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Growth of cell populations via one-parameter semigroups of positive operators. (English) Zbl 0643.92017
Mathematics applied to science, Mem. E.D. Conway, Proc. Conf., Tulane Univ., New Orleans/La. 1986, 79-105 (1988).
[For the entire collection see Zbl 0628.00002.] The asymptotic behaviour of solutions of a mathematical model for the growth of cell populations is studied. The mathematical equation describing the growth of the cell population is transformed into the following abstract Cauchy problem: $$ (1)\quad du(t)/dt=A u(t),\quad u(0)=u\sb 0\in D(A)\subset E, $$ where $E=L\sb 1(\alpha /2,\beta /2)$ and $\alpha$ and $\beta$ are the minimal and maximal cell size, respectively. After recalling some results on well-posedness of abstract Cauchy problems and positive semigroups, it is shown that (1) is well-posed and that the semigroup generated by A, $(T(t))\sb{t>0}$, is positive. Using these results and other properties of $(T(t))\sb{t>0}$ it is shown that if the growth rate g(x) is such that $2g(x)>g(2x)$ for some $x\in (\alpha /2,\beta /2)$ then T(t) converges in the operator norm to a one- dimensional strictly positive projection as $t\to +\infty$. On the contrary, when $2g(x)=g(2x)$ for all $x\in (\alpha /2,\beta /2)$, $(T(t))\sb{t>0}$ considered in $L\sb 1(\alpha /2,\beta /2)$ with a suitable weighted norm, converges to a rotation semigroup with period $\int\sp{\alpha}\sb{\alpha /2}g\sp{-1}(t)dt$.
Reviewer: S.Totaro

92D25Population dynamics (general)
47D03(Semi)groups of linear operators
47H07Monotone and positive operators on ordered topological linear spaces