Solving a system of an arbitrary number of ODEs, at steady-state

[kalish1]

I have constructed a system of an arbitrary number of ordinary differential equations that describes my model at steady-state. There are ($i+1$) ODEs, with $i$ arbitrary.

Goal: I want to solve the resulting algebraic system (all equations set to $0$) and obtain an analytical expression that captures the dependence of $S_T = \sum\limits_{j=1}^{i+1} S_j$ on $\alpha$.

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Here is the system:

$$ Q-\left(\dfrac{\alpha}{X} + r_1\right)S_1+\dfrac{\beta}{Y}\sum\limits_{j=2}^{i+1} S_j = 0 \\
\dfrac{\alpha}{X}(S_1-S_i)-\left(\dfrac{\beta}{Y}+r_1\right)\sum\limits_{j=2}^{i} S_j = 0 \\
\dfrac{\alpha}{X}S_i-\left(\dfrac{\beta}{Y}+\sigma r_1 \right)S_{i+1} = 0
$$

The first equation is the ODE for $S_1$, the last is for $S_{i+1}$, and the middle is for $S_2$ to $S_{i}$. I know beforehand that the parameters $Q, \alpha, X, r_1, \beta, Y$ and all $S_j$'s are positive real numbers and $\sigma > 1.$

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Work in progress:
Note that $$\frac{d}{dt}S_T = \frac{d}{dt} \left(\sum\limits_{j=1}^{i+1}S_j\right) = Q - r_1 \sum\limits_{j=1}^{i} S_i - \sigma r_1 S_{i+1} = Q - r_1 S_T + r_1 (1-\sigma) S_{i+1},$$ implying that $$S_T = \frac{Q}{r_1} + (1-\sigma)S_{i+1} - \frac{1}{r_1} \dot{S}_T.$$

Now notice that upon solving for $S_1$ in the first and second ODE equations respectively, we obtain:

$$S_1 = \dfrac{X}{Y(X r_1 + \alpha)}\left(QY + \beta \sum\limits_{j=2}^{i+1}S_j\right)$$

and

$$S_1 = \dfrac{X(\beta+Y r_1)}{\alpha Y} \sum\limits_{j=2}^{i} S_j + S_i.$$

Question: How can I find $S_{i+1}$ explicitly so that I can (hopefully) solve the linear ODE for $S_T$?

Thanks in advance for help.

This question has been crossposted here: http://math.stackexchange.com/questions/1537553/solving-a-system-of-an-arbitrary-number-of-odes-at-steady-state

 

[jvicens]

Thank you for sharing your progress and your question. It seems like you are on the right track in your approach to solving this system of ODEs. However, finding an explicit expression for $S_{i+1}$ may not be possible in this case. This is because the system is nonlinear and it may not be possible to find an analytical solution for $S_{i+1}$.

In this case, it may be more useful to use numerical methods to solve the system. You can use a numerical solver to solve the system of ODEs and obtain values for $S_1$ to $S_{i+1}$ at steady state. From there, you can use these values to calculate $S_T$ and see how it depends on $\alpha$. This approach may be more feasible and accurate compared to trying to find an explicit expression for $S_{i+1}$.

I hope this helps. Good luck with your research!