American Institute of Mathematical Sciences

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March  2015, 20(2): 683-701. doi: 10.3934/dcdsb.2015.20.683

Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps

Xiaoling Zou 1, and  Ke Wang 2,

1. 

Department of Mathematics, Harbin Institute of Technology, Weihai 264209
2. 

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209

Received  January 2014 Revised  August 2014 Published  January 2015

Optimization problem for a stochastic N-dimensional competitive Lotka-Volterra system is studied in this paper. The considered system is driven by both white noise and jumping noise, and the jumping noise is modeled by a stochastic integral with respect to a Poisson counting measure generated by a Poisson point process. For two types of objective functions, namely, time-averaged yield and sustained yield, the optimal harvesting efforts as well as the corresponding maximum yields are given respectively. Moreover, almost sure equivalence between these two objective functions is proved by ergodic method. This paper provides us a new idea to study the stochastic optimal harvesting problem with sustained yield, and this idea can be popularized to other stochastic systems.
Keywords: Brownian motion, sustained yield., Fokker-Planck equation, time-averaged yield, jumping noise, stationary probability density.
Mathematics Subject Classification: Primary: 60H10, 93E20; Secondary: 92B05, 37N2.
Citation: Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683
References:
[1]

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[2]

V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments,, Springer-Verlag, (2007).   Google Scholar

[3]

D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

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L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).   Google Scholar

[5]

L. Arnold, Random Dynamical Systems,, Springer, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

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J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

[7]

J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise,, J. Math. Anal. Appl., 391 (2012), 363.  doi: 10.1016/j.jmaa.2012.02.043.  Google Scholar

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J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463.  doi: 10.1126/science.197.4302.463.  Google Scholar

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J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).   Google Scholar

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C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets,, Physica A., 387 (2008), 3837.  doi: 10.1016/j.physa.2008.01.078.  Google Scholar

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H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999).  doi: 10.1007/b97846.  Google Scholar

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D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation,, J. Math. Anal. Appl., 390 (2012), 582.  doi: 10.1016/j.jmaa.2011.12.049.  Google Scholar

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W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model,, Appl. Math. Comput., 218 (2011), 157.  doi: 10.1016/j.amc.2011.05.079.  Google Scholar

[22]

X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching,, J. Comput. Appl. Math., 232 (2009), 427.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

[23]

X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation,, Discret. Contin. Dyn. S., 24 (2009), 523.  doi: 10.3934/dcds.2009.24.523.  Google Scholar

[24]

R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.  doi: 10.1080/17442508008833146.  Google Scholar

[25]

A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).   Google Scholar

[26]

E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Math. Biosci., 145 (1997), 47.  doi: 10.1016/S0025-5564(97)00029-1.  Google Scholar

[27]

X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).  doi: 10.1533/9780857099402.  Google Scholar

[28]

X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

[29]

X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics,, Stochastic Process. Appl., 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[30]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model,, J. Math. Anal. Appl., 287 (2003), 141.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

[31]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[32]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001).   Google Scholar

[33]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[34]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations,, Stochastic Process. Appl., 116 (2006), 370.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar

[35]

D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[36]

M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment,, Appl. Stochastic Models Bus. Ind., 19 (2003), 43.  doi: 10.1002/asmb.490.  Google Scholar

[37]

R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering,, Springer, (2005).   Google Scholar

[38]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.   Google Scholar

[39]

K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).   Google Scholar

[40]

C. Zhu and G. Yin, On competitive lotka-volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

[41]

C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[42]

X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps,, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557.  doi: 10.1016/j.cnsns.2013.09.010.  Google Scholar

show all references

References:
[1]

L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation,, Math Biosci., 152 (1998), 63.  doi: 10.1016/S0025-5564(98)10018-4.  Google Scholar

[2]

V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments,, Springer-Verlag, (2007).   Google Scholar

[3]

D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[4]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).   Google Scholar

[5]

L. Arnold, Random Dynamical Systems,, Springer, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

[6]

J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

[7]

J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise,, J. Math. Anal. Appl., 391 (2012), 363.  doi: 10.1016/j.jmaa.2012.02.043.  Google Scholar

[8]

I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires,, Rev. Math. Pures. Appl., 4 (1959), 267.   Google Scholar

[9]

J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463.  doi: 10.1126/science.197.4302.463.  Google Scholar

[10]

J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).   Google Scholar

[11]

C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets,, Physica A., 387 (2008), 3837.  doi: 10.1016/j.physa.2008.01.078.  Google Scholar

[12]

C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources,, John Wiley and Sons Inc., (1990).   Google Scholar

[13]

H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999).  doi: 10.1007/b97846.  Google Scholar

[14]

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

[15]

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

[16]

G. Hu and K. Wang, Stability in distribution of competitive lotka-volterra system with markovian switching,, Appl. Math. Model., 35 (2011), 3189.  doi: 10.1016/j.apm.2010.12.025.  Google Scholar

[17]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Amsterdam, (1981).   Google Scholar

[18]

D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation,, J. Math. Anal. Appl., 390 (2012), 582.  doi: 10.1016/j.jmaa.2011.12.049.  Google Scholar

[19]

F. C. Klebaner, Introduction to Stochastic Calculus With Applications,, Imperial College Press, (2005).  doi: 10.1142/p386.  Google Scholar

[20]

H. Kunita, Itô's stochastic calculus: Its surprising power for applications,, Stochastic Process. Appl., 120 (2010), 622.  doi: 10.1016/j.spa.2010.01.013.  Google Scholar

[21]

W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model,, Appl. Math. Comput., 218 (2011), 157.  doi: 10.1016/j.amc.2011.05.079.  Google Scholar

[22]

X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching,, J. Comput. Appl. Math., 232 (2009), 427.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

[23]

X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation,, Discret. Contin. Dyn. S., 24 (2009), 523.  doi: 10.3934/dcds.2009.24.523.  Google Scholar

[24]

R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.  doi: 10.1080/17442508008833146.  Google Scholar

[25]

A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).   Google Scholar

[26]

E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Math. Biosci., 145 (1997), 47.  doi: 10.1016/S0025-5564(97)00029-1.  Google Scholar

[27]

X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).  doi: 10.1533/9780857099402.  Google Scholar

[28]

X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

[29]

X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics,, Stochastic Process. Appl., 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[30]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model,, J. Math. Anal. Appl., 287 (2003), 141.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

[31]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[32]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001).   Google Scholar

[33]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[34]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations,, Stochastic Process. Appl., 116 (2006), 370.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar

[35]

D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[36]

M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment,, Appl. Stochastic Models Bus. Ind., 19 (2003), 43.  doi: 10.1002/asmb.490.  Google Scholar

[37]

R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering,, Springer, (2005).   Google Scholar

[38]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.   Google Scholar

[39]

K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).   Google Scholar

[40]

C. Zhu and G. Yin, On competitive lotka-volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

[41]

C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[42]

X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps,, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557.  doi: 10.1016/j.cnsns.2013.09.010.  Google Scholar

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