# American Institute of Mathematical Sciences

July  2019, 24(7): 3439-3451. doi: 10.3934/dcdsb.2018327

## Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics

 a. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China b. School of Computer Science and Network Security, Dongguan University of Technology, Dongguan, 523808, Guangdong, China c. College of Finance and Statistics, Hunan University, Changsha, 410079, Hunan, China d. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada

* Corresponding author: Lin Wang

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: The work of DZ was partially supported by the National Natural Science Foundation of China (No. 11501193) and the China Post Doctorial Fund (No. 2015M582335). LW was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

An inhibitory uptake function is incorporated into the discrete, size-structured nonlinear chemostat model developed by Arino et al. (Journal of Mathematical Biology, 45(2002)). Different from the model with a monotonically increasing uptake function, we show that the inhibitory kinetics can induce very complex dynamics including stable equilibria, cycles and chaos (via the period-doubling cascade). In particular, when the nutrient concentration in the input feed to the chemostat $S^0$ is larger than the upper break-even concentration value $\mu$, the model exhibits three types of bistability allowing a stable equilibrium to coexist with another stable equilibrium, or a stable cycle or a chaotic attractor.

Citation: Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327
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##### References:
Bifurcation diagram of the limiting system (7). Here $f(s) = \frac{as}{1+0.1s+0.01s^2}, E = 0.1, S^{0} = 100, U_{0} = 70$ with $a\in[0.049, 0.059]$ Thus $aS^{0}>1$ and $S^{0}> \mu$. A cascade of period-doublings to chaos occurs as $a$ increases
Left: a numerical solution of the limiting system (7) with $U_0 = 60$; Right: a numerical solution of system (11) with $(U_0, S_0) = (60,6)$. Here $f(s) = \frac{0.05s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$
Numerical solutions of system (11) with $f(s) = \frac{0.05s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to E_0 = (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t) \to E_1 = (S^0-\lambda, \lambda)$ as $t\to\infty$, initial condition was $(U_0, S_0) = (80,6)$
Numerical solutions of system (11) with $f(s) = \frac{0.054s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t)$ approaches a stable $2-cycle$, initial condition was $(U_0, S_0) = (80,6)$
Numerical solutions of system (11) with $f(s) = \frac{0.059s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t)$ approaches a chaotic attractor, initial condition was $(U_0, S_0) = (80,6)$
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