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

• * Corresponding author: Lin Wang
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.

Mathematics Subject Classification: Primary: 39A30, 92D25.

 Citation: • • Figure 1.  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

Figure 2.  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 $f(s) = \frac{0.05s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$

Figure 3.  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 $(U_0, S_0) = (80,6)$

Figure 4.  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 $(U_0, S_0) = (80,6)$

Figure 5.  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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