The Annals of Applied Probability

An explicitly spatial version of the Lotka-Volterra model with interspecific competition

Claudia Neuhauser and Stephen W. Pacala

Full-text: Open access

PDF File (208 KB)

Abstract

We consider a spatial stochastic version of the classical Lotka-Volterra model with interspecific competition.

The classical model is described by a set of ordinary differential equations, one for each species. Mortality is density dependent, including both intraspecific and interspecific competition. Fecundity may depend on the type of species but is density independent. Depending on the relative strengths of interspecific and intraspecific competition and on the fecundities, the parameter space for the classical model is divided into regions where either coexistence, competitive exclusion or founder control occur.

The spatial version is a continuous time Markov process in which individuals are located on the d-dimensional integer lattice. Their dynamics are described by a set of local rules which have the same components as the classical model.

Our main results for the spatial stochastic version can be summarized as follows. Local competitive interactions between species result in (1) a reduction of the parameter region where coexistence occurs in the classical model, (2) a reduction of the parameter region where founder control occurs in the classical model, and (3) spatial segregation of the two species in parts of the parameter region where the classical model predicts coexistence.

Article information

Source
Ann. Appl. Probab. Volume 9, Number 4 (1999), 1226-1259.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1029962871

Digital Object Identifier
doi:10.1214/aoap/1029962871

Mathematical Reviews number (MathSciNet)
MR1728561

Zentralblatt MATH identifier
0948.92022

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92B05: General biology and biomathematics

Keywords
Competition contact process interacting particle system voter model

Citation

Neuhauser, Claudia; Pacala, Stephen W. An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. 9 (1999), no. 4, 1226--1259. doi:10.1214/aoap/1029962871. http://projecteuclid.org/euclid.aoap/1029962871.


Export citation

References

  • [1] Antonovics, J. and Levin, D. A. (1980). The ecological and genetic consequences of density dependent regulation in plants. Annual Review of Ecology and Sy stematics 11 411-452.
  • [2] Bramson, M., Wan-ding, Ding and Durrett, R. (1991). Annihilating branching processes. Stochastic. Process Appl. 37 1-17.
    Mathematical Reviews (MathSciNet): MR92f:60178
    Zentralblatt MATH: 0745.60085
    Digital Object Identifier: doi: 10.1016/0304-4149(91)90056-I
  • [3] Bramson, M. and Neuhauser, C. (1994). Survival of one-dimensional cellular automata under random perturbations. Ann. Probab. 22 244-263.
    Mathematical Reviews (MathSciNet): MR95b:60125
    Zentralblatt MATH: 0793.60107
    Digital Object Identifier: doi: 10.1214/aop/1176988858
    Project Euclid: euclid.aop/1176988858
  • [4] Bramson, M. and Neuhauser, C. (1997). Coexistence for a cataly tic surface reaction model. Ann. Appl. Probab. 7 565-614.
    Mathematical Reviews (MathSciNet): MR99a:60113
    Zentralblatt MATH: 0881.92037
    Digital Object Identifier: doi: 10.1214/aoap/1034801245
    Project Euclid: euclid.aoap/1034801245
  • [5] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581-588.
    Mathematical Reviews (MathSciNet): MR49:8690
    Zentralblatt MATH: 0272.60072
    Digital Object Identifier: doi: 10.1093/biomet/60.3.581
    JSTOR: links.jstor.org
  • [6] Durrett, R. (1991). A new method for proving the existence of phase transitions. In Spatial Stochastic Processes (K. S. Alexander and J. S. Watkins, eds.) 141-170. Birkh¨auser, Boston.
    Mathematical Reviews (MathSciNet): MR93c:60158
    Zentralblatt MATH: 0825.60052
  • [7] Durrett, R. (1995). Ten lectures on particle sy stems. Lectures on Probability Theory, Ecole d'Et´e de Probabilit´es de Saint-Flour XXIII-1993. Springer, New York.
  • [8] Durrett, R. (1998). Stochastic spatial models. Preprint.
  • [9] Durrett, R. and Levin, S. A. (1994). The importance of being discrete (and spatial). Theoret. Population Biol. 46 363-394.
    Zentralblatt MATH: 0846.92027
  • [10] Durrett, R. and Neuhauser, C. (1994). Particle sy stems and reaction diffusion equations. Ann. Probab. 22 289-333.
    Mathematical Reviews (MathSciNet): MR95d:60159
    Zentralblatt MATH: 0799.60093
    Digital Object Identifier: doi: 10.1214/aop/1176988861
    Project Euclid: euclid.aop/1176988861
  • [11] Durrett, R. and Neuhauser, C. (1997). Coexistence results for some competition models. Ann. Appl. Probab. 7 10-45.
    Mathematical Reviews (MathSciNet): MR98g:60178
    Zentralblatt MATH: 0873.92020
    Digital Object Identifier: doi: 10.1214/aoap/1034625251
    Project Euclid: euclid.aoap/1034625251
  • [12] Griffeath, D. (1978). Limit theorems for nonergodic set-valued Markov processes. Ann. Probab. 6 379-387.
    Mathematical Reviews (MathSciNet): MR58:7926
    Digital Object Identifier: doi: 10.1214/aop/1176995524
  • [13] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Sy stems. Lecture Notes in Math. 724. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR80i:60132
    Zentralblatt MATH: 0412.60095
  • [14] Harris, T. E. (1972). Nearest neighbor Markov interaction processes. Adv. Math 9 66-89.
    Zentralblatt MATH: 0267.60107
    Mathematical Reviews (MathSciNet): MR307392
    Digital Object Identifier: doi: 10.1016/0001-8708(72)90030-8
  • [15] Holley, R. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting particle sy stems and the voter model. Ann. Probab. 3 643-663.
    Mathematical Reviews (MathSciNet): MR402985
    Digital Object Identifier: doi: 10.1214/aop/1176996306
  • [16] Holley, R. and Stroock, D. (1979). Dual processes and their applications to infinite interacting sy stems. Adv. Math. 32 149-174.
    Mathematical Reviews (MathSciNet): MR80m:60104
    Zentralblatt MATH: 0459.60097
    Digital Object Identifier: doi: 10.1016/0001-8708(79)90040-9
  • [17] Levin, S. A. and Pacala, S. W. (1997). Theories of simplification and scaling of spatially distributed processes. In Spatial Ecology: The Role of Space in Population Dy namics and Interspecific Interactions (D. Tilman and P. Kareiva, eds.) 271-295. Princeton Univ. Press.
  • [18] Liggett, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
    Mathematical Reviews (MathSciNet): MR86e:60089
  • [19] Lotka, A. J. (1932). The growth of mixed populations: two species competing for a common food supply. Journal of the Washington Academy of Sciences 22 461-469.
  • [20] MacArthur, R. H. (1972). Geographical Ecology. Patterns in the Distribution of Species. Harper and Row, New York.
  • [21] Neuhauser, C. (1992). Ergodic theorems for the multity pe contact process. Probab. Theory Related Fields 91 467-506.
    Mathematical Reviews (MathSciNet): MR93c:60162
    Zentralblatt MATH: 0739.60100
    Digital Object Identifier: doi: 10.1007/BF01192067
  • [22] Neuhauser, C. (1994). A long range sexual reproduction process. Stochastic Process Appl. 53 193-220.
    Mathematical Reviews (MathSciNet): MR95i:60082
    Zentralblatt MATH: 0810.60097
    Digital Object Identifier: doi: 10.1016/0304-4149(94)90063-9
  • [23] Neuhauser, C. (1999). The ancestral graph and gene genealogy under frequency-dependent selection. Theoret. Population Biol. To appear.
    Zentralblatt MATH: 0956.92026
  • [24] Neuhauser, C. and Sudbury, A. (1993). The biased annihilating branching process. Adv. Appl. Probab. 25 24-38.
    Mathematical Reviews (MathSciNet): MR94b:60118
    Zentralblatt MATH: 0777.60097
    Digital Object Identifier: doi: 10.2307/1427494
    JSTOR: links.jstor.org
  • [25] Pacala, S. W. (1989). Plant population dy namic theory. In Perspectives in Ecological Theory (J. Roughgarden, R. M. May and S. A. Levin, eds.) 54-67. Princeton Univ. Press.
    Mathematical Reviews (MathSciNet): MR1055089
  • [26] Pacala, S. W. (1997). Dy namics of plant communities. In Plant Ecology, 2nd ed. (M. C. Crawley, ed.) 532-555. Blackwell, Oxford.
  • [27] Pacala, S. W., Canham, C. D., Saponara, J. Silander, J. A., Kobe, R. K. and Ribbens,
  • E. (1996). Forest models defined by field measurements II. Estimation, error analysis and dy namics. Ecological Monographs 66 1-44.
  • [28] Pacala, S. W. and Levin, S. A. (1997). Biologically generated spatial pattern and the coexistence of competing species. In Spatial Ecology: The Role of Space in Population Dy namics and Interspecific Interactions (D. Tilman and P. Kareiva, eds.) 204-232. Princeton Univ. Press.
  • [29] Pacala, S. W. and Silander, J. A. Jr. (1990). Field tests of neighborhood population dy namic models of two annual weed species. Ecological Monographs 60 113-134.
  • [30] Rees, M. (1996). Evolutionary ecology of seed dormancy and seed size. Philos. Trans. Roy. Soc. London Ser. B 351 1299-1308.
  • [31] Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246-290.
    Mathematical Reviews (MathSciNet): MR42:3856
    Zentralblatt MATH: 0312.60060
    Digital Object Identifier: doi: 10.1016/0001-8708(70)90034-4
  • [32] Thrall, P. and Antonovics, J. (1995). Theoretical and empirical studies of meta-populations: population and genetic dy namics of the Silene-Ustilago sy stem. Canadian Journal of Botany 73 S1249-S1258.
  • [33] Tilman, D. (1982). Resource Competition and Community Structure. Princeton Univ. Press.
  • [34] Volterra, V. (1926). Variations and Fluctuations of the Number of Individuals in an Animal Species Living Together. In Animal Ecology (R. N. Chapman, ed.). (Reprinted in 1931. McGraw Hill, New York.)

The Institute of Mathematical Statistics

The Annals of Applied Probability

Follow this journal

Email TOC RSS TOC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more