zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. (English) Zbl 1225.92059
Summary: Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.
MSC:
92D40Ecology
60H10Stochastic ordinary differential equations
60H30Applications of stochastic analysis
65C20Models (numerical methods)
References:
[1]Alonso, D., Pascual, M., & McKane, J. A. (2007). Stochastic Amplification in Epidemics. J. R. Soc. Interface, 4, 575–582. · doi:10.1098/rsif.2006.0192
[2]Arnold, L. (1974). Stochastic differential equations: theory and applications. Wiley: New York.
[3]Bahar, A., & Mao, X. (2004). Stochastic delay Lotka–Volterra model. J. Math. Anal. Appl., 292, 364–380. · Zbl 1043.92034 · doi:10.1016/j.jmaa.2003.12.004
[4]Bandyopadhyay, M., & Chattopadhyay, J. (2005). Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity, 18, 913–936. · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022
[5]Beddington, J. R., & May, R. M. (1977). Harvesting natural populations in a randomly fluctuating environment. Science, 197, 463–465. · doi:10.1126/science.197.4302.463
[6]Braumann, C. A. (2002). Variable effort harvesting models in random environments: generalization to density-dependent noise intensities. Math. Biosci., 177& 178, 229–245. · Zbl 1003.92027 · doi:10.1016/S0025-5564(01)00110-9
[7]Braumann, C. A. (2007). Itô versus Stratonovich calculus in random population growth. Math. Biosci., 206, 81–107. · Zbl 1124.92039 · doi:10.1016/j.mbs.2004.09.002
[8]Braumann, C. A. (2008). Growth and extinction of populations in randomly varying environments. Comput. Math. Appl., 56, 631–644. · Zbl 1155.92347 · doi:10.1016/j.camwa.2008.01.006
[9]Cattiaux, P. Méléard, S. (2009). Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol., doi: 10.1007/s00285-009-0285-4 .
[10]Chattopadhyay, J. (1996). Effect of toxic substances on a two species competitive system. Ecol. Model., 84, 287–289. · doi:10.1016/0304-3800(94)00134-0
[11]Du, N. H., & Sam, V. H. (2006). Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. Appl., 324, 82–97. · Zbl 1107.92038 · doi:10.1016/j.jmaa.2005.11.064
[12]Freedman, H. I., & Shukla, J. B. (1991). Models for the effect of toxicant in single-species and predator-prey systems. J. Math. Biol., 30, 15–30. · Zbl 0825.92125 · doi:10.1007/BF00168004
[13]Gard, T. C. (1984). Persistence in stochastic food web models. Bull. Math. Biol., 46, 357–370.
[14]Gard, T. C. (1986). Stability for multispecies population models in random environments. Nonlinear Anal., 10, 1411–1419. · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[15]Gard, T. C. (1988). Introductions to stochastic differential equations. New York.
[16]Gard, T. C. (1992). Stochastic models for toxicant-stressed populations. Bull. Math. Biol., 54, 827–837.
[17]Gillespiea, D. T. (2000). The chemical Langevin equation. J. Chem. Phys., 113, 297–306. · doi:10.1063/1.481811
[18]Hallam, T. G., Clark, C. E., & Lassider, R. R. (1983a). Effects of toxicant on population: a qualitative approach I. Equilibrium environmental exposure. Ecol. Model., 8, 291–304. · doi:10.1016/0304-3800(83)90019-4
[19]Hallam, T. G., Clark, C. E., & Jordan, G. S. (1983b). Effects of toxicant on population: a qualitative approach II. First Order Kinetics. J. Math. Biol., 18, 25–37. · Zbl 0548.92019 · doi:10.1007/BF00275908
[20]Hallam, T. G., & Deluna, J. L. (1984). Effects of toxicant on populations: a qualitative approach III. Environmental and food chain pathways. J. Theor. Biol., 109, 411–429. · doi:10.1016/S0022-5193(84)80090-9
[21]Hallam, T. G., & Ma, Z. (1986). Persistence in population models with demographic fluctuations. J. Math. Biol., 24, 327–339. · Zbl 0606.92022 · doi:10.1007/BF00275641
[22]Hallam, T. G., & Ma, Z. (1987). On density and extinction in continuous population model. J. Math. Biol., 25, 191–201. · Zbl 0641.92011 · doi:10.1007/BF00276389
[23]Hardin, G. (1960). The competitive exclusion principle. Science, 131, 1292–1297. · doi:10.1126/science.131.3409.1292
[24]He, J., & Wang, K. (2007). The survival analysis for a single-species population model in a polluted environment. Appl. Math. Model., 31, 2227–2238. · Zbl 1128.92046 · doi:10.1016/j.apm.2006.08.017
[25]He, J., & Wang, K. (2009). The survival analysis for a population in a polluted environment. Nonlinear Anal. Real World Appl., 10, 1555–1571. · Zbl 1160.92041 · doi:10.1016/j.nonrwa.2008.01.027
[26]Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43, 525–546. · Zbl 0979.65007 · doi:10.1137/S0036144500378302
[27]Hsu, S. B., Luo, T. K., & Waltman, P. (1995). Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitorm. J. Math. Biol., 34, 225–238. · Zbl 0835.92028 · doi:10.1007/BF00178774
[28]Ikeda, N., & Watanabe, S. (1977). A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 619–633.
[29]Jensen, A. L., & Marshall, J. S. (1982). Application of a surplus production model to assess environmental impacts on exploited populations of Daphnia pluex in the laboratory. Environ. Pollut., 28, 273–280. · doi:10.1016/0143-1471(82)90143-X
[30]Li, X., & Mao, X. (2009). Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst., 24, 523–545. · Zbl 1161.92048 · doi:10.3934/dcds.2009.24.523
[31]Liu, H., & Ma, Z. (1991). The threshold of survival for system of two species in a polluted environment. J. Math. Biol., 30, 49–51. · Zbl 0745.92028 · doi:10.1007/BF00168006
[32]Liu, M., & Wang, K. (2009). Survival analysis of stochastic single-species population models in polluted environments. Ecol. Model., 220, 1347–1357. · doi:10.1016/j.ecolmodel.2009.03.001
[33]Liu, M., & Wang, K. (2010). Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. J. Theor. Biol., 264, 934–944. · doi:10.1016/j.jtbi.2010.03.008
[34]Ludwig, D. (1975). Persistence of dynamical systems under random perturbations. SIAM. Rev., 17, 605–640. · Zbl 0312.60040 · doi:10.1137/1017070
[35]Luna, J. T., & Hallam, T. G. (1987). Effects of toxicants on population: a qualitative approach IV. Resources-consumer-toxicant models. Ecol. Model., 35, 249–273. · doi:10.1016/0304-3800(87)90115-3
[36]Luo, Q., & Mao, X. (2007). Stochastic population dynamics under regime switching. J. Math. Anal. Appl., 334, 69–84. · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032
[37]Ma, Z., Song, B., & Hallam, T. G. (1989). The threshold of survival for systems in a fluctuating environment. Bull. Math. Biol., 51, 311–323.
[38]Mao, X., Marion, G., & Renshaw, E. (2002). Environmental Brownian noise suppresses explosions in populations dynamics. Stoch. Process. Appl., 97, 95–110. · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[39]Mao, X., Sabanis, S., & Renshaw, E. (2003). Asymptotic behaviour of the stochastic Lotka–Volterra model. J. Math. Anal. Appl., 287, 141–156. · Zbl 1048.92027 · doi:10.1016/S0022-247X(03)00539-0
[40]Mao, X. (2005). Delay population dynamics and environmental noise. Stoch. Dyn., 5, 149–162. · Zbl 1093.60033 · doi:10.1142/S021949370500133X
[41]Mao, X., & Yuan, C. (2006). Stochastic differential equations with Markovian switching. London: Imperial College Press.
[42]May, R. M. (2001). Stability and complexity in model ecosystems. Princeton University Press: Princeton.
[43]McKane, A. J., & Newman, T. J. (2005). Predator-prey cycles from resonant amplification of demographic stochasticity. Phys. Rev. Lett., 94, 218102. · doi:10.1103/PhysRevLett.94.218102
[44]Nelson, S. A. (1970). The problem of oil pollution of the sea. In Advances in marine biology (pp. 215–306). London: Academic Press.
[45]Øsendal, B. (1998). Stochastic differential equations: an introduction with applications (5th ed.). Berlin: Springer.
[46]Pang, S., Deng, F., & Mao, X. (2008). Asymptotic properties of stochastic population dynamics, Dynamics of Continuous. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15, 603–620.
[47]Rudnicki, R., & Pichor, K. (2007). Influence of stochastic perturbation on prey-predator systems. Math. Biosci., 206, 108–119. · Zbl 1124.92055 · doi:10.1016/j.mbs.2006.03.006
[48]Samanta, G. P., & Maiti, A. (2004). Dynamical model of a single-species system in a polluted environment. J. Appl. Math. Comput., 16, 231–242. · Zbl 1052.92056 · doi:10.1007/BF02936164
[49]Shukla, J. B., & Dubey, B. (1996). Simultaneous effect of two toxicants on biological species: a mathematical model. J. Biol. Syst., 4, 109–130. · doi:10.1142/S0218339096000090
[50]Shukla, J. B., Freedman, H. I., Pal, V. N., Misra, O. P., Agarwal, M., & Shukla, A. (1989). Degradation and subsequent regeneration of a forestry resource: a mathematical model. Ecol. Model., 44, 219–229. · doi:10.1016/0304-3800(89)90031-8
[51]Thomas, D. M., Snell, T. W., & Joffer, S. M. (1996). A control problem in a polluted environment. Math. Biosci., 133, 139–163. · Zbl 0844.92026 · doi:10.1016/0025-5564(95)00091-7
[52]Turelli, M. (1977). Random environments and stochastic calculus. Theor. Pop. Biol., 12, 140–178. · Zbl 0444.92013 · doi:10.1016/0040-5809(77)90040-5
[53]Zhu, C., & Yin, G. (2009). On competitive Lotka–Volterra model in random environments. J. Math. Anal. Appl., 357, 154–170. · Zbl 1182.34078 · doi:10.1016/j.jmaa.2009.03.066