Elsevier

Applied Mathematics and Computation

Volume 316, 1 January 2018, Pages 115-137
Applied Mathematics and Computation

Dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations

https://doi.org/10.1016/j.amc.2017.08.019Get rights and content

Highlights

  • A hybrid stochastic bioeconomic system with double time delays is proposed.

  • Existence and properties of singularity induced bifurcation are investigated.

  • Existence and uniqueness of the global positive solution are studied.

  • Existence of stochastic Hopf bifurcation and stochastic stability are discussed.

  • Dynamic analysis of solutions of the hybrid stochastic system are investigated.

Abstract

In this paper, we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton, where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Stochastic fluctuations are incorporated into the proposed system in form of Gaussian white noises to depict stochastic environmental factors in plankton system. For deterministic system without double time delays, existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting, and state feedback controllers are designed to eliminate singularity induced bifurcation. For deterministic system with double time delays, positivity and uniform persistence of solutions are studied, and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated. For stochastic system without double time delays, stochastic stability and existence of stochastic Hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory. For stochastic system with double time delays, existence and uniqueness of global positive solution are investigated, and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate Lyapunov functions. Numerical simulations are carried out to validate theoretical analysis.

Introduction

In marine ecosystem, some specific phytoplankton population release toxin substances and reduce predation pressure from zooplankton population in plankton system [2], [6], [17]. Furthermore, some practical observations from real marine ecosystem reveal that liberation process of toxin substances is not instantaneous but delayed by certain maturation duration of toxin producing phytoplankton in marine ecosystem [2], [3], [6], [11]. In recent years, some mathematical models composed of delayed differential equations have been utilized to discuss complex dynamics and stability analysis of plankton system with toxin producing phytoplankton [22], [23], [24], [25], [27], [28], [31], [32], [38], [46]. The oscillatory situations and Hopf bifurcation phenomenon around interior equilibrium due to variation of time delay are discussed in [22], [23], [24], [32], [38]. The destabilizing effect of time delay and some sufficient conditions associated with persistence of the interior equilibrium are studied in [23], [25], [27], [28], [46]. Based on field-collected samples, a phytoplankton zooplankton system with toxin liberation delay τ > 0 is established in [31] to reflect population dynamics depending on maturation delay of toxin producing phytoplankton, which is as follows, {P˙(t)=rP(t)(1P(t)K)βP(t)Z(t)α+P(t),Z˙(t)=β1P(t)Z(t)α+P(t)dZ(t)ρP(tτ)Z(t),where P(t) and Z(t) denotes toxin producing phytoplankton population density and zooplankton population density, respectively. r represents intrinsic growth rate of toxin producing phytoplankton, d stands for natural decay rate of zooplankton, K represents environmental carrying capacity of toxin producing phytoplankton. β is supreme consumption rate of zooplankton, β1 denotes biomass conversion rate such that 0 < β1 < β. ρ represents toxin substances liberation rate. Supposing that rate of toxin substance liberation is less than rate of biomass consumed by zooplankton, it is assumed that ρ < β1. Complex dynamical behavior and local stability analysis around the interior equilibrium are discussed in [31].

It is well known that some zooplankton such as aretes, jellyfish and krill may be commercially exploited from the view points of human medical and physical need [13], [16], [18], [35], [40]. In recent years, dynamic effects of harvest effort on zooplankton population have drawn a great deal of attentions [26], [30], [36], [39], which show that an appropriate commercial harvest effort may also guarantee sustainable development of harvested zooplankton population in marine system. From the perspective of economy theory viewpoint, it should be noted that commercial harvesting fluctuates due to variation of economic interest of commercial harvesting [13]. However, little research work has been made on dynamic effects of economic interest of commercial harvesting on population dynamics. Gordon [1] proposed common-resource properties theory from an economic viewpoint, which is as follows: NetEconomicRevenue=TotalRevenue(TR)TotalCost(TC).

Furthermore, uncertainty in population growth is usually regarded as one of important themes in plankton system. Especially, some stochastic fluctuations are considered to be dynamic effect of environmental stochasticity, which refers to temperature and nutrient density within surrounding marine environment show oscillation around specific average state in plankton system [12]. Hence, population dynamics of plankton system can not be totally depicted and investigated by utilizing deterministic mathematical model [12]. Recently, Gaussian white noises are proved to efficiently depict rapidly fluctuating phenomena arising from marine ecosystem in the real world [5], [19].

Based on the above analysis, some hypotheses are proposed as follows:

  • (H1)

    In this paper, it is assumed that zooplankton is commercially harvested, E(t) represents commercial harvesting on zooplankton, w represents unit price of harvested zooplankton, c and v denotes cost and economic interest of commercial harvesting, respectively. Hence, it follows from Eq. (2) that {TR=wE(t)Z(t),TC=cE(t),v=E(t)(wZ(t)c).

  • (H2)

    In this paper, it is assumed that the liberation of toxin substances is not instantaneous but mediated by some time lag which is required for toxin producing phytoplankton maturation. The reproduction of zooplankton after predating toxin producing phytoplankton is not instantaneous but will be mediated by some time lag required for zooplankton gestation. We will extend the work in [31] by incorporating discrete time delay for gestation of zooplankton population into system (1). τ1 ≥ 0 denotes gestation delay for zooplankton, and τ2 ≥ 0 represents maturation delay for toxin producing phytoplankton.

  • (H3)

    In this paper, population growth of toxin producing phytoplankton and zooplankton affected by environmental stochasticity are all assumed to be stochastic process rather than a deterministic process. Gaussian white noises will be introduced into the proposed system to depict stochastic environmental factors in plankton system.

By using hypotheses (H1)-(H3), a double delayed bioeconomic phytoplankton zooplankton system with stochastic fluctuations is as follows, {P˙(t)=rP(t)(1P(t)K)βP(t)Z(t)α+P(t)+ω11P(t)ξ1(t)+ω12ξ2(t),Z˙(t)=β1P(tτ1)Z(tτ1)α+P(tτ1)dZ(t)ρP(tτ2)Z(t)E(t)Z(t)+ω21Z(t)ξ1(t)+ω22ξ2(t),0=E(t)(wZ(t)c)v,where ωjk (j,k=1,2) are nonnegative constants, ξ1(t) and ξ2(t) represents multiplicative stochastic excitation and external stochastic excitation related to marine environment, respectively. ξj(t) (j=1,2) denotes independent Gaussian white noise such that E[ξj(t)]=0 (j=1,2). For discrete time t1 ≠ t2, E[ξj(t1)ξj(t2)]=δ(t2t1) (j=1,2), and E[ξj(t1)ξk(t2)]=0 (j,k=1,2,jk), where δ denotes Dirac delta function.

Other parameters share the same interpretations mentioned in (1) and hypotheses (H1)–(H3). The initial conditions for system (4) are as follows, P(θ)>0,Z(θ)>0,E(0)>0,θ[τm,0],where τm=max{τ1,τ2}.

Recently, some delayed bioeconomic systems with stochastic fluctuations are constructed in [33], [37], [43] to discuss combined dynamic effects of economic interest and stochastic fluctuations on population dynamics. The proposed systems in [33], [37], [43] are constructed by several differential equations with stochastic fluctuations and an algebraic equation, which is established based on Eq. (2). Compared with the previously related harvested plankton systems proposed in [26], [30], [35], [36], [40] and the references therein, advantages of delayed bioeconomic systems proposed in [33], [37], [43] are that they can not only discuss coexistence mechanism of commercially harvested marine system under stochastic fluctuations, but also provide a straightforward way to investigate complex dynamics with the dynamic changes of economic interest of commercial harvesting. However, time delays such as gestation delay and maturation delay for interacting population in [33], [37], [43] are assumed to be the same discrete value. Biological characteristics among interacting population are not considered in [33], [37], [43], which contradicts to the practical observations in the real world. Furthermore, asymptotical stability of interior equilibrium is not discussed in [33], [37], [43]. Dynamic effects of time delay and stochastic fluctuations on epidemic system with single time delay are investigated in [42], [45], while dynamic effects of economic interest of commercial harvesting and double time delays on population dynamics are not studied in [42], [45]. Dynamic effects of multiple time delays on hybrid bioeconomic prey predator system are investigated in [29], [34], [44], and combined dynamic effects of time delay and diffusion term on hybrid bioeconomic prey predator system are investigated in [39], [41]. However, dynamic effect of stochastic fluctuation and asymptotical stability of solutions of stochastic system are not studied in [29], [34], [39], [41], [44].

To author’s best knowledge, combined dynamic effects of double time delays and stochastic fluctuations on bioeconomic plankton system have not been investigated in present related research. In the second section of this paper, complex dynamical behavior and stability analysis of deterministic form of system (4) are discussed. In absence of double time delays, existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting. Three state feedback controllers are designed to eliminate singularity induced bifurcation and stabilize the controlled deterministic system around corresponding interior equilibrium. In presence of double time delays, positivity and uniform persistence of solutions are studied, sufficient conditions for asymptotic stability of interior equilibrium are investigated. In the third section of this paper, complex dynamical behavior and stability analysis of system (4) are discussed. In absence of double time delays, stochastic stability and existence of stochastic Hopf bifurcation are investigated based on singular boundary theory of diffusion process and invariant measure theory. In presence of double time delays, existence and uniqueness of global positive solution are investigated, and asymptotic behaviors of the interior equilibrium are studied by constructing some appropriate Lyapunov functions. In the fourth section of this paper, numerical simulations are carried out to show consistency with theoretical analysis. Finally, this paper ends with a discussion.

Section snippets

Qualitative analysis of deterministic system

Firstly, positivity of solutions and uniform persistence of system (4) without stochastic fluctuations are investigated with initial conditions (5), which can be found in the following Lemmas 2.1 and 2.2.

Lemma 2.1

Whenωjk=0(j,k=1,2), any solutions of system (4) with initial conditions (5) are positive.

Proof

Based on fundamental theory of functional differential equations [9], when ωjk=0(j,k=1,2), system (4) has a unique solution (P(t), Z(t), E(t)) satisfying initial conditions (5).

By using standard arguments

Qualitative analysis of stochastic system

If (τ1, τ2, v) ∈ R2 (R2 has been defined in Theorem 2.6 of this paper), then it follows from implicit function theorem [4] and the third equation of system (4) that E(t)=vwZ(t)c. Hence, system (4) is transformed as follows: {P˙(t)=rP(t)(1P(t)K)βP(t)Z(t)α+P(t)+ω11P(t)ξ1(t)+ω12ξ2(t),Z˙(t)=β1P(tτ1)Z(tτ1)α+P(tτ1)dZ(t)ρP(tτ2)Z(t)vZ(t)wZ(t)c+ω21Z(t)ξ1(t)+ω22ξ2(t).

Existence and uniqueness of global positive solution of system (39) with initial conditions (5) will be discussed in the

Numerical simulation

Numerical simulations are carried out to support theoretical findings in this paper. Parameters values of system (4) are partially from [31], r=6,K=0.8,α=1.4,β=0.9,β1=0.6,d=0.065,ρ=0.06,w=0.5 and c=1 with appropriate unit.

Discussion

It is well known that plenty of phytoplankton population produce toxin substances for reducing predation pressure from zooplankton population, such as Amphidinium carterae, Chrysochromulina polyepis, Dinophysis sp., Gambierdiscus toxicus and Pseudonitzschia sp. [15]. In the plankton system, the liberation of toxin substances is not an instantaneous process but mediated by some time lag which is required for toxin producing phytoplankton maturation. The reproduction of zooplankton after

Acknowledgements

Authors would like to express their gratitude to editor and anonymous reviewers for their valuable comments and suggestions. Without the expert comments made by editor and anonymous reviewers, this paper would not be of this quality. This work is supported by National Natural Science Foundation of China, grant nos. 61673099 and 61104003. Research Program for Liaoning Excellent Talents in University, grant no. LJQ2014027. Hebei Natural Science Foundation, grant no. F2015501047. Fundamental

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