(The linked picture should be very helpful in understanding this question!)
I am working on a population model using ordinary differential equations. I have a 1D system made of 7 populations, where transitions occur in one direction between adjacent populations, starting from a population with $100\%$ of the maximum value of one parameter and ending at a population with $0\%$ (i.e. if we call the parameter $a$, then we could have values e.g. $a_1 = 100, a_2 = 83, \dots, a_7 = 0$). Each transition between adjacent populations occurs at the same rate $\sigma$.
I extended this to make a 2D system of 4x4 populations so we can have two parameters $b$ and $c$ taking the values $100,66,33,0$. Here, each population is coloured by its distance away from the first population, where the average of the two parameters $\frac{b+c}{2}$ can take exactly the possible values of $a$ from the 1D system. (So if $b = 66, c = 33$, then the average is $50$, so we label this green to match the $a = 50$ state in the 1D case.)
In the 2D system, the transition rates are assigned so that the sum of the transition rates out of one 'colour family' of cells is exactly equal to $\sigma$ in the 1D system. For instance, the purple population $(100,100)$ transitions to each of the dark blue populations $(100,66)$ and $(66,100)$ at a rate $\frac{\sigma}{2}$. In my head, I have been imagining the transition between these populations as a 'flow', so that when comparing the 1D and 2D systems, the flow in and out of each colour is the same.
To test the models, I ran simulations where I set a parameter $k$ to be equal to $a$ in the 1D system and $\frac{b+c}{2}$ in the 2D system, with all other parameters kept equal between the two. Both simulations started with all population sizes set to $0$ except the 'original' $100\%$ purple population. My expectation was that the population dynamics for each colour family should be identical between the two systems but this was not the case. Instead, in the 2D system, the purple population dies out sooner, and the intermediate populations arise sooner.
The exact implementation of this system is more complex than I have described here. After spending a week dissecting my code, I have been unable to identify any errors in the code or in the implementation of this transition scheme, which has led me to question whether my fundamental assumptions about these two schemes being identical is incorrect.
The point of my question is therefore to work out whether my intuition about 'flow rates' being equivalent between 1D and 2D with 'grouped families' is incorrect (can this idea be applied in general?) or if it is likely that the discrepancy is instead arising from some other aspect of my model.