Exploitative competition in a chemostat for two complementary, and possibly inhibitory, resources.

*(English)*Zbl 0609.92035The following model of two populations \((x_ i)\) of microorganisms competing for two complementary resources is considered:
\[
S'(t)=[S^ 0-S(t)]D-\sum^{2}_{i=1}x_ i(t)f_ i(S(t),R(t))/y_{Si},
\]
\[
R'(t)=[R^ 0-R(t)]D-\sum^{2}_{i=1}x_ i(t)f_ i(S(t),R(t))/y_{Ri},
\]
\[
x_ i'(t)=x_ i(t)[-D+f_ i(S(t),R(t))],
\]
\[
S(0)=S_ 0\geq 0,\quad R(0)=R_ 0\geq 0,\quad x_ i(0)=x_{i0}>0,\quad i=1,2,
\]
where S and R denote the resources, and
\[
f_ i(S,R):=\min (p_ i(S),q_ i(R)),\quad i=1,2.
\]
Generalizing the results by S. Hsu, K.-S. Cheng and S. P. Hubbell, SIAM J. Appl. Math. 41, 422-444 (1981; Zbl 0498.92014), it turned out that the form of the functions is irrelevant, only general regularity assumptions and meaningful qualitative assumptions are needed to obtain qualitatively the same results.

An alternative method is also developed that makes it possible to generalize further the results for more general consumption- and conversion functions. A graphical method is also presented to characterize the set of critical points.

It is shown that given monotonous kinetics the model permits regular dynamics in the sense that all solutions tend to equilibria. When at least one of the competitors is inhibited by high concentrations of the substrate then stable periodic solutions may appear. In this case it may also happen that coexistence is possible and neither competitor can survive in the absence of its rival.

An alternative method is also developed that makes it possible to generalize further the results for more general consumption- and conversion functions. A graphical method is also presented to characterize the set of critical points.

It is shown that given monotonous kinetics the model permits regular dynamics in the sense that all solutions tend to equilibria. When at least one of the competitors is inhibited by high concentrations of the substrate then stable periodic solutions may appear. In this case it may also happen that coexistence is possible and neither competitor can survive in the absence of its rival.

Reviewer: J.Tóth

##### MSC:

92D40 | Ecology |

34C25 | Periodic solutions to ordinary differential equations |

92Cxx | Physiological, cellular and medical topics |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

##### Keywords:

exploitative competition; chemostat; populations of microorganisms; monotone increasing uptake functions; complementary resources; critical points; equilibria; stable periodic solutions; coexistence
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\textit{G. J. Butler} and \textit{G. S. K. Wolkowicz}, Math. Biosci. 83, 1--48 (1987; Zbl 0609.92035)

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