# American Institute of Mathematical Sciences

February  2022, 15(2): 245-263. doi: 10.3934/dcdss.2020468

## Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species

 Department of Mathematical Sciences, College of Science, UAE University, Al Ain, 15551, UAE

* Corresponding author: F.A. Rihan (frihan@uaeu.ac.ae)

Received  April 2020 Revised  September 2020 Published  February 2022 Early access  November 2020

Fund Project: This work supported by UPAR-Project (Code # G00003440)

Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter ${\mathcal R}_0^s$ that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.

Citation: Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 245-263. doi: 10.3934/dcdss.2020468
##### References:
 [1] J. Alebraheem and Y. A. Hasan, Dynamics of a two predator–one prey system, Computational and Applied Mathematics, 33 (2014), 767-780.  doi: 10.1007/s40314-013-0093-8. [2] A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380.  doi: 10.1016/j.jmaa.2003.12.004. [3] Y. Bai and Y. Li, Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator, Advances in Difference Equations, 2019 (2019), 1-20.  doi: 10.1186/s13662-019-1979-6. [4] J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, Journal of Mathematical Analysis and Applications, 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043. [5] G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199.  doi: 10.1016/S0377-0427(00)00468-4. [6] E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1. [7] B. Dubey and A. Kumar, Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679.  doi: 10.1007/s11071-019-04951-5. [8] M. F. Elettreby, Two-prey one-predator model, Chaos, Solitons & Fractals, 39 (2009), 2018-2027.  doi: 10.1016/j.chaos.2007.06.058. [9] T. C. Gard, Persistence in stochastic food web models, Bulletin of Mathematical Biology, 46 (1984), 357-370.  doi: 10.1016/S0092-8240(84)80044-0. [10] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [11] R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [12] P. E. Kloeden and T. Shardlow, The Milstein scheme for stochastic delay differential equations without using anticipative calculus, Stochastic Analysis and Applications, 30 (2012), 181-202.  doi: 10.1080/07362994.2012.628907. [13] S. Kundu and S. Maitra, Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species, Nonlinear Dynamics, 92 (2018), 627-643.  doi: 10.1007/s11071-018-4079-3. [14] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [15] R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146. [16] Z. Liu and R. Tan, Impulsive harvesting and stocking in a monod–haldane functional response predator–prey system, Chaos, Solitons & Fractals, 34 (2007), 454-464.  doi: 10.1016/j.chaos.2006.03.054. [17] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic predator–prey model with herd behavior, Journal of the Franklin Institute, 355 (2018), 8177-8193.  doi: 10.1016/j.jfranklin.2018.09.013. [18] A. J. Lotka, Elements of Physical Biology, Baltimore: Williams & Wilkins Co., 1925. [19] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402. [20] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [21] X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0. [22] J. D. Murray, Mathematical Biology, Springer New york, 1993. doi: 10.1007/b98869. [23] R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 25 (2014), 30-53.  doi: 10.1016/j.mbs.2014.02.008. [24] R Rakkiyappan, G. Velmurugan, F. A. Rihan and and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39.  doi: 10.1002/cplx.21618. [25] F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator–prey systems with {Holling type-II} functional response, Nonlinear Dynamics, 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8. [26] F. A. Rihan, H. J. Alsakaji and C. Rajivganthi, Stability and Hopf bifurcation of three-species prey-predator system with time delays and Allee effect, Complexity, 2020 (2020), 7306412. doi: 10.1155/2020/7306412. [27] F. A. Rihan, A. A. Azamov and H. J. Al-Sakaji, An inverse problem for delay differential equations: Parameter estimation, nonlinearity, sensitivity, Applied Mathematics & Information Sciences, 12 (2018), 63-74.  doi: 10.18576/amis/120106. [28] F. A. Rihan and H. J. Alsakaji, Persistence and extinction for stochastic delay differential model of prey-predator system with hunting cooperation in predators, Advances in Difference Equations, 124 (2020), 1-22.  doi: 10.1186/s13662-020-02579-z. [29] F.A. Rihan, H.J. Alsakaji and C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 1-20.  doi: 10.1186/s13662-020-02964-8. [30] F. A. Rihan, C. Rajivganthi and P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 2017 (2017), Art. ID 5394528, 11 pp. doi: 10.1155/2017/5394528. [31] 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. [32] G. Tang, S. Tang and R. A. Cheke, Global analysis of a holling type II predator–prey model with a constant prey refuge, Nonlinear Dynamics, 76 (2014), 635-647.  doi: 10.1007/s11071-013-1157-4. [33] D. A. Vasseuri and P. Yodzis, The color of environmental noise, Ecology, 85 (2004), 1146-1152. [34] V. Volterra, Variazioni e Fluttuazioni Del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. [35] 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. [36] X. Zhao and Z. Zeng, Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123310, 17 pp. doi: 10.1016/j.physa.2019.123310.

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##### References:
 [1] J. Alebraheem and Y. A. Hasan, Dynamics of a two predator–one prey system, Computational and Applied Mathematics, 33 (2014), 767-780.  doi: 10.1007/s40314-013-0093-8. [2] A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380.  doi: 10.1016/j.jmaa.2003.12.004. [3] Y. Bai and Y. Li, Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator, Advances in Difference Equations, 2019 (2019), 1-20.  doi: 10.1186/s13662-019-1979-6. [4] J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, Journal of Mathematical Analysis and Applications, 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043. [5] G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199.  doi: 10.1016/S0377-0427(00)00468-4. [6] E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1. [7] B. Dubey and A. Kumar, Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679.  doi: 10.1007/s11071-019-04951-5. [8] M. F. Elettreby, Two-prey one-predator model, Chaos, Solitons & Fractals, 39 (2009), 2018-2027.  doi: 10.1016/j.chaos.2007.06.058. [9] T. C. Gard, Persistence in stochastic food web models, Bulletin of Mathematical Biology, 46 (1984), 357-370.  doi: 10.1016/S0092-8240(84)80044-0. [10] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [11] R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [12] P. E. Kloeden and T. Shardlow, The Milstein scheme for stochastic delay differential equations without using anticipative calculus, Stochastic Analysis and Applications, 30 (2012), 181-202.  doi: 10.1080/07362994.2012.628907. [13] S. Kundu and S. Maitra, Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species, Nonlinear Dynamics, 92 (2018), 627-643.  doi: 10.1007/s11071-018-4079-3. [14] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [15] R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146. [16] Z. Liu and R. Tan, Impulsive harvesting and stocking in a monod–haldane functional response predator–prey system, Chaos, Solitons & Fractals, 34 (2007), 454-464.  doi: 10.1016/j.chaos.2006.03.054. [17] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic predator–prey model with herd behavior, Journal of the Franklin Institute, 355 (2018), 8177-8193.  doi: 10.1016/j.jfranklin.2018.09.013. [18] A. J. Lotka, Elements of Physical Biology, Baltimore: Williams & Wilkins Co., 1925. [19] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402. [20] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [21] X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0. [22] J. D. Murray, Mathematical Biology, Springer New york, 1993. doi: 10.1007/b98869. [23] R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 25 (2014), 30-53.  doi: 10.1016/j.mbs.2014.02.008. [24] R Rakkiyappan, G. Velmurugan, F. A. Rihan and and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39.  doi: 10.1002/cplx.21618. [25] F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator–prey systems with {Holling type-II} functional response, Nonlinear Dynamics, 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8. [26] F. A. Rihan, H. J. Alsakaji and C. Rajivganthi, Stability and Hopf bifurcation of three-species prey-predator system with time delays and Allee effect, Complexity, 2020 (2020), 7306412. doi: 10.1155/2020/7306412. [27] F. A. Rihan, A. A. Azamov and H. J. Al-Sakaji, An inverse problem for delay differential equations: Parameter estimation, nonlinearity, sensitivity, Applied Mathematics & Information Sciences, 12 (2018), 63-74.  doi: 10.18576/amis/120106. [28] F. A. Rihan and H. J. Alsakaji, Persistence and extinction for stochastic delay differential model of prey-predator system with hunting cooperation in predators, Advances in Difference Equations, 124 (2020), 1-22.  doi: 10.1186/s13662-020-02579-z. [29] F.A. Rihan, H.J. Alsakaji and C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 1-20.  doi: 10.1186/s13662-020-02964-8. [30] F. A. Rihan, C. Rajivganthi and P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 2017 (2017), Art. ID 5394528, 11 pp. doi: 10.1155/2017/5394528. [31] 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. [32] G. Tang, S. Tang and R. A. Cheke, Global analysis of a holling type II predator–prey model with a constant prey refuge, Nonlinear Dynamics, 76 (2014), 635-647.  doi: 10.1007/s11071-013-1157-4. [33] D. A. Vasseuri and P. Yodzis, The color of environmental noise, Ecology, 85 (2004), 1146-1152. [34] V. Volterra, Variazioni e Fluttuazioni Del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. [35] 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. [36] X. Zhao and Z. Zeng, Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123310, 17 pp. doi: 10.1016/j.physa.2019.123310.
Shows numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), when $\tau_1 = 1.25$, $\tau_2 = 0.6$ and $\tau_3 = 0.5$, with noise intensities $\sigma_1^2 = 0.08$, $\sigma_2^2 = 0.1$, $\sigma_3^2 = 0.06$, and parameter values: $r_1 = 0.2$, $r_2 = 0.6$, $K_1 = 0.7$, $K_2 = 0.8$, $\alpha_1 = 0.3$, $\alpha_2 = 0.6$, $\alpha_3 = 0.8$, $\beta = 0.1$, $\delta = 0.8$, $a_1 = 1$, $a_2 = 1.4$. For ${\mathcal R}_0^s>1$, the stochastic model has a unique ergodic stationary distribution $\pi(.)$ of stochastic system (2)
Numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with parameter values given in Example 2, with noise intensities: $\sigma_1^2 = 0.03$, $\sigma_2^2 = 0.02$ and $\sigma_3^2 = 1.4$. When ${\mathcal R}_0^s<1$, we can clearly see that the predator goes to extinct
Shows numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with the same parameter values of Figure 2, but with noise intensities $\sigma_1^2 = 1.2$, $\sigma_2^2 = 1.2$ and $\sigma_3^2 = 0.5$. When $r_1<\frac{\sigma_1^2}{2}$, $r_2<\frac{\sigma_2^2}{2}$ and ${\mathcal R}_0^s<1$, we can clearly see that all the species goes to extinct. A strong intensity of noise can be a cause for extinction of the prey species, which will also drive predator population to extinct
Numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with parameter values of Figure 1, but with $\tau_1 = 10$, $\tau_2 = 0.1$ and $\tau_3 = 0.1$, under the noise intensities $\sigma_1^2 = 0.2$, $\sigma_2^2 = 0.2$ and $\sigma_3^2 = 0.2$. Clearly, time-delays can lead to Hopf-type bifurcations of deterministic systems
Shows the effect of white noise to prevent the explosion of the population. When $\beta = 0.5$, with the same parameter values of Figure 1, there is an explosion of population with deterministic model (left); While the noise prevent such explosion of the population (right)
One biological meaning for the parameters of model (2)
 Parameters Description $r_1$, $r_2$ intrinsic growth rate for x and y $k_1$, $k_2$ carrying capacity for x and y $\alpha_1$, $\alpha_2$ rate of predation of preys x and y $\beta$ rate of cooperation of preys x and y against predator z $\delta$ Predator death rate $\alpha_3$ rate of intra-species competition within the predators $a_1$, $a_2$ transformation rate of predator to preys $x$ and $y$.
 Parameters Description $r_1$, $r_2$ intrinsic growth rate for x and y $k_1$, $k_2$ carrying capacity for x and y $\alpha_1$, $\alpha_2$ rate of predation of preys x and y $\beta$ rate of cooperation of preys x and y against predator z $\delta$ Predator death rate $\alpha_3$ rate of intra-species competition within the predators $a_1$, $a_2$ transformation rate of predator to preys $x$ and $y$.
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