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doi: 10.3934/dcdsb.2019247

Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays

School of Mathematics, Renmin University of China, Beijing 100872, China

* Corresponding author: Kexin Wang

Received  April 2019 Revised  July 2019 Published  November 2019

Fund Project: The first author is supported by NSFC grant No. 71531012

In this work a stochastic Holling-Ⅱ type predator-prey model with infinite delays and feedback controls is investigated. By constructing a Lyapunov function, together with stochastic analysis approach, we obtain that the stochastic controlled predator-prey model admits a unique global positive solution. We then utilize graphical method and stability theorem of stochastic differential equations to investigate the globally asymptotical stability of a unique positive equilibrium for the stochastic controlled predator-prey system. If the stochastic predator-prey system is globally stable, then we show that using suitable feedback controls can alter the position of the unique positive equilibrium and retain the stable property. If the predator-prey system is destabilized by large intensities of white noises, then by choosing the appropriate values of feedback control variables, we can make the system reach a new stable state. Some examples are presented to verify our main results.

Citation: Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019247
References:
[1]

M. Aizerman and F. Gantmacher, Absolute Stability of Regulator Systems, Holden Day, San Francisco, 1964.  Google Scholar

[2]

L. ArnoldW. Horsthemke and J. Stucki, The influence of external real and white noise on the Lotka-Volterra model, Biomedical J., 21 (1979), 451-471.  doi: 10.1002/bimj.4710210507.  Google Scholar

[3]

L. ChangG. SunZ. Wang and Z. Jin, Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput., 256 (2015), 540-550.  doi: 10.1016/j.amc.2015.01.052.  Google Scholar

[4]

F. Chen, Global stability of a single species model with feedback control and distributed time delay, Appl. Math. Comput., 178 (2006), 474-479.  doi: 10.1016/j.amc.2005.11.062.  Google Scholar

[5]

L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334.  doi: 10.1016/j.aml.2009.03.005.  Google Scholar

[6]

L. ChenF. Chen and L. Chen, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 246-252.  doi: 10.1016/j.nonrwa.2008.10.056.  Google Scholar

[7]

L. Chen and J. Chen, Nonlinear Biodynamics Systems, Science Press of China, Beijing, 1993. Google Scholar

[8]

P. Chesson and R. Warner, Environmental variability promotes coexistence in lottery competitive systems, Amer. Natur., 117 (1981), 923-943.  doi: 10.1086/283778.  Google Scholar

[9]

Y. Fan and L. Wang, Global asymptotical stability of a Logistic model with feedback control, Nonlinear Anal. Real World Appl., 11 (2010), 2686-2697.  doi: 10.1016/j.nonrwa.2009.09.016.  Google Scholar

[10]

T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Differential Equations, 22 (2010), 299-324.  doi: 10.1007/s10884-010-9166-1.  Google Scholar

[11]

S. Gakkhar and A. Singh, Complex dynamics in a prey-predator system with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2011), 914-929.  doi: 10.1016/j.cnsns.2011.05.047.  Google Scholar

[12]

T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.  doi: 10.1007/BF02462011.  Google Scholar

[13]

T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.  doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[14]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications, 74, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[15]

K. Gopalsamy and P. Weng, Feedback regulation of logistic growth, Internat. J. Math. Math. Sci., 16 (1993), 177-192.  doi: 10.1155/S0161171293000213.  Google Scholar

[16]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[17]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[18]

C. S. Holling, The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[19]

C. JiD. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.  doi: 10.1016/j.jmaa.2009.05.039.  Google Scholar

[20]

T. K. Kara and A. Batabyal, Stability and bifurcation of a prey-predator model with time delay, Comptes Rendus Biologies, 332 (2009), 642-651.  doi: 10.1016/j.crvi.2009.02.002.  Google Scholar

[21]

W. Krajewski and U. Viaro, Locating the equilibrium points of a predator-prey model by means of affine state feedback, J. Franklin. Inst., 345 (2008), 489-498.  doi: 10.1016/j.jfranklin.2008.02.001.  Google Scholar

[22] S. Lefschetz, Stability of Nonlinear Control Systems, Mathematics in Science and Engineering, 13, Academic Press, New York, 1965.  doi: 10.1002/zamm.19660460515.  Google Scholar
[23]

A. Levin, Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

[24]

S. LiJ. Wu and Y. Dong, Effects of a degeneracy in a diffusive predator-prey model with Holling II functional response, Nonlinear Anal. Real World Appl., 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.  Google Scholar

[25]

Z. LiM. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays, Nonlinear Anal. Real World Appl., 14 (2013), 402-413.  doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[26]

M. Liu and K. Wang, Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3115-3123.  doi: 10.1016/j.cnsns.2011.09.021.  Google Scholar

[27]

M. Liu and K. Wang, Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1114-1121.  doi: 10.1016/j.cnsns.2010.06.015.  Google Scholar

[28]

Q. LiuY. Liu and X. Pan, Global stability of a stochastic predator-prey system with infinite delays, Appl. Math. Comput., 235 (2014), 1-7.  doi: 10.1016/j.amc.2014.02.091.  Google Scholar

[29]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997.  Google Scholar

[30]

X. Mao, Stochastic stabilisation and destabilisation, Syst. Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[31]

X. MaoS. Sabais and E. Renshaw, Asymptotic behavior of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

[32]

R. M. 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

[33]

E. A. McGehee and E. Peacock-López, Turing patterns in a modified Lotka-Volterra model, Phys. Lett. A, 342 (2005), 90-98.  doi: 10.1016/j.physleta.2005.04.098.  Google Scholar

[34]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[35]

Y. Saito, The necessary and sufficient condition for global stability of a Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268 (2002), 109-124.  doi: 10.1006/jmaa.2001.7801.  Google Scholar

[36]

Y. TakeuchiN. H. DubN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[37]

Q. WangZ. JiZ. WangM. Ding and H. Zhang, Existence and attractivity of a periodic solution for a ratio-dependent Leslie system with feedback controls, Nonlinear Anal. Real World Appl., 12 (2011), 24-33.  doi: 10.1016/j.nonrwa.2010.05.032.  Google Scholar

[38] K. Yang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, Boston, 1993.   Google Scholar
[39]

K. Yang, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[40]

K. YangZ. MiaoF. Chen and X. Xie, Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls, J. Math. Anal. Appl., 435 (2016), 874-888.   Google Scholar

[41]

R. YangM. Liu and C. Zhang, A delayed-diffusive predator-prey model with a ratio-dependent functional response, Commun. Nonlinear Sci. Numer. Simul., 53 (2017), 94-110.  doi: 10.1016/j.cnsns.2017.04.034.  Google Scholar

[42]

F. Yin and Y. Li, Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput., 153 (2004), 475-484.  doi: 10.1016/S0096-3003(03)00648-9.  Google Scholar

[43]

Q. Zhang, X. Wen, D. Jiang and Z. Liu, The stability of a predator-prey system with linear mass-action functional response perturbed by white noise, Adv. Differ. Equ., (2016). doi: 10.1186/s13662-016-0776-8.  Google Scholar

show all references

References:
[1]

M. Aizerman and F. Gantmacher, Absolute Stability of Regulator Systems, Holden Day, San Francisco, 1964.  Google Scholar

[2]

L. ArnoldW. Horsthemke and J. Stucki, The influence of external real and white noise on the Lotka-Volterra model, Biomedical J., 21 (1979), 451-471.  doi: 10.1002/bimj.4710210507.  Google Scholar

[3]

L. ChangG. SunZ. Wang and Z. Jin, Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput., 256 (2015), 540-550.  doi: 10.1016/j.amc.2015.01.052.  Google Scholar

[4]

F. Chen, Global stability of a single species model with feedback control and distributed time delay, Appl. Math. Comput., 178 (2006), 474-479.  doi: 10.1016/j.amc.2005.11.062.  Google Scholar

[5]

L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334.  doi: 10.1016/j.aml.2009.03.005.  Google Scholar

[6]

L. ChenF. Chen and L. Chen, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 246-252.  doi: 10.1016/j.nonrwa.2008.10.056.  Google Scholar

[7]

L. Chen and J. Chen, Nonlinear Biodynamics Systems, Science Press of China, Beijing, 1993. Google Scholar

[8]

P. Chesson and R. Warner, Environmental variability promotes coexistence in lottery competitive systems, Amer. Natur., 117 (1981), 923-943.  doi: 10.1086/283778.  Google Scholar

[9]

Y. Fan and L. Wang, Global asymptotical stability of a Logistic model with feedback control, Nonlinear Anal. Real World Appl., 11 (2010), 2686-2697.  doi: 10.1016/j.nonrwa.2009.09.016.  Google Scholar

[10]

T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Differential Equations, 22 (2010), 299-324.  doi: 10.1007/s10884-010-9166-1.  Google Scholar

[11]

S. Gakkhar and A. Singh, Complex dynamics in a prey-predator system with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2011), 914-929.  doi: 10.1016/j.cnsns.2011.05.047.  Google Scholar

[12]

T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.  doi: 10.1007/BF02462011.  Google Scholar

[13]

T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.  doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[14]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications, 74, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[15]

K. Gopalsamy and P. Weng, Feedback regulation of logistic growth, Internat. J. Math. Math. Sci., 16 (1993), 177-192.  doi: 10.1155/S0161171293000213.  Google Scholar

[16]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[17]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[18]

C. S. Holling, The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[19]

C. JiD. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.  doi: 10.1016/j.jmaa.2009.05.039.  Google Scholar

[20]

T. K. Kara and A. Batabyal, Stability and bifurcation of a prey-predator model with time delay, Comptes Rendus Biologies, 332 (2009), 642-651.  doi: 10.1016/j.crvi.2009.02.002.  Google Scholar

[21]

W. Krajewski and U. Viaro, Locating the equilibrium points of a predator-prey model by means of affine state feedback, J. Franklin. Inst., 345 (2008), 489-498.  doi: 10.1016/j.jfranklin.2008.02.001.  Google Scholar

[22] S. Lefschetz, Stability of Nonlinear Control Systems, Mathematics in Science and Engineering, 13, Academic Press, New York, 1965.  doi: 10.1002/zamm.19660460515.  Google Scholar
[23]

A. Levin, Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

[24]

S. LiJ. Wu and Y. Dong, Effects of a degeneracy in a diffusive predator-prey model with Holling II functional response, Nonlinear Anal. Real World Appl., 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.  Google Scholar

[25]

Z. LiM. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays, Nonlinear Anal. Real World Appl., 14 (2013), 402-413.  doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[26]

M. Liu and K. Wang, Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3115-3123.  doi: 10.1016/j.cnsns.2011.09.021.  Google Scholar

[27]

M. Liu and K. Wang, Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1114-1121.  doi: 10.1016/j.cnsns.2010.06.015.  Google Scholar

[28]

Q. LiuY. Liu and X. Pan, Global stability of a stochastic predator-prey system with infinite delays, Appl. Math. Comput., 235 (2014), 1-7.  doi: 10.1016/j.amc.2014.02.091.  Google Scholar

[29]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997.  Google Scholar

[30]

X. Mao, Stochastic stabilisation and destabilisation, Syst. Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[31]

X. MaoS. Sabais and E. Renshaw, Asymptotic behavior of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

[32]

R. M. 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

[33]

E. A. McGehee and E. Peacock-López, Turing patterns in a modified Lotka-Volterra model, Phys. Lett. A, 342 (2005), 90-98.  doi: 10.1016/j.physleta.2005.04.098.  Google Scholar

[34]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[35]

Y. Saito, The necessary and sufficient condition for global stability of a Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268 (2002), 109-124.  doi: 10.1006/jmaa.2001.7801.  Google Scholar

[36]

Y. TakeuchiN. H. DubN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[37]

Q. WangZ. JiZ. WangM. Ding and H. Zhang, Existence and attractivity of a periodic solution for a ratio-dependent Leslie system with feedback controls, Nonlinear Anal. Real World Appl., 12 (2011), 24-33.  doi: 10.1016/j.nonrwa.2010.05.032.  Google Scholar

[38] K. Yang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, Boston, 1993.   Google Scholar
[39]

K. Yang, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[40]

K. YangZ. MiaoF. Chen and X. Xie, Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls, J. Math. Anal. Appl., 435 (2016), 874-888.   Google Scholar

[41]

R. YangM. Liu and C. Zhang, A delayed-diffusive predator-prey model with a ratio-dependent functional response, Commun. Nonlinear Sci. Numer. Simul., 53 (2017), 94-110.  doi: 10.1016/j.cnsns.2017.04.034.  Google Scholar

[42]

F. Yin and Y. Li, Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput., 153 (2004), 475-484.  doi: 10.1016/S0096-3003(03)00648-9.  Google Scholar

[43]

Q. Zhang, X. Wen, D. Jiang and Z. Liu, The stability of a predator-prey system with linear mass-action functional response perturbed by white noise, Adv. Differ. Equ., (2016). doi: 10.1186/s13662-016-0776-8.  Google Scholar

Figure 1.  The parabola $ l_1 $ (green line) and the hyperbola $ l_2 $(blue line) of 9 provided that the condition 7 holds
Figure 2.  The region $ R_0 $ where positive equilibria $ (x^{*}, y^{*}) $ will occur under feedback controls
Figure 3.  Dynamic behavior of the solution $ (x(t), y(t))^{T} $ of system 20 with the initial condition $ (\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0] $
Figure 4.  The region $ R_0 $ where positive equilibria of system 20 will occur under feedback controls
Figure 5.  Dynamic behavior of the solution $ (x(t), y(t))^{T} $ of system 20 with small perturbations $ \sigma_1 = \sigma_2 = 0.1 $ and the initial condition $ (\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0] $
Figure 6.  Dynamic behavior of the solution $ (x(t), y(t), u_1(t), u_2(t))^{T} $ of system 21 with the initial condition $ (\varphi_1(\theta), \varphi_2(\theta), u_1(0), u_2(0)) = (1.2 e^{\theta}, 0.8 e^{\theta}, 1,1), \theta \in (-\infty, 0] $
Figure 7.  Dynamic behavior of the solution $ (x(t), y(t))^{T} $ of system 20 with big perturbations $ \sigma_1 = \sigma_2 = 1 $ and the initial condition $ (\varphi_1(\theta), \varphi_2(\theta)) = (1.2 e^{\theta}, 0.8 e^{\theta}), \theta \in (-\infty, 0] $
Figure 8.  Dynamic behavior of the solution $ (x(t), y(t), u_1(t), u_2(t))^{T} $ of system 22 with the initial condition $ (\varphi_1(\theta), \varphi_2(\theta), u_1(0), u_2(0)) = (1.2 e^{\theta}, 0.8 e^{\theta}, 1,1), \theta \in (-\infty, 0] $
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