# American Institute of Mathematical Sciences

## 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

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.

Citation: Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020192
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A curve of coexistence equilibria
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 $D_2{ = }1.3$
Before and after a Hopf bifurcation: $D{ = }D_1{ = }D_2{ = }1$ and Holling type Ⅱ rate functions
Before and after Hopf bifurcation: $D{ = }1$, $D_1{ = }1.2$ and $D_2{ = }1.3$ and Holling type Ⅱ rate functions
Real part using rate functions (23)
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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