I'm trying to better understand what this question is asking. The critical substance approaches 1 as time approaches infinity. But I have no clue what critical substance is and no idea what the value of this limit signifies. Any help?
" One of the first models of bacteria movement was developed by Evelyn Keller and Lee Segel, as reported in their paper "Traveling Bands of Chemotactic Bacteria: A Theoretical Analysis."  Keller and Segel used partial differential equations to model bacteria swimming in a tube. The bacteria traveled in bands toward nutrients and oxygen, coined "critical substance." "
[i]"Q(t) is said to represent the physiological state of the cells, being proportional to the per cell concentration of a [b]critical substance produced by the cells.[/b]"[/i]
(emphasis added by me)
It seems that there is some disagreement on the usage of the term. The first source says that the critical substance is nutrients that the bacteria feed on, and the second says that the critical substance is produced by the bacteria. You should probably ask whoever assigned that problem to you which one is meant.
Okay, now that we have some clarification for us biologists (sorry, the equations meant nothing to me), it sounds like "critical substance" is just being used as some vague term to describe anything that could affect bacterial behavior. Bacteria will move along nutrient gradients, and they secrete substances for communication. I don't see the reason for someone to be so vague and not just define the specific substance they are interested in measuring responses to rather than giving it some useless name like "critical substance."
What units are being used? The critical substance approaches 1 what? 1%, a ratio of 1, 1 arbitrary unit on a scale of 0 to 1? A number with no units is meaningless.
I'm sorry. This was just a extra cred question the math prof threw at us. Unfortunately not much info was given. This is mainly a question about the Baranyi Population Model. This question goes as follows:
A population model that is currently used in food science research is the Baranyi Population Model. This model tries to account for the way certain critical substances affect bacterial growth. This model can be reduced to the two initial value problems.........I've already solved it and found the equation for critical substance, q(t), as outlined in redirected post to the math forum.
I agree a unitless number is meaningless. Unless it is some sort of ratio. But a ratio to what? It there anyway of figuring out the meaning just from the context?
EDIT: It also seems that the limit only ranges from 0 to 1.
As for taking the limit of q(t)/(1+q(t)) instead of simply q(t), I have no idea why it was given in this form, Tom.
It sounds a bit out of touch with actual biology. Based on what Tom and you have provided, let's say it's something secreted by the cells into a culture medium. As time progresses, the concentration of that will increase. Bacterial growth rate is exponential, so unless something inhibits the production of that substance with increased bacterial population, the secretion of that substance would also occur exponentially. However, once nutrient availability becomes depleted, bacterial growth will plateau and then decrease as they die off. The substance, assuming it doesn't have anywhere else to go, would also plateau in secretion, but would remain fixed, or maybe continue increasing more slowly as the surviving bacteria continue to secrete it.
The only way to continue producing it for an "infinite" amount of time would be to have an open system where nutrients could continuously be added, but adding nutrients and having an open system would also mean that whatever substance was being secreted could also dissipate out of the system, and would be diluted by whatever was bringing in the nutrients. At some point, you might reach a stable equilibrium, with a steady supply of nutrients, growth of bacteria that were able to move out of the system so overpopulation does not occur, and the amount of whatever substance is being produced equlibrates with the amount being lost. So, unless 1 represents an equilibrium state in an open system, or represents the maximal amount of a substance that can be secreted into a closed system before all the bacteria die...but that would be a finite time limit, not an infinite one...I really don't know what that's all supposed to mean.
It may be a good example of the usual outcome of attempts to mathematically model biological systems...they don't work, because there are too many variables, and all the model tells you is that you're still missing some or they don't behave as you predicted.