# American Institute of Mathematical Sciences

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

 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.

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. [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.

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.
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
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 $P$
Time-series plots of model (3) without time-delay and only with different noises $\sigma_1, \, \sigma_2$, and other parametric values are $\tau = 0, \, c = 0.9, \, b = 0.7, \, d = 0.3, \, m = 0.1$
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)
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)
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
 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
 [1] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [2] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [3] Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108 [4] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [5] Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737 [6] Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138 [7] R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423 [8] Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244 [9] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [10] Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173 [11] Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536 [12] Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061 [13] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [14] J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059 [15] Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128 [16] Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095 [17] Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048 [18] Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046 [19] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [20] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

2017 Impact Factor: 1.23