Perhaps it is somewhat difficult to give you a straight answer to your questions because it is not totally clear what you try to do/what your models do. At least not to me. You present 4 equations. Equations (1) and (2) seem to be balance equations for some quantity B_i.
- Is index i in B_i the species type? E.g., if you have two species, could B_1 be “hare” and B_2 be “fox”? In your simulation code, you indicate 50 species. Does that mean B_1, …, B_{50}?
- Are Eqs. (1) and (2) two possible choices of models, or will they occur simultaneously?
- Are r_i and K_i in Eq. (1) constants? What about e_i and x_i in Eq. (2)?
- In your Eqs. (1) and (2), when you write \sum_i F_{ij}B_i, do you mean that, or do you mean \sum_j F_{ij} B_j. Likewise, in Eq. (2), do you mean \sum_j e_j F_{ij}B_i or do you mean \sum_j e_j F_{ij} B_j?
Your Eq. (3) seems to be some sort of “Michaelis-Menten” saturation function (or: rather a Hill model). Your code indicates that q is a Hill exponent.
- Is H a constant? Independent of i/j?
- Is T_i some “temperature”?
- What is a_i in the numerator of the expression for F_{ij}? A constant? Or related to a_{ij} in Eq. (4)?
- What is B in the numerator of the expression for F_{ij}? Any relationship to B_i?
- What is the sum index in the numerator of the expression for F_{ij}? Do you mean \sum_i?
In your Eq. (4), this looks like a “reaction expression” with an Arrhenius temperature dependence. I assume E is “activation energy” – not per mol, but per individual, and that E/k has the unit of temperature. The term \frac{T_0 - T}{T T_0} = \frac{1}{T} - \frac{1}{T_0} supports the impression that this is an Arrhenius temperature dependence. Still:
- Is d a “reaction constant”, or is it the differential operator so that dm_i is the differential of m_i? Probably the first, but using d always creates uncertainty.
OK – my questions reflect uncertainty in the problem that you consider. I assume this is some on-going research you do, so perhaps you don’t want to reveal every detail.
Now, looking at your code, I assume that Feeding returns F_{ij} – although I don’t understand why you introduce identifier x within that function (is it related to x_i in Eq. (1)).
Also, I don’t understand what functions Basal and Consumer are meant to do, related to your Eqs. (1)-(2), and what is FoodWeb, etc.
OK – it would be easier to help if the problem was described somewhat clearer. It will still be possible to help without understanding your model, but it is necessary to understand what you want to achieve.