American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delayed population models with Allee effects and exploitation
  • MBE Home
  • This Issue
  • Next Article
    Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach
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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yoshiaki Muroya, Teresa Faria. Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3089-3114. doi: 10.3934/dcdsb.2018302
[2]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75
[3]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
[4]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[5]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[6]

Ting-Hui Yang, Weinian Zhang, Kaijen Cheng. Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2867-2881. doi: 10.3934/dcdsb.2016077
[7]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[8]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[9]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[10]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[11]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
[12]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137
[13]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[14]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[15]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
[16]

Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014
[17]

Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5633-5671. doi: 10.3934/dcdsb.2019076
[18]

Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477
[19]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[20]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy