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2015, 12(1): 71-81. doi: 10.3934/mbe.2015.12.71

Dynamics of competitive systems with a single common limiting factor

Ryusuke Kon 1,

1. 

Faculty of Engineering, University of Miyazaki, Gakuen Kibanadai Nishi 1-1, Miyazaki 889-2192, Japan

Received  April 2014 Revised  October 2014 Published  December 2014

The concept of limiting factors (or regulating factors) succeeded in formulating the well-known principle of competitive exclusion. This paper shows that the concept of limiting factors is helpful not only to formulate the competitive exclusion principle, but also to obtain other ecological insights. To this end, by focusing on a specific community structure, we study the dynamics of Kolmogorov equations and show that it is possible to derive an ecologically insightful result only from the information about interactions between species and limiting factors. Furthermore, we find that the derived result is a generalization of the preceding work by Shigesada, Kawasaki, and Teramoto (1984), who examined a certain Lotka-Volterra equation in a different context.
Keywords: nonlinear complementary problem, saturated equilibrium, Lotka-Volterra equation., P-function, P-matrix.
Mathematics Subject Classification: Primary: 34D20; Secondary: 92B0.
Citation: Ryusuke Kon. Dynamics of competitive systems with a single common limiting factor. Mathematical Biosciences & Engineering, 2015, 12 (1) : 71-81. doi: 10.3934/mbe.2015.12.71
References:
[1]

R. A. Armstrong and R. McGehee, Coexistence of species competing for shared resources, Theoretical Population Biology, 9 (1976), 317-328. doi: 10.1016/0040-5809(76)90051-4.

[2]

R. A. Armstrong and R. McGehee, Coexistence of two competitors on one resource, Journal of Theoretical Biology, 56 (1976), 499-502. doi: 10.1016/S0022-5193(76)80089-6.

[3]

R. A. Armstrong and R. McGehee, Competitive exclusion, The American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553.

[4]

M. Hirsch and H. Smith, Monotone dynamical systems, In A. Canada, P. Drabek, and A. Fonda, editors, Ordinary Differential Equations, Elsevier, II (2005), 239-357.

[5]

J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mountain J. Math., 20 (1990), 1017-1031. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1988). doi: 10.1216/rmjm/1181073059.

[6]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press Cambridge, 1988.

[7]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[8]

R. D. Holt, J. Grover and D. Tilman, Simple rules for interspecific dominance in systems with exploitative and apparent competition, American Naturalist, 144 (1994), 741-771. doi: 10.1086/285705.

[9]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423. doi: 10.1086/282676.

[10]

D. Logofet, Matrices and Graphs: Stability Problems in Mathematical Ecology, CRC Press, Boca Raton, Florida, 1993.

[11]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, Journal of Differential Equations, 23 (1977), 30-52. doi: 10.1016/0022-0396(77)90135-8.

[12]

J. Moré and W. Rheinboldt, On P- and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra and its Applications, 6 (1973), 45-68. doi: 10.1016/0024-3795(73)90006-2.

[13]

J. J. Moré, Classes of functions and feasibility conditions in nonlinear complementarity problems, Mathematical Programming, 6 (1974), 327-338. doi: 10.1007/BF01580248.

[14]

F. Scudo and J. Ziegler, Lecture Notes in Biomathematic, volume 22 of Lecture notes in Biomathematics, Sprinter, 1978.

[15]

N. Shigesada, K. Kawasaki and E. Teramoto, The effects of interference competition on stability, structure and invasion of a multispecies system, J. Math. Biol., 21 (1984), 97-113. doi: 10.1007/BF00277664.

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. American Mathematical Society, 1995.

[17]

Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098.

[18]

Y. Takeuchi and N. Adachi, Existence of stable equilibrium point for dynamical systems of Volterra type, J. Math. Anal. Appl., 79 (1981), 141-162. doi: 10.1016/0022-247X(81)90015-9.

[19]

Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X.

[20]

Y. Takeuchi, N. Adachi and H. Tokumaru, The stability of generalized Volterra equations, J. Math. Anal. Appl., 62 (1978), 453-473. doi: 10.1016/0022-247X(78)90139-7.

show all references

References:
[1]

R. A. Armstrong and R. McGehee, Coexistence of species competing for shared resources, Theoretical Population Biology, 9 (1976), 317-328. doi: 10.1016/0040-5809(76)90051-4.

[2]

R. A. Armstrong and R. McGehee, Coexistence of two competitors on one resource, Journal of Theoretical Biology, 56 (1976), 499-502. doi: 10.1016/S0022-5193(76)80089-6.

[3]

R. A. Armstrong and R. McGehee, Competitive exclusion, The American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553.

[4]

M. Hirsch and H. Smith, Monotone dynamical systems, In A. Canada, P. Drabek, and A. Fonda, editors, Ordinary Differential Equations, Elsevier, II (2005), 239-357.

[5]

J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mountain J. Math., 20 (1990), 1017-1031. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1988). doi: 10.1216/rmjm/1181073059.

[6]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press Cambridge, 1988.

[7]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[8]

R. D. Holt, J. Grover and D. Tilman, Simple rules for interspecific dominance in systems with exploitative and apparent competition, American Naturalist, 144 (1994), 741-771. doi: 10.1086/285705.

[9]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423. doi: 10.1086/282676.

[10]

D. Logofet, Matrices and Graphs: Stability Problems in Mathematical Ecology, CRC Press, Boca Raton, Florida, 1993.

[11]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, Journal of Differential Equations, 23 (1977), 30-52. doi: 10.1016/0022-0396(77)90135-8.

[12]

J. Moré and W. Rheinboldt, On P- and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra and its Applications, 6 (1973), 45-68. doi: 10.1016/0024-3795(73)90006-2.

[13]

J. J. Moré, Classes of functions and feasibility conditions in nonlinear complementarity problems, Mathematical Programming, 6 (1974), 327-338. doi: 10.1007/BF01580248.

[14]

F. Scudo and J. Ziegler, Lecture Notes in Biomathematic, volume 22 of Lecture notes in Biomathematics, Sprinter, 1978.

[15]

N. Shigesada, K. Kawasaki and E. Teramoto, The effects of interference competition on stability, structure and invasion of a multispecies system, J. Math. Biol., 21 (1984), 97-113. doi: 10.1007/BF00277664.

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. American Mathematical Society, 1995.

[17]

Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098.

[18]

Y. Takeuchi and N. Adachi, Existence of stable equilibrium point for dynamical systems of Volterra type, J. Math. Anal. Appl., 79 (1981), 141-162. doi: 10.1016/0022-247X(81)90015-9.

[19]

Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X.

[20]

Y. Takeuchi, N. Adachi and H. Tokumaru, The stability of generalized Volterra equations, J. Math. Anal. Appl., 62 (1978), 453-473. doi: 10.1016/0022-247X(78)90139-7.

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