Dynamical Systems in Biology

Tutorial 0

Exercise 1: matrix manipulation

There are a couple of pre-defined vectors and numbers in R, e.g., pi, letters, LETTERS, etc..

• Create a 26 x 26 matrix C where each row contains all letters from A to Z. Do not use a loop!
• Change all “V”-entries to the square numbers 1 to 676. Determine by the == operator which row contains the number 100.
• Restrict the matrix C to a 9 x 9 square matrix containing only the letters E, …, I and W, …, Z.
• Generate a new matrix E with entries TRUE and FALSE. The entries should be TRUE if the corresponding letter of the matrix C appears before K in the alphabet.

Exercise 2: functions and loops

Write a function f <- function(n) which returns a vector for the integers 1,…,n with entries TRUE or FALSE for even or odd numbers, respectively.

• Implement the function by looping over i=1,…,n. Have a look at is.integer() or the %% operator for testing if a number is odd or even (?is.integer, help("%%")).
• A key feature of R is vectorization, enabling the calculation of operations on vectors and matrices and making loops mostly unnecessary: implement the function without using a loop.

Exercise 3: Newton method

Given the ODE for the Verhulst model \[\dot{x} = f(x,\vec{p}) = p_1 x \left(1-\frac{x}{p_2}\right)\] find the points \(\bar{x}\) where \(\dot{\bar{x}} = 0\) numerically.

• Write a function fg <- function(x,p) returning the vector \(\big(f(x,\vec p), f'(x, \vec p)\big)\).
• Implement the newton method \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) in a function newton <- function(x0,prec) with initial value x0 and precision prec. Stop the iterations when \(|x_{n+1} - x_n| <\) prec is reached.

For an illustration of Newton’s method see: https://en.wikipedia.org/wiki/Newton's_method#/media/File:NewtonIteration_Ani.gif