The von Bertalanffy growth model is used to predict the length L(t) of a fish ov

Question

The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L_infinity is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L_infinity - L, the length yet to be achieved. (a) Formulate and solve a differential equation to find an expression for L(t). (b) For the North Sea haddock it has been determined that L_infinity = 53 cm, L(0) = 10 cm, and the constant of proportionality is 0.2. What does the expression for L(t) become with these data?

Explanation / Answer

19)

a)dL/dt =k(L -L)

seperate the variables

dL/(L -L) =kdt

integrtate on both sides

dL/(L -L) =kdt

-ln|(L -L)| =kt +c

ln|(L -L)| =-kt +c

(L -L) =e-kt +c

(L -L) =Ce-kt

L=L -Ce-kt

so L(t)=L -Ce-kt

b)L=53cm ,L(0)=10cm ,k=0.2

L(0)=L -Ce-k*0

10=53 -Ce-0.2*0

10=53 -C

C=43

L(t)=L -Ce-kt

L(t)=53 -43e-0.2t

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