump and values plays significant roles to determine the magnitude of oscillation of population dynamics. It follows from the detailed numerical comparison results between Figures 5 and 6 that oscillation magnitude of dynamical responses increases as Lévy jump and values increase for the fixed environmental external stochastic excitation and values. On the other hand, oscillation magnitudes of dynamical responses increase as external stochastic excitation and values increase for the fixed environmental Lévy jump and values.
5. Conclusion
In this paper, a double delayed hybrid stochastic bioeconomic system with commercial harvesting and Lévy jumps is established, which extends work done in [18] by incorporating double time delays and stochastic fluctuations. The dynamical model proposed in [18] is composed of ordinary differential equations, which are utilized to study the interaction mechanism of the prey-predator system with single time delay. Compared with the system established in [18], an algebraic equation is introduced into system (3), which concentrates on the dynamic effect of the economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to the variation of economic interests. Furthermore, two discrete time delays, which represent the maturation delay of the prey and gestation delay of the predator, are incorporated into system (3). Population growth of the prey and predator population affected by environmental stochastic fluctuations is assumed to be a stochastic process. Gaussian white noises and Lévy jumps are incorporated into system (3) to describe stochastic surrounding environmental factors. For the deterministic system, positivity of solutions and uniform persistence of system are discussed in Lemmas 1 and 2, respectively. Some sufficient conditions associated with double time delays are derived to discuss the asymptotic stability of interior equilibrium in Theorem 1. For the stochastic system, existence and uniqueness of global positive solution are studied in Theorem 2. By using the invariant measure theory and singular boundary theory of the diffusion process, the existence of stochastic Hopf bifurcation and stochastic stability is investigated in the absence of double time delays and Lévy jumps, which can be found in Theorems 3 and 4, respectively. By constructing appropriate Lyapunov functions, the asymptotic dynamic behavior of the proposed hybrid stochastic system with double time delays and Lévy jumps is discussed in Theorem 5.
From the practical perspective of viewpoints, positivity of the proposed hybrid stochastic double delayed bioeconomic system is relevant to the prey and predator population survival for a long duration under a commercially harvested environment. Generally speaking, the natural resources for the prey and predator population survival are under sever intraspecies competition. Therefore, permanence of the proposed hybrid stochastic double delayed bioeconomic system biologically means that there are certain positively finite upper constraints and positively finite lower constraints for the prey and predator population, which may, to a great extent, avoid population extinction and overpopulation. Furthermore, permanence of the proposed hybrid stochastic double delayed bioeconomic system biologically also means that there exist positively finite upper constraints and positively finite lower constraints for commercial harvesting amount on the prey population. The practical interpretations are introduced as follows: with the purpose of maintaining sustainable development of commercially harvested prey population resources, commercial harvesting should be regulated within certain harvesting range by formulating some constructive policies.
From the practical perspective of viewpoints, if the proposed hybrid stochastic double delayed bioeconomic system is unstable in terms of probability, then it practically interprets that the prey and predator population density will not eventually arrive at an ecological balance level with a high probability. The biological interpretations show that such phenomenon is not advantageous for a sustainable development of interacting populations under a commercial harvest effect. If the boundary is attractively natural and stochastic Hopf bifurcation does not occur, then it practically means that the interacting population densities may dramatically increase with a high probability during short time and it may reduce the corresponding survival resources and space, which are beyond the environment carrying capacity and destroy the ecological balance state with a high probability.
From the resource management of viewpoints, the numerical simulations reveal that population density may remain an ideal level by controlling double time delays within certain constraints and the critical values of time delays may increase as economic interests of commercial harvesting increase. The sustainable development of commercially harvested population can be indirectly achieved by formulating relevant policy to regulate an economic interest within some appropriate ranges.
Recently, some hybrid bioeconomic systems with time delay and stochastic fluctuations are established in [27–30] to investigate the combined dynamic effects of stochastic fluctuation and commercial harvesting on population dynamics. These proposed systems in [27–30] are also constructed by several differential equations with stochastic fluctuations and an algebraic equation. Compared with the previously bioeconomic systems proposed in [19–23, 31] and the references therein, such delayed bioeconomic systems [27–30] can not only discuss coexistence and interaction mechanism of delayed bioeconomic system under stochastic environmental fluctuations but also investigate population dynamics due to variations of the economic interest of commercial harvesting. However, biological characteristics among interacting populations are not considered in [27–29]; time delays such as gestation delay and maturation delay for interacting populations in [27–29] are assumed to be the same discrete value, which contradicts to the reality in the real world. Asymptotical stability of interior equilibrium and dynamic effects of Lévy jumps on population dynamics are not discussed in [27–29]. It is proved that Lévy noise can efficiently depict sudden and severe environmental perturbations arising in the real world [24, 25], while these phenomena can not be described better by Brownian motion. Although the dynamic effects of double time delays have been investigated in [30], the combined dynamic effects of multiple time delays and Lévy jumps on population dynamics have not been investigated in [30]. Dynamic effects of multiple time delays on the hybrid bioeconomic prey-predator system are investigated in [15, 32]. However, the dynamic effect of Lévy jumps and asymptotical stability of solutions of the stochastic system are not studied in [15, 32]. To the authors’ best knowledge, population dynamics of the hybrid bioeconomic system with double time delays and Lévy jumps have not been investigated. Compared with the related work, we can investigate combined dynamic effects of double time delays and Lévy jumps on population dynamics by analyzing stability analysis and stochastic dynamical behavior of system (3) in this paper; these analytical findings make this paper have some positive and new features.
Appendix
A. Proof of Theorem 1
Proof 8. Let , by using Lemma 2 and computing the upper right derivative of along the solution of system (24), it can be obtained that
where have been defined in Lemma 2.
Furthermore, let
where .
By virtue of (A.1) and computing the upper right derivative of along the solution of system (24), it can be obtained that
Let , by using Lemma 2 and computing the upper right derivative of along the solution of system (24), it can be obtained that
where have been defined in Lemma 2.
Furthermore, let
By virtue of (A.1) and computing the upper right derivative of along the solution of system (24), it can be obtained that
By defining , it is easy to show that . By virtue of (A.3), (A.6), and simple computations, it can be obtained that
where are defined as follows:
where have been defined in Lemma 2 and has been defined in (A.3).
By using the mean value theorem [35], for , , and it can be obtained that
If , then it is easy to derive that and is as follows:
where satisfies the following inequalities and hold.
B. Proof of Theorem 2
Proof 9. For system (25), it is easy to show that Lipschitz conditions hold. Hence, there exists a unique local solution for , where and represents the explosion time [38].
Subsequently, we will show that , which implies that solution is global, by assuming that is sufficiently large such that and for . With the purpose of facilitating the following analysis, the stopping time [38] for any is defined as follows:
Let represent an empty set and . It is easy to show that increases as increases through , by defining , which derives that almost surely. Hence, if almost surely, then we can show and the solution for all almost surely.
If , then there exists a pair of constants and such that . Hence, there exists some , and holds for all .
Define a -function as follows:
By assuming that are sufficiently small, by utilizing Lemma 2 of this paper and Itô’s formula (5), it can be obtained that
In order to facilitate the following analysis, is defined as follows:
where are defined in Lemma 2.
If and are positive constants, then
Consequently, it follows from (B.5) that
By integrating both sides of (B.6) from to , has been defined in (25) and then expectations can be computed as follows:
which follows that
When , we define ; it is easy to derive that based on the fact holds for all . Furthermore, it can be obtained that or equals to either or which holds for any , which follows that is no less than either or .
Consequently, it derives that
It follows from (B.8) that
where represents indicator function of . When , it derives that
which is a contradiction.
Based on the above analysis, it can be obtained that and will not explode in a finite time almost surely.
C. Proof of Theorem 5
Proof 10. Firstly, we construct the following function:
By using simple computations, it can be obtained that
where satisfies the following inequalities:
Based on (C.3), is defined as follows,
By using simple computations, it can be obtained that
Secondly, we construct the following function:
By using similar arguments in (C.3) and (C.5), it follows from simple computations that
Let ; it follows from (C.5) and (C.7) that
If and are positive constants and is sufficiently small, then
where are defined in Lemma 2 of this paper.
By integrating both sides of (C.9) from to and deriving expectation, it is easy to show that
It follows from (C.10) that
where is defined as follows:
Thirdly, we construct the following function:
By using simple computations, it can be obtained that
where satisfies the following inequalities:
If and are positive constants and are sufficiently small, then
Based on (C.16), is defined as follows,
By using simple computations, it can be obtained that
where have been defined in Lemma 2.
Fourthly, we construct the following function
By using similar arguments in (C.16) and (C.18), it follows from simple computations that
If and are positive constants and is sufficiently small, then
where have been defined in Lemma 2.
According to (C.21), is defined as follows:
It follows from simple computations that
Let ; it follows from (C.18) and (C.23) that
By integrating both sides of (C.24) from to and deriving expectation, it gives that
It follows from (C.25) that
where is defined as follows:
Conflicts of Interest
All authors of this article declare that there is no conflict of interests regarding the publication of this article. They 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 review of this article.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 61673099), Hebei Province Natural Science Foundation (Grant no. F2015501047), and Fundamental Research Funds for the Central Universities of Education Ministry (Grant no. N162304006).