zbMATH — the first resource for mathematics

The effects of interference competition on stability, structure and invasion of a multi-species system. (English) Zbl 0568.92017
Consider a Lotka-Volterra model for N competing species: \[ dx_ i/dt=(\epsilon_ i-\sum^{n}_{j=1}\mu_{ij}x_ j)x_ i,\quad i=1,2,...,N, \] where \(x_ i\) is the population density of the i th species; \(\epsilon_ i(>0)\) is its intrinsic growth rate; \(\mu_{ii}\) and \(\mu_{ij}\) (i\(\neq j)\) are the coefficients of intra-and interspecific competitions, respectively. The paper assumes that the competition coefficients can be written in the form \(\mu_{ij}=\sigma_ i\alpha_ i\) for \(i=j\) or \(\mu_{ij}=\sigma_ i\beta_ j\) for \(i\neq j.\)
All the equilibrium points of the model are obtained explicitly in terms of the parameters, and these equilibria are classified according to the concept of sector stability. Thus survival or extinction of species at a stable equilibrium point can be determined analytically.
The result of the analysis is extended to the successional processes of a community. A criterion for invasion of a new species is obtained and it is also shown that there are some characteristic quantities which show directional changes as succession proceeds.
Reviewer: J.Keesling

92D25 Population dynamics (general)
34C99 Qualitative theory for ordinary differential equations
92D40 Ecology
Full Text: DOI
[1] Case, T. J., Casten, R. G.: Global stability and multiple domains of attraction in ecological systems. Am. Nat. 113, 705-714 (1979)
[2] Connell, J. H.: The influence of interspecific competition and other factors on the distribution of the barnacle Chthamlus stellatus. Ecology 42, 710-723 (1961)
[3] Dayton, P. K.: Competition, disturbance and community organization: the provision and subsequent utilization of space in a rocky intertidal community. Ecol. Monogr. 41, 351-389 (1971)
[4] Gilpin, M. E.: Stability of feasible predator-prey systems. Nature 245, 137-139 (1975)
[5] Gilpin, M. E., Case, T. J.: Multiple domains of attraction in competition communities. Nature 261, 40-42 (1976)
[6] Goh, B. S.: Management and Analysis of Biological Populations. New York: Elsevier 1980
[7] Goh, B. S., Jennings, L. S.: Feasibility and stability in randomly assembled Lotka-Volterra models. Ecol. Modelling 3, 63-71 (1977)
[8] Ikeda, M., ?iljak, D. D.: Lotka-Volterra equations: decomposition, stability, and structure. J. Math. Biology 9, 68-83 (1980) · Zbl 0436.92018
[9] Levin, S. A.: Community equilibria and stability, and an extension of the competitive exclusion principle. Am. Nat. 104, 413-423 (1970)
[10] MacArthur, R. H.: Species packing and competitive equilibrium among many species. Theoret. Population Biology 1, 1-11 (1970)
[11] May, R.: Stability and Complexity in Model Ecosystems. Princeton, N. J.: Princeton University Press 1973
[12] Miller, R. S.: Pattern and process in competition. Adv. Ecol. Res. 4, 1-74 (1967)
[13] Robinson, J. V., Ladde, G. S.: Feasibility constraints on the elastic expansion of ecosystem models. J. Theoret. Biology 97, 277-287 (1982)
[14] Roughgarden, J.: Theory of Population Genetics and Evolutionary Ecology: An Introduction. New York: MacMillan 1979
[15] ?iljak, D. D.: When is a complex ecosystem stable? Mathematical Biosciences 25, 25-50 (1975) · Zbl 0316.92014
[16] Strobeck, C.: N species competition. Ecology 54, 651-654 (1973)
[17] Whittaker, R. H.: Communities and Ecosystems. tNew York: MacMillan 1970 · Zbl 0203.26303
[18] Yodzis, P.: Competition for space and the structure of ecological communities. In: Lecture Notes in Biomathematics. Levin, S.A. (e.d.), vol. 25, Berlin-Heidelberg-New York: Springer-Verlag 1978 · Zbl 0387.92009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.