Limited exponential growth: Introduction
The von Bertalanffy model of length growth of fish
Modelling length growth of fish The length growth of fish is often described by the following mathematical model, introduced by Ludwig von Bertalanffy (1901-1972) in the context of research into aquaculture and wild life. The same model was previously introduced by Eilhard Mitscherlich (1874-1956) in the context of plant and agricultural research. The reasoning in the context of longitudinal growth of fish is as follows.
Suppose \(L(t)\) is the average fish length of a particular species at an age of \(t\) months. We may assume that the length growth ends and that the fish reach an average adult height, which we will refer to as \(L_v\). Also, it's not crazy to assume that the growth rate decreases as the length comes closer to the adult height. The simplest is, to assume that the growth rate is proportional to the length difference \(L_v-L(t)\); this corresponds to the following differential equation \[\frac{\dd L}{\dd t} =r\cdot (L_v-L)\] where \(r\) and \(L_v\) are positive constants. This differential equation is exactly solvable.
The von Bertanalffy model of length growth The von Bertanalffy model of length growth consists of the differential equation \[\frac{\dd L}{\dd t} =r\cdot (L_v-L)\] where \(r\) and \(L_v\) are positive constants. The exact solution of this differential equation is \[L(t)= L_v\cdot \bigl(1-e^{-r\cdot t}\bigr)+L_0\cdot e^{-r\cdot t}\] where \(L_0=L(0)\).
This solution can be found by the method of separation of variables. For math enthusiasts we present here an elementary proof, on the basis of transforming the problem into the differential equation of unlimited growth.
Start with a new quantity \(y\) via \[y=L-L_v\] Then: \[\begin{aligned}\frac{\dd y}{\dd t}&=\frac{\dd(L-L_v)}{\dd t} & \blue{\text{substitution}}\\[0.25cm]&=\frac{\dd L}{\dd t} & \blue{\text{calculation rules for differentiation}}\\[0.25cm]&= r\cdot (L_v-L) & \blue{\text{definition of the model}}\\[0.25cm]&= -r\cdot y & \blue{\text{rewriting in terms of }y}\end{aligned}\] The quantity \(y\) therefore satisfies the differential equation of exponential decay, \(y'(t)=-r\cdot y(t)\), and the general solution is \[y(t)=c\cdot e^{-r\cdot t}\] for a certain constant \(c\). But then we know that the length \(L(t)\) can be expressed as follows: \[L(t)=L_v+c\cdot e^{-r\cdot t}\] Using \(t=0\) and because \(e^0=1\), we know that \[L(0)=L_v+c\] that is \[c=L_0-L_v\] where \(L_0=L(0)\). So we arrive at the following formula: \[\begin{aligned}L(t)&=L_v+(L_0-L_v)\cdot e^{-r\cdot t}\\[0.25cm]&= L_v\cdot \bigl(1-e^{-r\cdot t}\bigr)+L_0\cdot e^{-r\cdot t}\end{aligned}\]
In biology literature you often see that the remaining free parameter \(L(0)\) in the case \(L_0<L_v\), is placed differently into the formula: \(L_v-L_0= L_v\cdot e^{t_0}\) for a certain negative parameter \(t_0\), rather than \[L(t)=L_v\cdot \bigl(1-e^{-r\cdot (t-t_0)}\bigr)\] Parameter \(t_0\) then gives a theoretical age at which the length is equal to 0. Such a negative age at which length is equal to 0 is needed because animals have an initial length at birth. As the elapsed time \(t\) becomes larger, the exponential term \(e^{-r\cdot (t+t_0)}\) decreases and the length \(L\) is approaching the maximum value \(L_v\).
Length growth of pike As a concrete example, let's look at the length of a pike (Esox lucius, Linnaeus 1758). The length that a pike reaches depends on the temperature (longitude), availability of food, and the growth rates of the previous (juvenile) stages of life. Under average growth conditions in the Netherlands, the pike reaches a length of 22 cm in the first year; at the end of the fourth year of life, the pike has reached a mean length of 55 cm. Usually, females live 12 to 13 years and males 5 to 6 years (Raat, 1998). Exceptionally, pike can reach ages of 25 up to 30 years. Craig (1966) reports the following values for the parameters of the growth model: \(L_v=87.0\;\mathrm{cm}\) and \(r=0.188\;\mathrm{year}^{-1}\), at an initial length \(L_0\) taken to be equal in this model to \(5.6\;\mathrm{cm}\), which corresponds to a theoretical age of \(t_0=-0.357\;\mathrm{years}\), at which time the length is equal to 0 cm. The formula of Craig is therefore: \[L(t)=87.0\cdot \bigl(1-e^{-0.188\cdot (t+0.357)}\bigr)\] The growth in length of a pike does not go so rapidly: only after more than 15 years the average length is within 5% percent of the maximum length, according to this model.
