I have a large system (N>100) of equations
$\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$
where $\vec{P}$ is a vector of N functions of the variable t.
What is the correct name of these kind of equations in Mathematical Ecology?
What do you guys suggest to do to numerically solve this equation as efficiently as possible? I am not familiar at all with numerical packages or strategies.
PS1: I am currently using Mathematica method NDSolve (with the option "EquationSimplification"->"Solve" which I am not sure what it does).
PS2: For sake of completeness, the actual structure of this quadratic form is
$\frac{dP_i}{dt}= \sum_j [\alpha^1_{ij}(t) P_i(1-P_j)+\alpha^2_{ij}(t) (1-P_i)P_j+\alpha^3_{ij}(t) P_i P_j+\alpha^4_{ij}(t) (1-P_i)(1-P_j) ]$
Has this form a name? It is pretty common in semiconductor physics (non-equilibrium kinetic equation for Fermionic open quantum systems).