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A competition model with dynamically allocated toxin production in the unstirred chemostat

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    * Corresponding author 

The first author is supported by the NSF of China(11671243), the Shaanxi New-star Plan of Science and Technology(2015KJXX-21), and the Fundamental Research Funds for the Central Universities(GK201701001)

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  • This paper deals with a competition model with dynamically allocated toxin production in the unstirred chemostat. First, the existence and uniqueness of positive steady state solutions of the single population model is attained by the general maximum principle, spectral analysis and degree theory. Second, the existence of positive equilibria of the two-species system is investigated by the degree theory, and the structure and stability of nonnegative equilibria of the two-species system are established by the bifurcation theory. The results show that stable coexistence solution can occur with dynamic toxin production, which cannot occur with constant toxin production. Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting. Finally, numerical results illustrate that a wide variety of dynamical behaviors can be achieved for the system with dynamic toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles, which complement the analytic results.

    Mathematics Subject Classification: Primary: 35K57, 35K55, 35B32; Secondary: 35B50, 92D25.

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  • Figure 1.  We further fix $d=0.1, \alpha=0.2, \beta=1, $ and observe the effects of the parameter $c$: coexistence in the form of equilibria is observed in (b)(c) when $c=0.2, 0.3$ respectively; competitive exclusion occurs in (a)(d) when $c=0.01, 0.6$ respectively

    Figure 2.  We further fix $\alpha=0.8, \beta=0.001, c=0.2$, take the diffusion rates $d=0.4, 0.6, 0.65, 0.7, 0.9, 1.5$ in (a)-(f), and observe the effects of the diffusion rate $d$: (a) competition exclusion, (b)-(d) stable limit cycles, (e) stable positive equilibrium, (f) washout solution

    Figure 3.  We further fix $d=0.1, \alpha=0.2, \beta=1, $ and observe the effects of the parameter $c$: coexistence in the form of equilibria is observed in (b)(c) when $c=0.2, 0.3$ respectively; competitive exclusion occurs in (a)(d) when $c=0.01, 0.6$ respectively

    Figure 4.  Taking $d=0.1, \alpha=0.2, \beta=1, $ and $c=0.1$, bistable attractors occur, which are plotted in (a) and (b). Moreover, two positive equilibria appear (see (c) and (d))

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