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The Levins model subjected to the noise is employed to study the stability of a metapopulation. The analytic expressions of the stationary probability distribution function and the mean extinction time of the metapopulation are obtained according to the Fokker-Planck Equation. The results show that for the case of no correlation between the additive noise and the multiplicative noise (=0, is the intensity of correlation between multiplicative and additive noise), the increase of the additive noise intensity weakens the stability of a metapopulation; for the case of 0, enhances the stability of a metapopulation. For -(c-e-D)2/(4cD)1, can induce the resonance restrain phenomenon. Meantime, there exists a critical value of D. When D is lower than the critical value, the stability of the system is enhanced.
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Keywords:
- metapopulation /
- Levins model /
- noises /
- stability
[1] Levins R 1969 Bull. Entomol. Soc. Am. 15 237
[2] Levins R 1970 Lect. Notes. Math. 2 75
[3] Gilpin M E, Hanski I 1991 Metapopulation Dynamics:Empirical and Theoretical Investigations (London: Academic Press) p336
[4] Hanski I, Pakkala T, Kuussaari M, Lei G 1995 Oikos 72 21
[5] Moilanen A, Hanski I 1998 Ecology 79 2503.
[6] Hastings A, Harrison S 1994 Ann. Rev. Ecol. Syst. 25 167
[7] Harrison S 1991 Biol. J. Linn. Soc. 42 73
[8] Ovaskainen O, Sato K, Bascompte J, Hanski I 2002 J. Theor. Biol. 215 95
[9] Bascompte J 2001 J. Theor. Biol. 209 373
[10] Hanski I A 1994 J. Animal Ecol. 63 151
[11] Jung P, Hänggi P 1989 Europhys. Lett. 8 505
[12] Jia Y, Zhang X P, Hu X M and Li J R 2001 Phys. Rev. E 63 031107
[13] Wei X Q, Cao L, Wu D J 1995 Phys. Lett. A 207 338
[14] Wu D J, Cao L, Ke S Z 1994 Phys. Rev. E 50 2496
[15] Mei D C, Xie G Z,Cao L, Wu D J 1999 Phys. Rev. E 59 3880
[16] Mei D C, Xie G Z, Zhang L 2004 Eur. Phys. J. B 41 107
[17] Ai B Q, Wang X J, Liu G T, Liu L G 2003 Phys. Rev. E 67 22903
[18] Cai J C, Wang C J, Mei D C 2007 Chin. Phys. Lett. 24 1162
[19] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 61 010503]
[20] Wang C J 2012 Acta Phys. Sin. 61 050501 (in Chinese) [王参军 2012 61 050501]
[21] yJia Y, Li J R 1997 Phys. Rev. Lett. 78 994
[22] Vilar J M G, Solé R V 1998 Phys. Rev. Lett. 80 4099
[23] Wang C J, Wei Q, Zheng B B, Mei D C 2008 Acta Phys. Sin. 57 1375(in chinese) [王参军, 魏群, 郑宝兵, 梅冬成 2008 57 1375]
[24] Zhu S 1993 Phys. Rev. A 47 2405
[25] Zhang L Y, Cao L, Wu D J 2002 Chin. Phys. 13 353
[26] Xie C W, Mei D C 2004 Phys. Lett. A 323 421
[27] Cao L, Wu D J 1999 Phys. Lett. A 260 126
[28] Li J C, Mei D C 2008 Acta Phys. Sin. 57 (in chinese)6792[李江城, 梅冬成 2008 57 6792]
[29] Hofbauer J, Sigmund K 1998 Evolutionary Games and Population Dynamics (Cambridge: Cambridge University Press )p91
[30] May R M 1976 Nature 261 459
[31] Nie L R, Mei D C 2007 Europhys. Lett. 79 20005
[32] Li J H, Huang Z Q 1998 Phys. Rev. E 57 3917
[33] Nie L R, Mei D C 2007 Phys. Lett. A 371 111
[34] Mei D C, Xie G Z, Cao L and Wu D J 1999 Chin. Phys. Lett. 16 327
[35] Xie C W, Mei D C 2003 Chin. Phys. Lett. 20 813
[36] Bak P, Sneppen K 1993 Phys. Rev. Lett. 71 4083
[37] Ovaskainen O, Hanski I 2001 Theor. Popul. Biol. 60 281
[38] Lindenberg K, West B J 1986 J. Stat. Phys. 42 201
[39] Masoliver J, West B J, Lindenberg K 1987 Phys. Rev. A 35 3086
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[1] Levins R 1969 Bull. Entomol. Soc. Am. 15 237
[2] Levins R 1970 Lect. Notes. Math. 2 75
[3] Gilpin M E, Hanski I 1991 Metapopulation Dynamics:Empirical and Theoretical Investigations (London: Academic Press) p336
[4] Hanski I, Pakkala T, Kuussaari M, Lei G 1995 Oikos 72 21
[5] Moilanen A, Hanski I 1998 Ecology 79 2503.
[6] Hastings A, Harrison S 1994 Ann. Rev. Ecol. Syst. 25 167
[7] Harrison S 1991 Biol. J. Linn. Soc. 42 73
[8] Ovaskainen O, Sato K, Bascompte J, Hanski I 2002 J. Theor. Biol. 215 95
[9] Bascompte J 2001 J. Theor. Biol. 209 373
[10] Hanski I A 1994 J. Animal Ecol. 63 151
[11] Jung P, Hänggi P 1989 Europhys. Lett. 8 505
[12] Jia Y, Zhang X P, Hu X M and Li J R 2001 Phys. Rev. E 63 031107
[13] Wei X Q, Cao L, Wu D J 1995 Phys. Lett. A 207 338
[14] Wu D J, Cao L, Ke S Z 1994 Phys. Rev. E 50 2496
[15] Mei D C, Xie G Z,Cao L, Wu D J 1999 Phys. Rev. E 59 3880
[16] Mei D C, Xie G Z, Zhang L 2004 Eur. Phys. J. B 41 107
[17] Ai B Q, Wang X J, Liu G T, Liu L G 2003 Phys. Rev. E 67 22903
[18] Cai J C, Wang C J, Mei D C 2007 Chin. Phys. Lett. 24 1162
[19] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 61 010503]
[20] Wang C J 2012 Acta Phys. Sin. 61 050501 (in Chinese) [王参军 2012 61 050501]
[21] yJia Y, Li J R 1997 Phys. Rev. Lett. 78 994
[22] Vilar J M G, Solé R V 1998 Phys. Rev. Lett. 80 4099
[23] Wang C J, Wei Q, Zheng B B, Mei D C 2008 Acta Phys. Sin. 57 1375(in chinese) [王参军, 魏群, 郑宝兵, 梅冬成 2008 57 1375]
[24] Zhu S 1993 Phys. Rev. A 47 2405
[25] Zhang L Y, Cao L, Wu D J 2002 Chin. Phys. 13 353
[26] Xie C W, Mei D C 2004 Phys. Lett. A 323 421
[27] Cao L, Wu D J 1999 Phys. Lett. A 260 126
[28] Li J C, Mei D C 2008 Acta Phys. Sin. 57 (in chinese)6792[李江城, 梅冬成 2008 57 6792]
[29] Hofbauer J, Sigmund K 1998 Evolutionary Games and Population Dynamics (Cambridge: Cambridge University Press )p91
[30] May R M 1976 Nature 261 459
[31] Nie L R, Mei D C 2007 Europhys. Lett. 79 20005
[32] Li J H, Huang Z Q 1998 Phys. Rev. E 57 3917
[33] Nie L R, Mei D C 2007 Phys. Lett. A 371 111
[34] Mei D C, Xie G Z, Cao L and Wu D J 1999 Chin. Phys. Lett. 16 327
[35] Xie C W, Mei D C 2003 Chin. Phys. Lett. 20 813
[36] Bak P, Sneppen K 1993 Phys. Rev. Lett. 71 4083
[37] Ovaskainen O, Hanski I 2001 Theor. Popul. Biol. 60 281
[38] Lindenberg K, West B J 1986 J. Stat. Phys. 42 201
[39] Masoliver J, West B J, Lindenberg K 1987 Phys. Rev. A 35 3086
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