XU Wei, QI Lu-yuan, GAO Wei-ting. Effects of Noises and Habitat Complexity in the Prey-Predator Ecosystem[J]. Applied Mathematics and Mechanics, 2013, 34(2): 162-171. doi: 10.3879/j.issn.1000-0887.2013.02.006
 Citation: XU Wei, QI Lu-yuan, GAO Wei-ting. Effects of Noises and Habitat Complexity in the Prey-Predator Ecosystem[J]. Applied Mathematics and Mechanics, 2013, 34(2): 162-171.

# Effects of Noises and Habitat Complexity in the Prey-Predator Ecosystem

##### doi: 10.3879/j.issn.1000-0887.2013.02.006
• Rev Recd Date: 2013-01-31
• Publish Date: 2013-02-15
• To explore the habitat complexity and random environmental factors effects to the ecosystem, a new stochastic predation type ecosystem with habitat complexity was proposed. Both theoretical analysis and numerical analysis were explored. With the assumption of weak disturbances, the stationary probability density functions (PDF) for both species were obtained by applying the StratonovichKhasminskii averaging principle. The accuracy of the results obtained from theoretical method was demonstrated by those obtained from Monte Carlo simulation (MCS). The system mean first passage time(MFPT) was solved from the Pontryagin equation. The effects of the habitat complexity and the noise intensity were investigated via numerical calculations based on the data obtained from experiment. Results obtained show that: 1) the ecosystem with smaller habitat complexity is less stable when the system is disturbed by noises; 2) the stronger the noise intensities are, the less stable the ecosystem will be; 3) the ecosystem with bigger habitat complexity has longer MFPT; 4) the noise added to the prey natural increase rate has a bigger effect on the MFPT than that added to the predator natural death rate does.
•  [1] Lotka A J. Elements of Physical Biology [M].Baltimore: Williams and Wilkins Press, 1925. [2] Volterra V.Varizaioni e fluttuazioni del numero d’individui in specie d’animani conviventi[J]. Mem Acad Lincei,1926, 2: 31-113. [3] LI Li, ZHEN Jin.Pattern dynamics of a spatial predator-prey model with noise[J]. Nonlinear Dyn,2012, 67(3):1737-1744. [4] Haynes K J, Liebhold A M, Johnson D M.Elevational gradient in the cyclicity of a forest-defoliation insect[J]. Popul Ecol,2012, 54(2):239-250. [5] LI Anwei.Impact of noise on pattern formation in predator-prey model[J]. Nonlinear Dyn,2011, 66(4): 689-694. [6] Osuyama T.Behavioural states of predators stablize predatorprey dynamics[J].Theor Ecol,2012, 5(4): 605-610. [7] Sieber M, Hilker F M.The hydra effect in predatorprey models[J].J Math Biol,2012, 64(1): 341-360. [8] Cai G Q, Lin Y K.Stochastic modeling of ecosystem with two competing species[J]. Probabilistic Eng Mech,2012, 27(1):2-7. [9] May R M. Stability and Complexity in Model Ecosystems [M].Princeton: Princeton University Press, 1973. [10] Wu Y, Zhu W Q.Stochastic analysis of a pulse-type prey-predator model[J]. Phys Rev E,2008, 77: 041911. [11] Cai G Q, Lin Y K.Stochastic analysis of the LotkaVolterra model for ecosystems[J]. Phys Rev E,2004, 70: 041910. [12] Cai G Q, Lin Y K.Stochastic analysis of predatorprey type ecosystems[J]. Ecol Complex,2007, 4(4): 242-249. [13] Nelson W G, Bonsdorff E.Fish predation and habitat complexity: are complexity thresholds real?[J]. J Exp Mar Biol Ecol, 1990, 141(2/3): 183-194. [14] Savino J F, Stein R A.Behavioral interactions between fish predators and their prey: effects of plant density[J]. Animal Behavior, 1986, 37(2): 311-321. [15] Johnson D W.Predation, habitat complexity and variation in density dependent mortality of temperate reef fishes[J]. Ecology, 2006, 87(5): 1179-1188. [16] Christensen B, Persson L.Speciesspecific antipredatory behaviours: effects on prey choice in different habitats[J]. Behav Ecol Sociobiol, 1993, 32(1): 1-9. [17] Weis J S, Candelmo A.Pollutants and fish predator/prey behavious: a review of laboratory and field approaches[J]. Current Zoology,2012, 58(1): 9-20. [18] Bairagi N, Jana D.On the stability and Hopf bifurcation of a delayinduced predator-prey system with habitat complexity[J]. Appl Math Model, 2011, 35(7): 3255-3267. [19] Khasminskii R Z, Klebaner F C.Long term behavior of solutions of the LotkaVolterra system under small random perturbations[J]. Ann Appl Probab, 2001, 11(3) :952-963. [20] Holling C S.Some characteristics of simple types of predation and parasitism[J]. Canadian Entomologist, 1959, 91(7): 385-398. [21] 杨万利, 王铁宁.非线性动力学理论方法及应用[M].北京: 国防工业出版社, 2007.(YANG Wan-li, WANG Tie-ning. Nonlinear Dynamics Theory and Application [M].Beijing: Defense Industry Press, 2007.(in Chinese)) [22] Kuang Y, Freedman H I.Uniqueness of limit cycles in Gause-type models of predatorprey systems[J]. Math Biosci, 1988, 88(1): 67-84. [23] It K.On stochastic differential equations[J]. Memories Am Math Sco, 1951, 4: 289-302. [24] Lin Y K, Cai G Q. Probabilistic Structural Dynamics: Advanced Theory and Applications [M].New York: McGrawHill Press, 2004:127190; 419-465.

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