Question
The following model is an approximation of the discrete logistic model $x_{t+1} = f(x_t)$ where $$ f(x) = \begin{cases} \mu x, & 0 \le x \le 1/2,\\ \mu(1-x), & 1/2 \le x \le 1. \end{cases} $$ For which values of the parameter $\mu$ is this model well-defined, i.e., it can be used iteratively without being biologically invalid?
This question was asked in one of my textbook for my maths biology course, it really doesn't seem very hard however I'm unsure what it means for the model to be 'well-defined' or 'biologically invalid'. I understand what it means for a function to be well-defined however in a biological way I'm unsure, does this mean choose μ such that there exists steady states? or just that for any real x$_0$ f(x$_t$) is real as well? I most definitely don't want the answer for this, I would simply like explaining what 'well-defined' would mean/look like in this context.
To expand on my comment:
From a mathematical perspective, to use an iteration, we need $f: [0,1] \mapsto [0,1]$. Otherwise we end up with values that can't be plugged back into the original function! It's clear that we need $\mu \in [0,2]$ for this to be true.
From a biological perspective, there are a couple of things to consider. First, presumably $1$ is a maximum population: a carrying capacity of some sort, which is why the function maps back to $[0,1]$. In addition, we need to consider what the value of $\mu$ implies biologically.
For instance, $\mu=0$ implies a population that will go extinct immediately. But in fact, for all $\mu <1, x \to 0$ as $t \to \infty$; so any population with $\mu<1$ will dwindle and die.
If we want to examine a system that is changing in population over time but not have species die out, we need $1<\mu \le 2$. Note that $\mu=1$ implies a total steady state somewhere below $x=1/2$. In addition, $\mu=2$ is going to force the function to settle into a specific subsets of orbits, so it's possible the instructor wants $1 <\mu<2$ instead.
I think my answer would be that mathematically, the function is well-defined for $\mu \in [0,2]$, but that biologically it is only meaningful for $\mu \in (1,2)$. That is, we can't derive any information from a system where $\mu \le 1$ even if it's mathematically consistent.
But this is at least partially opinion; ecology and population biology was never my strong suit, I was in molecular bio and dealt with proteins. Hey would you like to talk about Michaelis-Menten kinetics? Wait, where are you going...