Ladybirds vs Aphids

Ladybirds vs Aphids

At t=0 we have a ladybird on the edge of a leaf at point A(0,10) in cm, and an aphid at point B(0,10).  The ladybird is in pursuit of the aphid.  In each time interval of 1 second the ladybird travels 1cm by heading towards the aphid following the shortest straight-line path.  However during this time interval the aphid moves 1cm along the branch in the positive x direction.

This is shown below:

We use the notation A_n and B_n to represent the position of the ladybird and the aphid respectively at time  t=n. The lengths of all lines from A_n to A_n+1 and from B_n to B_n+1 are 1.  This is shown below:

Finding the distance:

We first can find the distance from A_1 to B_1. This is the distance between the ladybird and the aphid after 1 second by simple Pythgoras:

To find the distance from A_2 to B_2, this distance between the 2 creatures when t = 2 is a little harder – and requires use of SOHCAHTOA, Pythagoras and the cosine rule:

Using geogebra

I then used Geogebra to work out some more distances:

I could then plot these distances (up to t = 9) on Desmos and notice that there was an exponential regression model that looked appropriate:

This gave a model for d(t) at time t as:

We can therefore see that as x approaches infinity, the exponential function approaches 0 and therefore d(t) approaches 4.5.  This means that the ladybird will never get with 4.5cm of the aphid.  The aphid is safe!  Clearly with more data there would be a greater accuracy for the model

Plotting the path of the ladybird

We can use the same data from Geogebra (this time for up to t = 22) to plot the position of the ladybird at discrete time intervals.  This time an exponential model regression did not fit the data well:

So, researching the expected solution to pursuit curves (which is what these paths are called) I found that the expected regression line would be approximated by:

(With p(x) a linear function in x).  Using Desmos this therefore gave:

And we can see that this is indeed a close fit:

We can now use this regression curve to see when the ladybird first gets to within 0.5cm of the branch:

We can see that the x-coordinate at this time is 11.8 and by counting the time (each dot) since t = 0, we have this occuring after 17 seconds.  There’s lots more to explore in pursuit curves – maybe for another post!