A nutrient-prey-predator model: Stability and bifurcations

Mary Ballyk; Ross Staffeldt; Ibrahim Jawarneh

doi:10.3934/dcdss.2020192

Article Contents

2020, Volume 13, Issue 11: 2975-3004. Doi: 10.3934/dcdss.2020192

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A nutrient-prey-predator model: Stability and bifurcations

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
^* Corresponding author: Ross Staffeldt

Received: April 2019

Revised: October 2019

Early access: December 2019

Published: July 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G10; Secondary: 34C23, 92D25, 34A34.

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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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References

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