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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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
School of Mathematics, Renmin University of China, Beijing 100872, China |
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 and Continuous Dynamical Systems - B,
2020, 25
(5)
: 1699-1714.
doi: 10.3934/dcdsb.2019247
![]()
References:
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M. Aizerman and F. Gantmacher, Absolute Stability of Regulator Systems, Holden Day, San Francisco, 1964. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
L. Chen and J. Chen, Nonlinear Biodynamics Systems, Science Press of China, Beijing, 1993. |
| [8] |
P. Chesson and R. Warner,
Environmental variability promotes coexistence in lottery competitive systems, Amer. Natur., 117 (1981), 923-943.
doi: 10.1086/283778. |
| [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. |
| [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. |
| [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. |
| [12] |
T. C. Gard,
Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.
doi: 10.1007/BF02462011. |
| [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. |
| [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. |
| [15] |
K. Gopalsamy and P. Weng,
Feedback regulation of logistic growth, Internat. J. Math. Math. Sci., 16 (1993), 177-192.
doi: 10.1155/S0161171293000213. |
| [16] |
D. J. Higham,
An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
| [17] |
C. S. Holling,
Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.
doi: 10.4039/Ent91385-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
S. Lefschetz, Stability of Nonlinear Control Systems, Mathematics in Science and Engineering, 13, Academic Press, New York, 1965.
doi: 10.1002/zamm.19660460515.
|
| [23] |
A. Levin,
Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.
doi: 10.1086/282900. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [29] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997. |
| [30] |
X. Mao,
Stochastic stabilisation and destabilisation, Syst. Control Lett., 23 (1994), 279-290.
doi: 10.1016/0167-6911(94)90050-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [38] |
K. Yang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, Boston, 1993.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
M. Aizerman and F. Gantmacher, Absolute Stability of Regulator Systems, Holden Day, San Francisco, 1964. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
L. Chen and J. Chen, Nonlinear Biodynamics Systems, Science Press of China, Beijing, 1993. |
| [8] |
P. Chesson and R. Warner,
Environmental variability promotes coexistence in lottery competitive systems, Amer. Natur., 117 (1981), 923-943.
doi: 10.1086/283778. |
| [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. |
| [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. |
| [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. |
| [12] |
T. C. Gard,
Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.
doi: 10.1007/BF02462011. |
| [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. |
| [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. |
| [15] |
K. Gopalsamy and P. Weng,
Feedback regulation of logistic growth, Internat. J. Math. Math. Sci., 16 (1993), 177-192.
doi: 10.1155/S0161171293000213. |
| [16] |
D. J. Higham,
An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
| [17] |
C. S. Holling,
Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.
doi: 10.4039/Ent91385-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
S. Lefschetz, Stability of Nonlinear Control Systems, Mathematics in Science and Engineering, 13, Academic Press, New York, 1965.
doi: 10.1002/zamm.19660460515.
|
| [23] |
A. Levin,
Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.
doi: 10.1086/282900. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [29] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997. |
| [30] |
X. Mao,
Stochastic stabilisation and destabilisation, Syst. Control Lett., 23 (1994), 279-290.
doi: 10.1016/0167-6911(94)90050-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [38] |
K. Yang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, Boston, 1993.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
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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