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

  • 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.

    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

    Related Papers:

  • 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.
    加载中
    [1] [ J. Arino, L. Wang and G. Wolkowicz, An alternative formulation for a delayed logistic equation, Journal of Theoretical Biology, 241 (2006), 109-119.
    [2] [ M. Bandyopadhyay, T. Saha and R. Pal, Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment, Nonlinear Analysis: Hybrid Systems, 2 (2008), 958-970.
    [3] [ Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502.
    [4] [ Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Applied Mathematics and Computation, 305 (2017), 221-240.
    [5] [ Y. Cai, Y. Kang, M. Banerjee and W. Wang, Complex dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonlinear Analysis: Real World Applications, 40 (2018), 444-465.
    [6] [Q. Han, D. Jiang and C. Ji, Analysis of a delayed stochastic predator-prey model in a polluted environment, Applied Mathematical Modelling, 38 (2014), 3067-3080.
    [7] [G. Harrison, Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48 (1986), 137-148.
    [8] [G. Harrison, Comparing predator-prey models to luckinbill's experiment with didinium and paramecium, Ecology, 76 (1995), 357-374.
    [9] [Y. Jin, Moment stability for a predator-prey model with parametric dichotomous noises, Chinese Physics B, 24 (2015), 060502-7.
    [10] [Y. Kuang, Delay Differental Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.
    [11] [B. Lian, S. Hu and Y. Fen, Stochastic delay Lotka-Volterra model, Journal of Inequalities and Applications, 2011 (2011), Art. ID 914270, 13 pp.
    [12] [M. Liu, C. Bai and Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete and Continuous Dynamical Systems, 37 (2017), 2513-2538.
    [13] [A. Maiti, M. Jana and G. Samanta, Deterministic and stochastic analysis of a ratio-dependent predator-prey system with delay, Nonlinear Analysis: Modelling and Control, 12 (2007), 383-398.
    [14] [X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
    [15] [X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.
    [16] [X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110.
    [17] [X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320.
    [18] [A. Martin and S. Ruan, Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43 (2001), 247-267.
    [19] [R. May, Time-delay versus stability in population models with two and three trophic levels, Ecology, 54 (1973), 315-325.
    [20] [R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 2001.
    [21] [J. Murray, Mathematical Biology, 3rd edition, Springer-Verlag, New York, 2003.
    [22] [F. Rao, W. Wang and Z. Li, Spatiotemporal complexity of a predator-prey system with the effect of noise and external forcing, Chaos, Solitons and Fractals, 41 (2009), 1634-1644.
    [23] [F. Rao and W. Wang, Dynamics of a Michaelis-Menten-type predation model incorporating a prey refuge with noise and external forces, Journal of Statistical Mechanics: Theory and Experiment, 3 (2012), P03014.
    [24] [F. Rao, C. Castillo-Chavez and Y. Kang, Dynamics of a diffusion reaction prey-predator model with delay in prey: Effects of delay and spatial components, Journal of Mathematical Analysis and Applications, 461 (2018), 1177-1214.
    [25] [T. Saha and M. Bandyopadhyay, Dynamical analysis of a delayed ratio-dependent prey-predator model within fluctuating environment, Applied Mathematics and Computation, 196 (2008), 458-478.
    [26] [G. Samanta, The effects of random fluctuating environment on interacting species with time delay, International Journal of Mathematical Education in Science and Technology, 27 (1996), 13-21.
    [27] [M. Vasilova, Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay, Mathematical and Computer Modelling, 57 (2013), 764-781.
    [28] [W. Wang, Y. Cai, J. Li and Z. Gui, Periodic behavior in a FIV model with seasonality as well as environment fluctuations, Journal of the Franklin Institute, 354 (2017), 7410-7428.
    [29] [X. Wang and Y. Cai, Cross-diffusion-driven instability in a reaction-diffusion Harrison predator-prey model, Abstract and Applied Analysis, 2013 (2013), Art. ID 306467, 9 pp.
    [30] [W. Wang, Y. Zhu, Y. Cai and W. Wang, Dynamical complexity induced by Allee effect in a predator-prey model, Nonlinear Analysis: Real World Applications, 16 (2014), 103-119.
    [31] [Y. Zhu, Y. Cai, S. Yan and W. Wang, Dynamical analysis of a delayed reaction-diffusion predator-prey system, Abstract and Applied Analysis, 2012 (2012), Art. ID 323186, 23 pp.

    © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(594) PDF downloads(731) Cited by(2)

Article outline

Figures and Tables

Figures(5)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog