Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components

Feng Rao; Carlos Castillo-Chavez; Yun Kang; Feng Rao; Carlos Castillo-Chavez; Yun Kang

doi:10.3934/mbe.2018064

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 6: 1401-1423. doi: 10.3934/mbe.2018064

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Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components

1.
School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, China
2.
Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, USA
3.
Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

Received: 10 December 2017 Accepted: 10 June 2018 Published: 01 December 2018
MSC : Primary: 34C25, 34F05; Secondary: 92B05

This paper investigates the complex dynamics of a Harrison-type predator-prey model that incorporating: (1) A constant time delay in the functional response term of the predator growth equation; and (2) environmental noise in both prey and predator equations. We provide the rigorous results of our model including the dynamical behaviors of a positive solution and Hopf bifurcation. We also perform numerical simulations on the effects of delay or/and noise when the corresponding ODE model has an interior solution. Our theoretical and numerical results show that delay can either remain stability or destabilize the model; large noise could destabilize the model; and the combination of delay and noise could intensify the periodic instability of the model. Our results may provide us useful biological insights into population managements for prey-predator interaction models.
- Prey-predator model,
- time delay,
- Hopf bifurcation,
- stochastic perturbations,
- stability
Citation: Feng Rao, Carlos Castillo-Chavez, Yun Kang. Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components[J]. Mathematical Biosciences and Engineering, 2018, 15(6): 1401-1423. doi: 10.3934/mbe.2018064

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Abstract

This paper investigates the complex dynamics of a Harrison-type predator-prey model that incorporating: (1) A constant time delay in the functional response term of the predator growth equation; and (2) environmental noise in both prey and predator equations. We provide the rigorous results of our model including the dynamical behaviors of a positive solution and Hopf bifurcation. We also perform numerical simulations on the effects of delay or/and noise when the corresponding ODE model has an interior solution. Our theoretical and numerical results show that delay can either remain stability or destabilize the model; large noise could destabilize the model; and the combination of delay and noise could intensify the periodic instability of the model. Our results may provide us useful biological insights into population managements for prey-predator interaction models.

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