I need some help finding a solution to a system of coupled non-linear differential equations that are similar to the Predator-Prey model (Lotka-Volterra equations), although this is a problem in atomic physics.

I am interested in the dynamical variables $[d_1(t),d_2(t),...d_{N_d}(t)]$ and $[a_1(t),a_2(t),...,a_{N_a}(t)]$ for time $t>0$.

The initial value of all variables is 1; that is, $d_i(0) = 1 ~ \forall i$ and $a_m(0) = 1 ~ \forall m$.

The time progression of these variables is governed by the following set of differential equations

\begin{eqnarray} \frac{\text{d}d_i(t)}{\text{d}t} &=& -\left(A_i+\sum\limits_m\Gamma_{i,m}a_m(t) \right)d_i(t) \\ \frac{\text{d}a_m(t)}{\text{d}t} &=& -\left(B_m+\sum\limits_i\Gamma_{i,m}d_i(t) \right)a_i(t) + B_m, \end{eqnarray}

where $A_i$, $B_m$, and $\Gamma_{i,m}$ are positive constants.

The above is the generalized problem and produces a set of $N_d+N_a$ coupled non linear differential equations. For simplicity I will work with a small number case. If I set $N_d=2$ and set $N_a=1$, I then get three coupled non-linear differential equations:

\begin{eqnarray} \frac{\text{d}d_1(t)}{\text{d}t} &=& -\left(A_1+\Gamma_{1,1}a_1(t) \right)d_1(t) \\ \frac{\text{d}d_2(t)}{\text{d}t} &=& -\left(A_2+\Gamma_{2,1}a_1(t) \right)d_2(t) \\ \frac{\text{d}a_1(t)}{\text{d}t} &=& -\left(B_1+\Gamma_{1,1}d_1(t)+\Gamma_{2,1}d_2(t) \right)a_1(t)+B_1. \end{eqnarray}

Does anyone have a method for solving the last three equations? I know how to numerically solve them using FDTD but would prefer a more elegant solution.

If anyone has any references in which they have seen any of the above two sets of equations, please reply.

Any help or suggestions would be much appreciated.


Ward Newman

University of Alberta


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