Stochastic persistence and stationary distribution in a HollingTanner type preypredator model

Physica A: Statistical Mechanics and its Applications

Volumn 391, Issue 4, 2012, Pages 1216-1233

  Mandal, Partha Sarathi ^a   Banerjee, Malay ^a  

^a INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

Author keywords

BeddingtonDeAngelis functional response; Global solution; It's formula; Persistence in mean; Stability; Stationary distribution

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; POPULATION STATISTICS; STOCHASTIC SYSTEMS;

BEDDINGTON-DE ANGELIS FUNCTIONAL RESPONSE; GLOBAL SOLUTIONS; IT'S FORMULAS; PERSISTENCE IN MEANS; STATIONARY DISTRIBUTION;

STOCHASTIC MODELS;

EID: 84655161375     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2011.10.019     Document Type: Article

References (96)

1. M.L. Rosenzweig Paradox of enrichment: destabilization of exploitation ecosystems in ecological time Science 171 1971 385 387
2. R.M. May Stability and Complexity in Model Ecosystems 2001 Princeton University Press NJ
3. T. Royama A comparative study of models for predation and parasitism Res. Popul. Ecol. 13 1971 1 91
4. R. Arditi, and L.R. Ginzburg Coupling in predatorprey dynamics: ratio-dependence J. Theoret. Biol. 139 1989 311 326
5. S.B. Hsu, and T.W. Hwang Global stability for a class of predatorprey systems SIAM J. Appl. Math. 55 1995 763 783
6. A.F. Nindjin, M.A. Aziz-Alaoui, and M. Cadivel Analysis of a predatorprey model with modified LeslieGower and Holling-type II schemes with time delay Nonlinear Anal. RWA 7 2006 1104 1118 (Pubitemid 43922468)
7. R. Xu, and M.A. Chaplain Persistence and global stability in a delayed predatorprey system with MichaelisMenten type functional response Appl. Math. Comput. 130 2002 441 455 (Pubitemid 34704469)
8. M. Kot Elements of Mathematical Biology 2001 Cambridge University Press Cambridge
9. J.D. Murray Mathematical Biology 1993 Springer-Verlag New York
10. P. Turchin Complex Population Dynamics 2003 Princeton University Press Princeton, NJ
11. H.I. Freedman Deterministic Mathematical Models in Population Ecology 1980 Marcel Dekker New York
12. S. Ruan, and D. Xiao Global analysis in a predatorprey system with nonmonotonic functional response SIAM J. Appl. Math. 61 2001 1445 1472 (Pubitemid 32825187)
13. R.E. Xiao, and S. Ruan Global dynamics of a ratio-dependent predatorprey system J. Math. Biol. 43 2001 268 290 (Pubitemid 33773345)
14. Y. Kuang Global stability of Gause-type predatorprey systems J. Math. Biol. 28 1990 463 474
15. Y. Kuang Nonuniqueness of limit cycles of Gause-type predatorprey systems Appl. Anal. 29 1988 269 287
16. Y. Kuang, and H.I. Freedman Uniqueness of limit cycles in Gause-type models of predatorprey systems Math. Biosci. 88 1988 67 84
17. A.D. Bazykin Nonlinear Dynamics of Interacting Populations 1998 World Scientific Singapore
18. H. Malchow, S.V. Petrovskii, and E. Venturino Spatiotemporal Pattern in Ecology and Epidemiology 2008 Chapman and Hall New York
19. P.H. Leslie, and J.C. Gower The properties of a stochastic model for the predatorprey type of interaction between two species Biometrika 47 1960 219 234
20. M.A. Aziz-Alaoui, and M. Daher-Okiye Boundedness and global stability for a predatorprey model with modified LeslieGower and Holling-type II schemes Appl. Math. Lett. 16 2003 1069 1075
21. A. Korobeinikov A Lyapunov function for LeslieGower predatorprey model Appl. Math. Lett. 14 2001 697 699 (Pubitemid 33675396)
22. E. Saez, and E. Gonzalez-Olivares Dynamics of predatorprey model SIAM J. Appl. Math. 59 1999 1867 1878
23. A. Gasull, R.E. Kooij, and J. Torregrosa Limit cycles in the HollingTanner model Publ. Mat. 41 1997 149 167
24. P.H. Leslie Some further notes on the use of matrices in population mathematics Biometrika 35 1948 213 245
25. E.C. Pielou An Introduction to Mathematical Ecology 1969 Wiley Interscience New York
26. Z. Liang, and H. Pan Qualitative analysis of a ratio-dependent HollingTanner model J. Math. Anal. Appl. 334 2007 954 964
27. C. Jost, O. Arino, and R. Arditi About deterministic extinction in ratio-dependent predatorprey model Bull. Math. Biol. 61 1999 19 32 (Pubitemid 29091080)
28. M. Bandyopadhyay, and J. Chattopadhyay Ratio-dependent predatorprey model: effect of environmental fluctuation and stability Nonlinearity 18 2005 913 936 (Pubitemid 40344893)
29. R. Arditi, and H. Saiah Empirical evidence of the role of heterogeneity in ratio-dependent consumption Ecology 73 1992 1544 1551 (Pubitemid 23396425)
30. R. Arditi, L.R. Ginzburg, and H.R. Akcakaya Variation in plankton densities among lakes: a case for ratio-dependent models Am. Nat. 138 1991 1287 1296
31. A.P. Gutierrez The physiological basis of ratio-dependent predatorprey theory: a metabolic pool model of Nicholson's blowflies as an example Ecology 73 1992 1552 1563 (Pubitemid 23397217)
32. T. Saha, and C.G. Chakrabarti Dynamical analysis of a delayed ratio-dependent HollingTanner predatorprey model J. Math. Anal. Appl. 358 2009 389 402
33. C.Y. Ji, D.Q. Jiang, and N.Z. Shi Analysis of a predatorprey moodel with moified LeslieGower and Holling-type II schemes with stochastic perturbation J. Math. Anal. Appl. 359 2009 482 498
34. M.A. Aziz-Alaoui Study of a LeslieGower type tritrophic population model Chaos Solitons Fractals 14 2002 1275 1293 (Pubitemid 34863130)
35. J.R. Beddington Mutual interference between parasites or predators and it's effect on searching efficiency J. Anim. Ecol. 44 1975 331 340
36. C. Cosner, D.L. Deangelis, J.S. Ault, and D.B. Olson Effects of spatial grouping on the functional response of predators Theor. Popul. Biol. 56 1999 65 75 (Pubitemid 29395865)
37. R.S. Cantrell, and C. Cosner On the dynamics of predatorprey models with the BeddingtonDeAngelis functional response J. Math. Anal. Appl. 257 2001 206 222 (Pubitemid 33642416)
38. D.L. DeAngelis, R.A. Goldstein, and R.V. ONeil A model for trophic inter-action Ecology 56 1975 881 892
39. S. Rinaldi, S. Muratori, and Y. Kuznetsov Multiple attractors, catastrophes and chaos in seasonally perturbed predatorprey communities Bull. Math. Biol. 55 1993 15 35
40. F. Doveri, M. Scheffer, S. Rinaldi, S. Moratori, and Y. Kuznetsov Seasonality and chaos in a plankton-fish model Theor. Popul. Biol. 43 1993 159 183 (Pubitemid 24429396)
41. G.C.W. Sabin, and D. Summers Chaos in a periodically forced predatorprey ecosystem model Math. Biosci. 113 1993 91 113
42. J. Vandermeer, L. Stone, and B. Blasius Categories of chaos and fractal basin boundaries in forced predatorprey models Chaos Solitons Fractals 12 2001 265 276 (Pubitemid 32025246)
43. S. Gakkhar, and B. Singh Dynamics of modified LeslieGower-type preypredator model with seasonally varying parameters Chaos Solitons Fractals 27 2006 1239 1255 (Pubitemid 41329817)
44. A. Gragnani, and S. Rinaldi Dynamics of modified LeslieGower-type preypredator model with seasonally varying parameters Bull. Math. Biol. 57 1995 701 712
45. J. Gani, and R.J. Swift An unexpected result in an approximate carrier-borne epidemic process Statist. Probab. Lett. 78 2008 2116 2120
46. E.J. Allen, L.J.S. Allen, and A. Arciniega Construction of equivalent stochastic differential equation models Stoch. Anal. Appl. 26 2008 274 297
47. L.J.S. Allen, and E.J. Allen A comparison of three different stochastic population models with regard to persistence time Theor. Popul. Biol. 64 2003 439 449 (Pubitemid 38042741)
48. A.S. Ackleh, L.J.S. Allen, and J. Carter Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion Theor. Popul. Biol. 71 2007 290 300 (Pubitemid 46528492)
49. E. Allen, Modeling with It Stochastic Differential Equations, Dordrecht, The Netherlands, 2007.
50. L.J.S. Allen An Introduction to Stochastic Processes with Applications to Biology 2003 Pearson Education Inc. New Jersey
51. T.G. Hallam, L.J. Svoboda, and T.C. Gard Persistence and extinction in three species LotkaVolterra competitive systems Math. Biosci. 46 1979 117 124 (Pubitemid 10221396)
52. M. Abundo A stochastic model for predatorprey systems: basic properties, stability and computer simulation J. Math. Biol. 29 1991 495 511
53. R. Mankin, T. Laas, A. Sauga, and A. Ainsaar Colored-noise-induced Hopf bifurcations in predatorprey communities Phys. Rev. E 74 2006
54. R. Mankin, T. Laasa, E. Soika, and A. Ainsaar Noise-controlled slowfast oscillations in predatorprey models with the Beddington functional response Eur. Phys. J. B 59 2007 259 269 (Pubitemid 47599030)
55. A. Maiti, and G.P. Samanta Deterministic and stochastic analysis of a prey-dependent predatorprey system Internat. J. Math. Ed. Sci. Tech. 36 2005 65 83 (Pubitemid 41654051)
56. A. Bahar, and X. Mao Stochastic delay LotkaVolterra model J. Math. Anal. Appl. 292 2004 364 380
57. X. Mao, S. Sabanis, and E. Renshaw Asymptotic behaviour of the stochastic LotkaVolterra model J. Math. Anal. Appl. 287 2003 141 156 (Pubitemid 37374125)
58. M. Bandyopadhyay, and C.G. Chakrabarti Deterministic and stochastic analysis of a nonlinear preypredator system J. Biol. Syst. 11 2003 161 172 (Pubitemid 37284043)
59. M. Carletti On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment Math. Biosci. 175 2002 117 131 (Pubitemid 34171162)
60. V.B. Kolmanovskii, and L.E. Shaikhet Some peculiarities of the general method of Lyapunov functionals construction Appl. Math. Lett. 15 2002 355 360
61. V.B. Kolmanovskii, and L.E. Shaikhet Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results Math. Comput. Modelling 36 2002 691 716
62. V.B. Kolmanovskii, and L.E. Shaikhet Control of Systems with Aftereffect Trans. Math. Monog. 1996 American Mathematical Society Providence, RI
63. M.C. Baishya, and C.G. Chakrabarti Non-equilibrium fluctuation in LotkaVolterra system Bull. Math. Biol. 49 1987 125 131
64. S. Banerjee, and C.G. Chakrabarti Stochastic dynamic modeling of damped LotkaVolterra system Syst. Anal. Modelling Simul. 30 1998 1 10
65. P.K. Tapaswi, and A. Mukhopadhyay Effects of environmental fluctuation on plankton allelopathy J. Math. Biol. 39 1999 39 58
66. S. Abbas, and M. Banerjee Stochastically perturbed allelopathic phytoplankton model Electron. J. Differential Equations 98 2010 1 15
67. R.M. Nisbet, and W.S.C. Gurney Modelling Fluctuating Populations 1982 Wiley Interscience New York
68. G.Q. Cai, and Y.K. Lin Stochastic analysis of predatorprey type ecosystems Ecol. Complex. 4 2007 242 249 (Pubitemid 47419445)
69. C.Y. Ji, D.Q. Jiang, and X.Y. Li Qualitative analysis of a stochastic ratio-dependent predatorprey system J. Comput. Appl. Math. 235 2011 1326 1341
70. C.Y. Ji, D.Q. Jiang, and N.Z. Shi A note on a predatorprey model with modified LeslieGower and Holling-type II schemes with stochastic perturbation J. Math. Anal. Appl. 377 2011 435 440
71. C.Y. Ji, and D.Q. Jiang Dynamics of a stochastic density dependent predatorprey system with BeddingtonDeAngelis functional response J. Math. Anal. Appl. 381 2011 441 453
72. G.P. Samanta Influence of environmental noises on the Gomatam model of interacting species Ecol. Model. 91 1996 283 291
73. M. Bandyopadhyay Gomatam model of interacting species: stochastic analysis of fluctuation and stability Bull. Calcutta Math. Soc. 97 3 2005 253 262
74. W. Khellaf, and N. Hamri Boundedness and global stability for a predatorprey system with the BeddingtonDeAngelis functional response Differ. Equ. Nonlinear Mech. 2010 10.1155/2010/813289
75. X. Mao, G. Marion, and E. Renshaw Environmental Brownian noise suppresses explosions in population dynamics Stochastic Process. Appl. 97 2002 95 110 (Pubitemid 40030718)
76. W. Horsthemke, and R. Lefever Noise Induced Transitions 1984 Springer-Verlag Berlin
77. T.C. Gard Introduction to Stochastic Differential Equations 1987 Marcel Dekker New York
78. C.W. Gardiner Handbook of Stochastic Methods 1983 Springer-Verlag New York
79. D.J. Higham An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Rev. 43 2001 525 546 (Pubitemid 32991214)
80. B. Spagnolo, and A.L. Barbera Role of the noise on the transient dynamics of an ecosystem of interacting species Physica A 315 2002 114 124 (Pubitemid 35412052)
81. B. Spagnolo, D. Valenti, and A. Fiasconaro Noise in ecosystems: a short review Math. Biosci. Eng. 1 2004 185 211
82. A. Friedman Stochastic Differential Equations and their Applications 1976 Academic Press New York
83. X. Mao Stochastic Differential Equations and Applications 1997 Horwood New York
84. L. Arnold Stochastic Differential Equations: Theory and Applications 1972 Wiley New York
85. N. Dalal, D. Greenhalgh, and X. Mao A stochastic model for internal HIV dynamics J. Math. Anal. Appl. 341 2 2008 1084 1101
86. V. Hutson, and J.S. Pym Applications of Functional Analysis and Operator Theory 1980 Academic Press London
87. A.N. Kolmogorov, and S.V. Fomin Introductory Real Analysis 1970 Dover Publications, Inc. New York
88. M. Liu, K. Wang, and Q. Wu Survival analysis of stochastic competitive models in a polluted envioronment and stochastic competitive exclusion principle Bull. Math. Biol. 2010 10.1007/s11538-010-9569-5
89. R.Z. Hasminskii Stochastic stability in differential equations Mechanics and Analysis Monogr. Textb. Mech. Solids Fluids vol. 7 1980 Sijthoff & Noordhoff Alphen aan den Rijn, The Netherlands
90. G. Strang Linear Algebra and its Applications 1988 Thomson Learning, Inc.
91. C. Zhu, and G. Yin Asymptotic properties of hybrid diffusion systems SIAM J. Control Optim. 46 2007 1155 1179
92. P.E. Kloeden, and E. Platen Numerical Solution of Stochastic Differential Equations 1999 Springer Berlin
93. R. Mankin, A. Sauga, A. Ainsaar, A. Haljas, and K. Paunel Colored-noise-induced discontinuous transitions in symbiotic ecosystems Phys. Rev. E 69 2004 061106
94. R. Mankin, A. Ainsaar, A. Haljas, and E. Reiter Trichotomous-noise- induced catastrophic shifts in symbiotic ecosystems Phys. Rev. E 65 2002 051108
95. A. d'Onofrio Bounded-noise-induced transitions in a tumor-immune system interplay Phys. Rev. E 81 2010 021923
96. G.Q. Cai, and Y. Suzuki Response of systems under non-Gaussian random excitations Nonlinear Dynam. 45 2005 95 108