https://www.flandershealth.us/microbiology/a-biased-random-walk.html
Im interested in simplest math, simulation friendly, that could describe how microbes move. Their movement is similar to a random walk, but with some bias towards their goal, that eventually allows them to move closer to their target. And their movement is continuous, or much larger than a grid per step.
First idea is something like this, for 1d:
x=x+abs(x-goal)*rand(-1,1)
This makes very unstable solution, it often walks much further away from the goal, but it seems to have some very weak bias towards the goal, so that it moves closer to the goal a bit more often. Making it more inert, to change x less per step, does not help to make it more reliable, and actually make it worse for some reason.
Real microbes seem to use variable time of movement in the same direction, and measure if they move closer to the goal continuously, im not sure how to implement it in a form of math, or if it will give any more reliable behaivor. Or if it is even different from my example.
Im interested in a solution that is similar to mine, that does not use tricks like measuring distance to the goal from two sensors, like big cells do. And Im interested in a solution that doesnt rely on PID-like behaivor, because Im interested to learn properties of these random walk solutions, as a way to make search for a goal less reliable on low noise on particular frequencies that PID-like solution may oscillate at.
Good answer that I would want to see would be simulation friendly, would avoid the use of PID-like solution, or two sensors measurments, would have stability enough to guarantee that random walk doesnt go further away from the goal than constant factor of initial distance. And ideally can operate even if some small random or even systematic error to measurment or comparison is added.
Or would link something similar to read, that connects math simulations and chemotaxis of microbes, in simple math, not perfect bio simulation that is too complex to use elsewhere.
P.S. I dont know math well, and I know next to nothing about this topic I try to describe. I would prefer simple answer if possible.
I don't know if this helps completely, but one of the sources I found details a method which (as I understand it) describes the movement of a microbe as being based on the change in concentration of a neighboring substance. This is all still stochastic, of course, because it relies heavily on probability, but other than that, there are still quite a few parameters you may have to tweak in order to get the desired behavior. In case you are curious as to where I got this information from, here is a link to the original article:
https://canvas.ucsc.edu/courses/1077/pages/modeling-bacterial-chemotaxis
Several other sources that make an attempt at the same problem are shown below:
https://brilliant.org/discussions/thread/solving-the-random-walk/ https://trace.tennessee.edu/cgi/viewcontent.cgi?article=6598&context=utk_graddiss https://pubmed.ncbi.nlm.nih.gov/18642048/ https://arxiv.org/abs/1806.08627