October 2017, 14(5&6): 1477-1498. doi: 10.3934/mbe.2017077

Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

3. 

Lamps and Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada

* Corresponding author: Sanling Yuan

Received  June 23, 2016 Accepted  September 27, 2016 Published  May 2017

Fund Project: The research is supported by the National Natural Science Foundation of China (11271260,11671260), Innovation program of Shanghai Municipal Education Committee (13ZZ116), Shanghai Leading Academic Discipline Project (No. XTKX2012), and the Hujiang Foundation of China (B14005)

In this paper, we investigate the dynamics of a delayed logistic model with both impulsive and stochastic perturbations. The impulse is introduced at fixed moments and the stochastic perturbation is of white noise type which is assumed to be proportional to the population density. We start with the existence and uniqueness of the positive solution of the model, then establish sufficient conditions ensuring its global attractivity. By using the theory of integral Markov semigroups, we further derive sufficient conditions for the existence of the stationary distribution of the system. Finally, we perform the extinction analysis of the model. Numerical simulations illustrate the obtained theoretical results.

Citation: Sanling Yuan, Xuehui Ji, Huaiping Zhu. Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1477-1498. doi: 10.3934/mbe.2017077
References:
[1]

S. AidaS. Kusuoka and D. Strook, On the support of Wiener functionals, Longman Scient. Tech., 284 (1993), 3-34.

[2]

T. AlkurdiS. Hille and O. Gaans, Ergodicity and stability of a dynamical system perturbed by impulsive random interventions, J. Math. Anal. Appl., 407 (2013), 480-494. doi: 10.1016/j.jmaa.2013.05.047.

[3]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York-London-Sydney, 1974.

[4]

I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.

[5]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (Ⅱ), Probab. Theory Related Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751.

[6]

A. Freedman, Stochastic differential equations and their applications, Stochastic Differential Equations, 77 (1976), 75-148. doi: 10.1007/978-3-642-11079-5_2.

[7]

S. Foguel, Harris operators, Israel J. Math., 33 (1979), 281-309. doi: 10.1007/BF02762166.

[8]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer-Verlag, New York, 1992. doi: 10.1007/978-94-015-7920-9.

[9]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoof & Noordhoof, Alphen aan den Rijn, The Netherlands, 1980.

[10]

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

[11]

D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027.

[12]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597. doi: 10.1016/j.jmaa.2007.08.014.

[13]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.

[14]

Y. Kuang, Delay differential equations with applications in population dynamics, in Mathematics in Science and Engineering, Academic Press, New York, 1993.

[15]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.

[16]

M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058.

[17]

M. Liu and K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63 (2012), 871-886. doi: 10.1016/j.camwa.2011.11.003.

[18]

Z. Ma and Y. Zhou, Qualitative and Stability Method of Ordinary Differential Equation, Science Press, Beijing, 2001.

[19]

M. MackeyM. Kamińska and R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theoret. Biol., 247 (2011), 84-96. doi: 10.1016/j.jtbi.2011.01.020.

[20]

X. Mao, Stochastic Differential Equations and their Applications, Horwood publishing, Chichester, England, 1997.

[21]

J. Norris, Simplified Malliavin calculus, in SLeminaire de probabilitiLes XX, Lecture Notes in Mathematics, Springer, New York, 1024 (1986), 101–130. doi: 10.1007/BFb0075716.

[22]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997), 56-74. doi: 10.1006/jmaa.1997.5609.

[23]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.

[24]

S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, Springer, Berlin, 205 (2006), 477–517. doi: 10.1007/1-4020-3647-7_11.

[25]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Math., 43 (1995), 245-262.

[26]

J. Yan, On the oscillation of impulsive neutral delay differential equations, Chinese Ann. Math., 21A (2000), 755-762.

show all references

References:
[1]

S. AidaS. Kusuoka and D. Strook, On the support of Wiener functionals, Longman Scient. Tech., 284 (1993), 3-34.

[2]

T. AlkurdiS. Hille and O. Gaans, Ergodicity and stability of a dynamical system perturbed by impulsive random interventions, J. Math. Anal. Appl., 407 (2013), 480-494. doi: 10.1016/j.jmaa.2013.05.047.

[3]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York-London-Sydney, 1974.

[4]

I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.

[5]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (Ⅱ), Probab. Theory Related Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751.

[6]

A. Freedman, Stochastic differential equations and their applications, Stochastic Differential Equations, 77 (1976), 75-148. doi: 10.1007/978-3-642-11079-5_2.

[7]

S. Foguel, Harris operators, Israel J. Math., 33 (1979), 281-309. doi: 10.1007/BF02762166.

[8]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer-Verlag, New York, 1992. doi: 10.1007/978-94-015-7920-9.

[9]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoof & Noordhoof, Alphen aan den Rijn, The Netherlands, 1980.

[10]

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

[11]

D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027.

[12]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597. doi: 10.1016/j.jmaa.2007.08.014.

[13]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.

[14]

Y. Kuang, Delay differential equations with applications in population dynamics, in Mathematics in Science and Engineering, Academic Press, New York, 1993.

[15]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.

[16]

M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058.

[17]

M. Liu and K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63 (2012), 871-886. doi: 10.1016/j.camwa.2011.11.003.

[18]

Z. Ma and Y. Zhou, Qualitative and Stability Method of Ordinary Differential Equation, Science Press, Beijing, 2001.

[19]

M. MackeyM. Kamińska and R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theoret. Biol., 247 (2011), 84-96. doi: 10.1016/j.jtbi.2011.01.020.

[20]

X. Mao, Stochastic Differential Equations and their Applications, Horwood publishing, Chichester, England, 1997.

[21]

J. Norris, Simplified Malliavin calculus, in SLeminaire de probabilitiLes XX, Lecture Notes in Mathematics, Springer, New York, 1024 (1986), 101–130. doi: 10.1007/BFb0075716.

[22]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997), 56-74. doi: 10.1006/jmaa.1997.5609.

[23]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.

[24]

S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, Springer, Berlin, 205 (2006), 477–517. doi: 10.1007/1-4020-3647-7_11.

[25]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Math., 43 (1995), 245-262.

[26]

J. Yan, On the oscillation of impulsive neutral delay differential equations, Chinese Ann. Math., 21A (2000), 755-762.

Figure 1.  Trajectories of impulsive stochastic system (5) with different $\sigma$ and $b_k$. Here $(S(0), x(0))=(1, 1)$ and $E^*(x^*, y^*)=(1.0909, 1.8182)$. System (5) is asymptotically stable
Figure 2.  Probability densities of $x(t)$ for system (5) based on $10000$ sample pathes, computed with the noise intensity $\sigma=0.03$, and for (a) differential initial value and (b) different iterative times. There exists stationary distribution and the density is stable
Figure 3.  The dynamics of the system (5) with different $\sigma$. (a) The population $x(t)$ is extinct with $\sigma=0.8$. (c) The population $x(t)$ is persistent with $\sigma=0.1$
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