To alleviate confusion around this, I decided to model the effect of a refugium/ATS and prove that growing algae in one location will have an effect on the amount of algae grown at a different location (and even has the ability to make it go to 0).
Consider the following setup of a display tank and a refugium/ATS with volumes of V0 and V1, respectively.
The Nutrients (N) in the tank are considered "well mixed", which means that the concentration is instantaneously equilibrated, and local effects are not considered, i.e. algae in the display has access to the identical nutrients in the refugium/ATS and vice versa. p0 and p1 are intrinsic rates of nutrient intake per unit volume for the display tank algae and fuge/ATS algae, respectively (these values are different because of lighting power/spectrum differences, etc. but could also be more general to measure of other factors too).
With this setup, we can model a set of differential equations that describes the change in nutrients in the water column, as well as nutrients that accumulate in algae from the display tank or fuge/ATS.
Here, N(t) is the amount of nutrients in the water column, A0 is the amount of algae in the display, and A1 is the amount of algae in the refugium (both A0 and A1 are in units of nutrients). We can also model a constant influx of nutrients to the tank (i.e. through feeding). In this model, I assume no outflow of nutrients, so the sum of N, A0, and A1 is always increasing by c*t.
Solving these differential equations for N, A0, and A1 yields the following:
Now the neat thing about this is that the growth of algae in the display is proportional to the ratio of intrinsic growth of the display and its volume by the total algal growth (p0V0 / (p0V0 + p1V1)). If we take this to the limits and have a significantly larger Fuge/ATS than the display tank, we can see the algae growth in the display tank goes to zero!
This is a proof that your model is wrong! Algae grown in one side of the tank *DOES* influence algae grown at another, and a sufficiently large fuge/ATS *CAN* completely outcompete algae in the display.
So, finishing our discussion: if a fuge/ATS doesn't work, it's NOT because algae grown at one location can't compete with another location.
Backing up though, how many of us have a fuge/ATS that's larger than our tank?? I doubt many! So what might these curves look like in practice? Well, I computed the nutrient/algae growth as a function of time from the above derivations and there's some informative data!
First, imagine we have a fuge the same size at the display, with identical lighting power/spectrum. Both A0 and A1 curves are the same (green and red respectively, though green is completely overlapped by red in this plot; free nutrients as black curve) and the ratio of algae in the display and in the fuge is 1 (seems reasonable!). BTW, code is in python and available on request!
Next, convincing ourselves that it's possible for a mega sized fuge (100x the display with a comparative advantage of 10x!) to completely out compete the display tank. A0/A1 ratio is 0.001(this value actually just ends up being the ratio of (P0V0)/(P1V1) so I'm dropping it from subsequent plots.
Okay, I know you're saying this isn't realistic! So what about a fuge that's half the size of the display (still a big fuge) with lights that give a 10x advantage (this is a guess but we can throw this parameter around). Doesn't look too bad! definitely helps, but this model predicts significant algae still growing in the display (green curve).
What about a significantly smaller ATS (5% of tank volume) with bigger advantages (50x comparative advantage to display)? Similar split.
What if we have an underpowered fuge/ATS? lets say same size as above but only a 10x advantage to the display?
Now the algae in the display starts to win! Though its important to notice that the display algae is still much lower than it would be without the fuge/ATS.
Now, I wouldn't read into exact values (a little arbitrary), but this shows how we can tune the knobs of our system and how it will influence the tank!