### Unlimited growth: Linear and quadratic growth

### Examples of linear growth

Linear growth is encountered in biology in selected growth stages or under controlled (lab) circumstances. For instance, the growth of fish larvae.

Length growth of a fish The average length *L* (in mm) of fish larvae of the white croaker (Genyonemus Lineatus), a fish from the family of umber fish, after *t* days, is well described by the mathematical formula \[L=-0.833+0.242t\] The growth rate is 0.242 millimetres per day. The average length after 10, respectively. 40 days is 1:59, respectively. 8.85 mm.

Source: Millar et al (2011), "Queen Fish (queenfish) and White Croacker (white croaker) Larval Growth Parametres" *SCIAENID Larval AGE AND GROWTH,* CalCOFI Vol.. 52, 75-79.

A second example, in this case of a linear decay, is a chemical reaction with zero-order reaction kinetics.

Zero-order reaction kinetics The gas dinitrogen monoxide \(\mathrm{N}_2\mathrm{O}\) (also known as nitrous oxide) dissolves into nitrogen gas \(\mathrm{N}_2\) and oxygen gas \(\mathrm{O}_2\) in the presence of finely divided gold powder, which acts as a catalyst: \[2\mathrm{N}_2\mathrm{O}\stackrel{\mathrm{Au}}{\longrightarrow}2\mathrm{N}_2+\mathrm{O}_2\]

It is striking, that the reaction rate does not depend on the amount of starting materials or reaction products, but it is constant. The reason for this, is that the gold surface can only accommodate a limited number of molecules and the dinitrogen monoxide concentration has no influence on it. The concentration \(C\) of dinitrogen monoxide as a function of time \(t\) can therefore be described as \[C'(t)=-k\] for certain constant \(k\). The solution of this differential equation gives a linear function \[C(t)=C_0-k\cdot t,\] where \(C_0\) is the concentration at time \(t=0\) is.

By the way, the term "zero-order" in the naming of the reaction kinetics does not refer to the order of the corresponding differential equation. The differential equation has order one. The addition of 'zero-order' refers to the right hand side of the differential equation, and indicates that the reaction rate does not depend on the concentration of the reactants.