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December  2018, 15(6): 1401-1423. doi: 10.3934/mbe.2018064

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

Feng Rao 1, , Carlos Castillo-Chavez 2,3, and  Yun Kang 2,3,,

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

* Corresponding author: Yun Kang

Received  December 10, 2017 Accepted  June 10, 2018 Published  September 2018

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.

Keywords: Prey-predator model, time delay, Hopf bifurcation, stochastic perturbations, stability.
Mathematics Subject Classification: Primary: 34C25, 34F05; Secondary: 92B05.
Citation: Feng Rao, Carlos Castillo-Chavez, Yun Kang. Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1401-1423. doi: 10.3934/mbe.2018064
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.  Google Scholar

[2]

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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.  Google Scholar

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G. Harrison, Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48 (1986), 137-148.  doi: 10.1007/BF02460019.  Google Scholar

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G. Harrison, Comparing predator-prey models to luckinbill's experiment with didinium and paramecium, Ecology, 76 (1995), 357-374.  doi: 10.2307/1941195.  Google Scholar

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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.  Google Scholar

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Y. Kuang, Delay Differental Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[15]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 2001. Google Scholar

[21]

J. Murray, Mathematical Biology, 3rd edition, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. Harrison, Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48 (1986), 137-148.  doi: 10.1007/BF02460019.  Google Scholar

[8]

G. Harrison, Comparing predator-prey models to luckinbill's experiment with didinium and paramecium, Ecology, 76 (1995), 357-374.  doi: 10.2307/1941195.  Google Scholar

[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.  Google Scholar

[10]

Y. Kuang, Delay Differental Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[15]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 2001. Google Scholar

[21]

J. Murray, Mathematical Biology, 3rd edition, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Phase portrait of model (2) and the parameters are taken as $c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$. The horizontal axis is prey population $N$ and the vertical axis is predator population $P$. The red dotted curve is the $N$-isoline $cP = (1-N)(mP+1)$ and the yellow solid curve is the $P$-isoline $bN = d(mP+1)$. Both $E_0 = (0, 0)$ and $E_1 = (1, 0)$ are saddle points, $E^* = (0.46, 0.64)$ is locally asymptotically stable
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Fig. 1. In the figures of time series, the red curve is the population of $N$ and the blue curve is the population of $P$">Figure 2.  The effects of the time delay $\tau$ on the dynamics of the DDE model (4) when $c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$ which are the same as in Fig. 1. In the figures of time series, the red curve is the population of $N$ and the blue curve is the population of $P$
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Figure 3.  Time-series plots of model (3) without time-delay and only with different noises $\sigma_1, \, \sigma_2$, and other parametric values are $\tau = 0, \, c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$
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Figure 4.  Time-series plots of the SDDE model (3) for different noise $\sigma_1, \, \sigma_2$ with time delay $\tau = 2.8 < \tau_0 = 3.46$, other parametric values are given as (16)
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Figure 5.  Time-series plots of the SDDE model (3) for different noise $\sigma_1, \, \sigma_2$ with $\tau = 3.9>\tau_0 = 3.46$, other parametric values are given as (16)
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Table 1.  The existence and stability of equilibria for model (2) where $N^* = \frac{b(m-c)+\sqrt{4bcdm+b^2(m-c)^2}}{2bm}, \, P^* = \frac{bN^*-d}{dm}$
Equilibrium Existence Condition Stability Condition
$(0, 0)$ Always exists Always saddle
$(1, 0)$ Always exists Sink if $d\geq b$;
Saddle if $d < b$
$(N^*, P^*)$ $d < b$ Always sink
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Equilibrium Existence Condition Stability Condition
$(0, 0)$ Always exists Always saddle
$(1, 0)$ Always exists Sink if $d\geq b$;
Saddle if $d < b$
$(N^*, P^*)$ $d < b$ Always sink
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