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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 |
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.
References:
[1] |
J. Arino, L. Wang and G. Wolkowicz,
An alternative formulation for a delayed logistic equation, Journal of Theoretical Biology, 241 (2006), 109-119.
doi: 10.1016/j.jtbi.2005.11.007. |
[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.
doi: 10.1016/j.nahs.2008.04.001. |
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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.
doi: 10.1016/j.jde.2015.08.024. |
[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.
doi: 10.1016/j.amc.2017.02.003. |
[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.
doi: 10.1016/j.nonrwa.2017.10.001. |
[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.
doi: 10.1016/j.apm.2013.11.014. |
[7] |
G. Harrison,
Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48 (1986), 137-148.
doi: 10.1007/BF02460019. |
[8] |
G. Harrison,
Comparing predator-prey models to luckinbill's experiment with didinium and paramecium, Ecology, 76 (1995), 357-374.
doi: 10.2307/1941195. |
[9] |
Y. Jin,
Moment stability for a predator-prey model with parametric dichotomous noises, Chinese Physics B, 24 (2015), 060502-7.
doi: 10.1088/1674-1056/24/6/060502. |
[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.
doi: 10.1155/2011/914270. |
[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.
doi: 10.3934/dcds.2017108. |
[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.
doi: 10.1533/9780857099402. |
[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.
doi: 10.1016/S0304-4149(01)00126-0. |
[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.
doi: 10.1016/j.jmaa.2004.09.027. |
[18] |
A. Martin and S. Ruan,
Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43 (2001), 247-267.
doi: 10.1007/s002850100095. |
[19] |
R. May,
Time-delay versus stability in population models with two and three trophic levels, Ecology, 54 (1973), 315-325.
doi: 10.2307/1934339. |
[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.
doi: 10.1007/b98869. |
[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.
doi: 10.1016/j.chaos.2008.07.005. |
[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.
doi: 10.1088/1742-5468/2012/03/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.
doi: 10.1016/j.jmaa.2018.01.046. |
[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.
doi: 10.1016/j.amc.2007.06.017. |
[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.
doi: 10.1080/0020739960270102. |
[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.
doi: 10.1016/j.mcm.2012.09.002. |
[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.
doi: 10.1016/j.jfranklin.2017.08.034. |
[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.
doi: 10.1016/j.nonrwa.2013.09.010. |
[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. |
show all references
References:
[1] |
J. Arino, L. Wang and G. Wolkowicz,
An alternative formulation for a delayed logistic equation, Journal of Theoretical Biology, 241 (2006), 109-119.
doi: 10.1016/j.jtbi.2005.11.007. |
[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.
doi: 10.1016/j.nahs.2008.04.001. |
[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.
doi: 10.1016/j.jde.2015.08.024. |
[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.
doi: 10.1016/j.amc.2017.02.003. |
[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.
doi: 10.1016/j.nonrwa.2017.10.001. |
[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.
doi: 10.1016/j.apm.2013.11.014. |
[7] |
G. Harrison,
Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48 (1986), 137-148.
doi: 10.1007/BF02460019. |
[8] |
G. Harrison,
Comparing predator-prey models to luckinbill's experiment with didinium and paramecium, Ecology, 76 (1995), 357-374.
doi: 10.2307/1941195. |
[9] |
Y. Jin,
Moment stability for a predator-prey model with parametric dichotomous noises, Chinese Physics B, 24 (2015), 060502-7.
doi: 10.1088/1674-1056/24/6/060502. |
[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.
doi: 10.1155/2011/914270. |
[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.
doi: 10.3934/dcds.2017108. |
[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.
doi: 10.1533/9780857099402. |
[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.
doi: 10.1016/S0304-4149(01)00126-0. |
[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.
doi: 10.1016/j.jmaa.2004.09.027. |
[18] |
A. Martin and S. Ruan,
Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43 (2001), 247-267.
doi: 10.1007/s002850100095. |
[19] |
R. May,
Time-delay versus stability in population models with two and three trophic levels, Ecology, 54 (1973), 315-325.
doi: 10.2307/1934339. |
[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.
doi: 10.1007/b98869. |
[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.
doi: 10.1016/j.chaos.2008.07.005. |
[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.
doi: 10.1088/1742-5468/2012/03/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.
doi: 10.1016/j.jmaa.2018.01.046. |
[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.
doi: 10.1016/j.amc.2007.06.017. |
[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.
doi: 10.1080/0020739960270102. |
[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.
doi: 10.1016/j.mcm.2012.09.002. |
[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.
doi: 10.1016/j.jfranklin.2017.08.034. |
[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.
doi: 10.1016/j.nonrwa.2013.09.010. |
[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. |




Equilibrium | Existence Condition | Stability Condition |
Always exists | Always saddle | |
Always exists | Sink if Saddle if | |
Always sink |
Equilibrium | Existence Condition | Stability Condition |
Always exists | Always saddle | |
Always exists | Sink if Saddle if | |
Always sink |
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