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

Kexin Wang

doi:10.3934/dcdsb.2019247

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May 2020, 25(5): 1699-1714. 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

Kexin Wang ^,

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.

Keywords: Global stability, feedback control, stochastic perturbation, predator-prey, infinite delay.

Mathematics Subject Classification: Primary: 34D23, 58J37; Secondary: 92D25, 93B52.

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, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247

References:

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[2]	L. Arnold, W. 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. Chang, G. Sun, Z. 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. Chen, F. 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. Ji, D. 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. Li, J. 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. Li, M. 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. Liu, Y. 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. Mao, S. 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. Takeuchi, N. H. Dub, N. 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. Wang, Z. Ji, Z. Wang, M. 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. Yang, Z. Miao, F. 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. Yang, M. 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. Arnold, W. 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. Chang, G. Sun, Z. 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. Chen, F. 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. Ji, D. 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. Li, J. 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. Li, M. 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. Liu, Y. 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. Mao, S. 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. Takeuchi, N. H. Dub, N. 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. Wang, Z. Ji, Z. Wang, M. 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. Yang, Z. Miao, F. 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. Yang, M. 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

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Figure 2. The region $ R_0 $ where positive equilibria $ (x^{*}, y^{*}) $ will occur under feedback controls

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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] $

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Figure 4. The region $ R_0 $ where positive equilibria of system 20 will occur under feedback controls

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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] $

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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] $

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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] $

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

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

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