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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))

  • [1] S. AbbasM. Banerjee and N. Hungerbuhler, Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model, J. Math. Anal. Appl., 367 (2010), 249-259.  doi: 10.1016/j.jmaa.2010.01.024.
    [2] M. AnD. L. LiuI. R. Johnson and J. V. Lovett, Mathematical modelling of allelopathy: Ⅱ. The dynamics of allelochemicals from living plants in the environment, Ecol. Modell., 161 (2003), 53-66.  doi: 10.1016/S0304-3800(02)00289-2.
    [3] B. L. Bassler, How bacteria talk to each other: regulation of gene expression by quorum sensing, Curr. Opinion Microbiol., 2 (1999), 582-587.  doi: 10.1016/S1369-5274(99)00025-9.
    [4] F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Addison-Wesley Longman, Harlow, UK, 1997.
    [5] J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Math. Biosci., 173 (2001), 55-84.  doi: 10.1016/S0025-5564(01)00078-5.
    [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. 
    [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearixed stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.
    [8] E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. 
    [9] E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. 
    [10] E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part Ⅰ, General existence results, Nonlinear Anal., 24 (1995), 337-357. 
    [11] P. FergolaE. Beretta and M. Cerasuolo, Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production, Nonliear Anal. Real World Appl., 7 (2006), 1081-1095.  doi: 10.1016/j.nonrwa.2005.10.001.
    [12] P. FergolaM. CerasuoloA. Pollio and G. Pinto, Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecol. Modell., 208 (2007), 205-214.  doi: 10.1016/j.ecolmodel.2007.05.024.
    [13] P. FergolaJ. Li and Z. Ma, On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ric. Mat., 60 (2011), 313-332.  doi: 10.1007/s11587-011-0108-y.
    [14] J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
    [15] S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490.  doi: 10.1007/BF03167323.
    [16] F. D. HulotP. J. Morin and M. Loreau, Interactions between algae and the microbial loop in experimental microcosms, Oikos, 95 (2001), 231-238.  doi: 10.1034/j.1600-0706.2001.950205.x.
    [17] J. López-Gómez and R. Parda, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Differential Integral Equations, 6 (1993), 1025-1031. 
    [18] H. NieN. Liu and J. H. Wu, Coexistence solutions and their stability of an unstirred chemostat model with toxins, Nonlinear Analysis: Real World Appl., 20 (2014), 36-51.  doi: 10.1016/j.nonrwa.2014.04.002.
    [19] H. Nie and J. H. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, International J. Bifurcation and Chaos, 16 (2006), 989-1009.  doi: 10.1142/S0218127406015246.
    [20] H. Nie and J. H. Wu, Asymptotic behavior on a competition model arising from an unstirred chemostat, Acta Mathematicae Applicatae Sinica, English Series, 22 (2006), 257-264.  doi: 10.1007/s10255-006-0301-z.
    [21] H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat, Discrete Contin. Dyn. Syst., 32 (2012), 303-329.  doi: 10.3934/dcds.2012.32.303.
    [22] H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin, European J. Appl. Math., 25 (2014), 481-510.  doi: 10.1017/S0956792514000096.
    [23] H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal., 89 (2010), 1141-1159.  doi: 10.1080/00036811003717954.
    [24] G. PintoA. PollioM. D. Greca and R. Ligrone, Linoleic acid-a potential allelochemical released by Eichhornia crassipes (Mart.) Solms in a continuous trapping apparatus, Allelopathy J., 2 (1995), 169-178. 
    [25] E. L. RiceAllelopathy, 2nd ed, Academic Press, Orlando, FL, . 
    [26] J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. 
    [27] H. L. Smith and  P. WaltmanThe Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
    [28] T. F. Thingstad and B. Pengerud, Fate and effect of allochthonus organic material in aquatic microbial ecosystems: an analysis based on chemostat theory, Mar. Ecol. Prog. Ser., 21 (1985), 47-62. 
    [29] M. X. WangNonlinear Elliptic Equations, Science Press, Beijing, 2010. 
    [30] J. H. WuH. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209-229.  doi: 10.1137/S0036139903423285.
    [31] J. H. WuH. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885.  doi: 10.1137/050627514.
    [32] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.
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