Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Liu, Chao; Lü, Na; Zhang, Qingling; Li, Jinna; Liu, Peiyong

doi:10.1016/j.amc.2016.01.039

Cited by 13 publications

(10 citation statements)

References 43 publications

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“…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…”

Section: Complexitymentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Modeling and Dynamic Analysis in a Hybrid Stochastic Bioeconomic System with Double Time Delays and Lévy Jumps

Liu

Wang

2018

Complexity

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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.

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Section: Complexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modeling and Dynamic Analysis in a Hybrid Stochastic Bioeconomic System with Double Time Delays and Lévy Jumps

Liu

Wang

2018

Complexity

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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Optimal Harvest Control in a Singular Prey-Predator Fishery Model with Maturation Delay and Gestation Delay

Liu¹,

Yu²,

Zhang³

2016

Discrete Dynamics in Nature and Society

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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 ( ) ( )/( 1 + ( ) ( )) + 2 ( ) ( )/( 2 + ( ) ( )), where 1 , 2 , 1 , 2 are all positive constants. Let represent cost of Hindawi Publishing Corporation

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“…According to common property resource economic theory in [14,24], when v = 0, there is a phenomenon of equilibrium state. An interior equilibrium is as follows:…”

Section: Local Stability Analysismentioning

confidence: 99%

Modelling and bifurcation analysis in a hybrid bioeconomic system with gestation delay and additive Allee effect

et al. 2018

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In this paper, we investigate a delayed differential algebraic prey-predator system, where commercial harvesting on predator and additive Allee effect on prey are considered. A discrete time delay is utilized to represent gestation delay of the predator population. Positivity of solutions and uniform persistence of system are discussed. In the absence of time delay, by taking economic interest as a bifurcation parameter, some sufficient conditions associated with additive Allee effect and economic interest are derived to show that the proposed system undergoes singularity-induced bifurcation around the interior equilibrium. In the presence of time delay, combined dynamic effects of time delay and additive Allee effect on population dynamics are discussed in the case of positive economic interest of commercial harvesting. Existence of Hopf bifurcation and local stability switch around the interior equilibrium are studied as gestation delay crosses the critical value. Furthermore, properties of Hopf bifurcation are investigated based on the center manifold theorem and the norm form of a delayed singular system. Existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. Numerical simulations are provided to show consistency with theoretical analysis.

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Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Cited by 13 publications

References 43 publications

Modeling and Dynamic Analysis in a Hybrid Stochastic Bioeconomic System with Double Time Delays and Lévy Jumps

Modeling and Dynamic Analysis in a Hybrid Stochastic Bioeconomic System with Double Time Delays and Lévy Jumps

Optimal Harvest Control in a Singular Prey-Predator Fishery Model with Maturation Delay and Gestation Delay

Modelling and bifurcation analysis in a hybrid bioeconomic system with gestation delay and additive Allee effect

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