By Walter E Thirring
Mathematical Physics, Nat. Sciences, Physics, arithmetic
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Extra info for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems
Steady states analysis, a˙ = 0 and x˙ = 0, reveals three different scenarios for (i) the uncatalyzed reaction, (ii) first order autocatalysis, and (iii) higher order autocatalysis. All three scenarios have the same limiting cases: At vanishing flow rate r the system approaches thermodynamic equilibrium with x¯ = k1 c0 /(k1 + k2 ) and a ¯ = k2 c0 /(k1 + k2 ), no reaction occurs at sufficiently large flow rates when the mean residence time is too small and then we have x¯ = 0 and a ¯ = c0 for lim r → ∞.
Fisher’s works together with the contributions of Sewall Green Wright (Wright, 1932, 1986) and John Burdon Sanderson Haldane (Haldane, 1924a,b) mark the beginning of population genetics being understood as a branch of applied mathematics that deals with evolutionary biology (Ewens, 1979; Hartl and Clark, 1997). The origin and the development of molecular biology in the second half of the twentieth century provided a new basis for modeling selection and evolution. , 1989). 10) and all other cases where the whole replication process is modeled by a single step only.
The flow rate r(t) is regulated such that the total concentrations of replicators is constant, c = c0 :3 N i=1 dci = 0 = dt N i=1 ci (a ki − r) = c0 (a · φ − r) , and r = a·φ . 19) is readily interpreted: In order to keep the concentration of replicators constant, the flow rate has to be raised when either the concentration of A or the mean replication rate of the population, φ, increases. The conservation relation c = c0 reduces the number of independent variables from N + 1 to N : a and N − 1 concentration variables cj .