November  2013, 18(9): 2331-2353. doi: 10.3934/dcdsb.2013.18.2331

On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response

1. 

Department of Mathematics and The National Center for Theoretical Science, National Tsing-Hua University, Hsinchu 30013, Taiwan

2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, United States

3. 

Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137

Received  June 2013 Revised  August 2013 Published  September 2013

In this paper we study a two-consumers-one-resource competing system with Beddington-DeAngelis functional response. The two consumers competing for a renewable resource have intraspecific competition among their own populations. Firstly we investigate the extinction and uniform persistence of the predators, local and global stability of the equilibria, and existence of Hopf bifurcation at the positive equilibrium. Then we compare the dynamic behavior of the system with and without interference effects. Analytically we study the competition of two identically species with different interference effects. We also study the relaxation oscillation in the case of interference effects. Finally we present extensive numerical simulations to understand the interference effects on the competition outcomes.
Citation: Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331
References:
[1]

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

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, Journal of Animal Ecology, 44 (1975), 331. doi: 10.2307/3866. Google Scholar

[3]

G. J. Butler and P. Waltman, Persistence in dynamical systems,, Journal of Differential Equations, 63 (1986), 255. doi: 10.1016/0022-0396(86)90049-5. Google Scholar

[4]

G. J. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems,, Proceedings of the American Mathematical Society, 96 (1986), 425. doi: 10.1090/S0002-9939-1986-0822433-4. Google Scholar

[5]

G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat,, Journal of Mathematical Biology, 12 (1981), 295. doi: 10.1007/BF00276918. Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 257 (2001), 206. doi: 10.1006/jmaa.2000.7343. Google Scholar

[7]

R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics,, Discrete and Continuous Dynamical Systems - B, 4 (2004), 527. doi: 10.3934/dcdsb.2004.4.527. Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. doi: 10.2307/1936298. Google Scholar

[9]

H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, Journal of Dynamics and Differential Equations, 6 (1994), 583. doi: 10.1007/BF02218848. Google Scholar

[10]

S. B. Hsu, Limiting behavior for competing species,, SIAM Journal on Applied Mathematics, 34 (1978), 760. doi: 10.1137/0134064. Google Scholar

[11]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

[12]

S. B. Hsu, S. Hubbell, and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,, SIAM Journal on Applied Mathematics, 32 (1977), 366. doi: 10.1137/0132030. Google Scholar

[13]

S. B. Hsu, S. P. Hubbell and P. Waltman, A contribution to the theory of competing predators,, Ecological Monographs, 48 (1978), 337. Google Scholar

[14]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators,, SIAM Journal on Applied Mathematics, 35 (1978), 617. doi: 10.1137/0135051. Google Scholar

[15]

G. Huisman and R. J. De Boer, A formal derivation of the "Beddington" functional response,, Journal of Theoretical Biology, 185 (1997), 389. doi: 10.1006/jtbi.1996.0318. Google Scholar

[16]

T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 281 (2003), 395. doi: 10.1016/S0022-247X(02)00395-5. Google Scholar

[17]

T.-W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 290 (2004), 113. doi: 10.1016/j.jmaa.2003.09.073. Google Scholar

[18]

J. P. Keener, Oscillatory coexistence in the chemostat: A codimension two unfolding,, SIAM Journal on Applied Mathematics, 43 (1983), 1005. doi: 10.1137/0143066. Google Scholar

[19]

W. Liu, D. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems,, Journal of Differential Equations, 188 (2003), 306. doi: 10.1016/S0022-0396(02)00076-1. Google Scholar

[20]

S. Muratori and S. Rinaldi, Remarks on competitive coexistence,, SIAM Journal on Applied Mathematics, 49 (1989), 1462. doi: 10.1137/0149088. Google Scholar

[21]

H. L. Smith, The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model,, SIAM Journal on Applied Mathematics, 42 (1982), 27. doi: 10.1137/0142003. Google Scholar

[22]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Vol. 118 of Graduate Studies in Mathematics, (2011). Google Scholar

show all references

References:
[1]

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

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, Journal of Animal Ecology, 44 (1975), 331. doi: 10.2307/3866. Google Scholar

[3]

G. J. Butler and P. Waltman, Persistence in dynamical systems,, Journal of Differential Equations, 63 (1986), 255. doi: 10.1016/0022-0396(86)90049-5. Google Scholar

[4]

G. J. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems,, Proceedings of the American Mathematical Society, 96 (1986), 425. doi: 10.1090/S0002-9939-1986-0822433-4. Google Scholar

[5]

G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat,, Journal of Mathematical Biology, 12 (1981), 295. doi: 10.1007/BF00276918. Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 257 (2001), 206. doi: 10.1006/jmaa.2000.7343. Google Scholar

[7]

R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics,, Discrete and Continuous Dynamical Systems - B, 4 (2004), 527. doi: 10.3934/dcdsb.2004.4.527. Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. doi: 10.2307/1936298. Google Scholar

[9]

H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, Journal of Dynamics and Differential Equations, 6 (1994), 583. doi: 10.1007/BF02218848. Google Scholar

[10]

S. B. Hsu, Limiting behavior for competing species,, SIAM Journal on Applied Mathematics, 34 (1978), 760. doi: 10.1137/0134064. Google Scholar

[11]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

[12]

S. B. Hsu, S. Hubbell, and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,, SIAM Journal on Applied Mathematics, 32 (1977), 366. doi: 10.1137/0132030. Google Scholar

[13]

S. B. Hsu, S. P. Hubbell and P. Waltman, A contribution to the theory of competing predators,, Ecological Monographs, 48 (1978), 337. Google Scholar

[14]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators,, SIAM Journal on Applied Mathematics, 35 (1978), 617. doi: 10.1137/0135051. Google Scholar

[15]

G. Huisman and R. J. De Boer, A formal derivation of the "Beddington" functional response,, Journal of Theoretical Biology, 185 (1997), 389. doi: 10.1006/jtbi.1996.0318. Google Scholar

[16]

T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 281 (2003), 395. doi: 10.1016/S0022-247X(02)00395-5. Google Scholar

[17]

T.-W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response,, Journal of Mathematical Analysis and Applications, 290 (2004), 113. doi: 10.1016/j.jmaa.2003.09.073. Google Scholar

[18]

J. P. Keener, Oscillatory coexistence in the chemostat: A codimension two unfolding,, SIAM Journal on Applied Mathematics, 43 (1983), 1005. doi: 10.1137/0143066. Google Scholar

[19]

W. Liu, D. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems,, Journal of Differential Equations, 188 (2003), 306. doi: 10.1016/S0022-0396(02)00076-1. Google Scholar

[20]

S. Muratori and S. Rinaldi, Remarks on competitive coexistence,, SIAM Journal on Applied Mathematics, 49 (1989), 1462. doi: 10.1137/0149088. Google Scholar

[21]

H. L. Smith, The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model,, SIAM Journal on Applied Mathematics, 42 (1982), 27. doi: 10.1137/0142003. Google Scholar

[22]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Vol. 118 of Graduate Studies in Mathematics, (2011). Google Scholar

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