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November  2020, 13(11): 2975-3004. doi: 10.3934/dcdss.2020192

A nutrient-prey-predator model: Stability and bifurcations

Mary Ballyk 1, , Ross Staffeldt 1,, and  Ibrahim Jawarneh 2,

1. 

Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA
2. 

Department of Mathematics, Faculty of Science, Al-Hussein Bin Talal University, Ma'an, P.O.Box (20), Jordan

* Corresponding author: Ross Staffeldt

Received  April 2019 Revised  October 2019 Published  December 2019

We model a nutrient-prey-predator system in a chemostat with general functional responses, using the input concentration of nutrient as the bifurcation parameter. We study changes in the existence and the stability of isolated equilibria, as well as changes in the global dynamics, as the nutrient concentration varies. The bifurcations of the system are analytically verified and we identify conditions under which an equilibrium undergoes a Hopf bifurcation and a limit cycle appears. Numerical simulations for specific functional responses illustrate the general results.

Keywords: Chemostat, Hopf bifurcation, coexistence equilibrium.
Mathematics Subject Classification: Primary: 37G10; Secondary: 34C23, 92D25, 34A34.
Citation: Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192
References:
[1]

M. Ballyk, I. Jawarneh and R. Staffeldt, A nutrient-prey-predator model: Stability and bifurcations, preprint, arXiv: 1812.09964. Google Scholar

[2]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

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S. Lang, Analysis II, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1969. Google Scholar

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J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, With Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar

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B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

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J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6.  Google Scholar

[11]

G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.  doi: 10.1016/0025-5564(89)90025-4.  Google Scholar

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T. Zhang and W. Wang, Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235.  doi: 10.1016/j.apm.2012.02.012.  Google Scholar

show all references

References:
[1]

M. Ballyk, I. Jawarneh and R. Staffeldt, A nutrient-prey-predator model: Stability and bifurcations, preprint, arXiv: 1812.09964. Google Scholar

[2]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[3]

H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci., 68 (1984), 213-231.  doi: 10.1016/0025-5564(84)90032-4.  Google Scholar

[4]

T. C. Gard, Mathematical analysis of some resource-prey-predator models: Application to an NPZ microcosm model, in Population Biology, Lecture Notes in Biomath., 52, Springer, Berlin, 1983,275–282. doi: 10.1007/978-3-642-87893-0_34.  Google Scholar

[5]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[6]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[7]

S. Lang, Analysis II, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1969. Google Scholar

[8]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, With Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar

[9]

B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

[10]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6.  Google Scholar

[11]

G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.  doi: 10.1016/0025-5564(89)90025-4.  Google Scholar

[12]

T. Zhang and W. Wang, Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235.  doi: 10.1016/j.apm.2012.02.012.  Google Scholar

Figure 1.  A curve of coexistence equilibria
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Figure 2.  Comparison of real parts. For the solid curve, $ D{ = }D_1{ = }D_2 = 1 $; for the curve of symbols, $ D{ = }1 $, $ D_1{ = }1.2 $ and $ D_2{ = }1.3 $
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Figure 3.  Before and after a Hopf bifurcation: $ D{ = }D_1{ = }D_2{ = }1 $ and Holling type Ⅱ rate functions
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Figure 4.  Before and after Hopf bifurcation: $ D{ = }1 $, $ D_1{ = }1.2 $ and $ D_2{ = }1.3 $ and Holling type Ⅱ rate functions
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Figure 5.  Real part using rate functions (23)
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Figure 6.  Before and after Hopf bifurcation: $ D{ = }1 $, $ D_1{ = }1.2 $, and $ D_2{ = }1.1 $ and using rate functions (23)
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