1. Von Bertalanffy model for fish growth Aim: To derive a semi-mechanistic mathematical model for growth based on an isometric morphological relationship (i.e. a constant body plan over time) Objectives: 1. Motivate problem using example of changes in weight of fish 2. Derivation of the model 3. Solution of a Bernoulli differential equation 4. Reinterpreting the mathematical solution in context

2. Jacob (Jacques) Bernoulli Born: 27 Dec 1654 in Basel, Switzerland Died: 16 Aug 1705 in Basel, Switzerland 1696: solved class of differential equations now bearing his name Karl Ludwig von Bertalanffy Born: 19 Sep 1901 in Atzgerdorf, Austria Died: 12 Jun 1972 in Aztgerdorf, Austria Today’s heroes 1938: formulated the model we study today

3. Bernoulli Equation – Extract from F.S.

4. Bernoulli is pretty famous The probability distribution used for Coin Tossing etc. …the key building block of the Binomial distribution (will study next term)

5. Bernoulli is pretty famous A crater on the moon

6. Bernoulli is pretty famous A “leminscate” (i.e. a figure 8)

7. Bernoulli is pretty famous A number of German and Austrian streets

8. But really the focus today: a guppy fish

9. Growth of fish

10. Bernoulli Equation – Extract from F.S.

11. Integrating Factors – Extract from F.S.

12. Growth of fish

15. Height Weight