# Influence of parameter variations on the solution of an ODE system

Hello community,

suppose we are given a system of ODEs \begin{align} x'(t) &=f(x(t),p) \newline x(0) &= x_0 \end{align}

where $f\in C^1(U,\mathbb{R}^n)$, $U\subseteq \mathbb{R}_{+}^n\times R^m$ is an open set and $p$ is a vector $(p_1,\ldots,p_m)$ of parameters.

Given a time point $t^\star>0$, I want to decide weather

$$\partial_{p_i}x_j(t^\star)>0$$

for given $i$ and $j$ with $x=(x_1,\ldots,x_n)$. Is there a theory I can use for answering this question?

Background:

I perform mathematical modeling of a biological system. The modeled system was only observed at the time point $t^\star$. In one case, a healthy organism was analyzed, resulting in a solution $x(t)$ of the system \begin{align} x'(t) &=f(x(t),p) \newline x(0) &= x_0 \end{align} for fixed parameters $p$. Then, a biologically altered organism was studied, resulting in a solution $y(t)$ of the system \begin{align} y'(t) &=f(y(t),\tilde p) \newline y(0) &= x_0 \end{align} with a changed parameter vector $\tilde p$ (note the unchanged initial conditions). By comparing $x(t^\star)$ and $y(t^\star)$, can we deduce $\mathrm{sgn}(p_i-\tilde p_i)$ for all $1\leq i \leq m$, i.e. how must the parameter vector $\tilde p$ have changed compared to $p$ to explain the change in the obervation $y(t^\star)$ compared to $x(t^\star)$? If one could compute $$\partial_{p_i}x_j(t^\star)>0,$$ then, knowing how the change of a certain parameter $p_i$ would influence the behaviour of the solution $x$ at $t^\star$, it would be possible to derive necessaey conditions on $p-\tilde p$ that can explain the change in the obervation $y(t^\star)$ compared to $x(t^\star)$.

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Maybe you can clarify what you mean by the statement "explain the change in observation y(t) compared to x(t)". Dependence on parameters is in a sense same as dependence on initial conditions, if the r.h.s. depends on smoothly on the parameters (Lipschitz is enough). So, you can get a bound on the difference between x and y in terms of a norm of $p−\tilde{p}$, if that is what you are after. –  Piyush Grover Apr 9 '13 at 20:58
Here's a nice document showing the bound, and an IVP which governs the partial derivatives you are interested in: math.byu.edu/~grant/courses/m634/f99/lec6.pdf –  Piyush Grover Apr 9 '13 at 21:01
Nonlinearism, thank you for your reply. Both $p$ and $\tilde p$ are unknown, so I want to deduce properties of $p-\tilde p$, such as $\mathrm{sgn}(p-\tilde p)$, that would imply $\mathrm{sgn}(x(t^\star)-y(t^\star))$. I hope that helps for clarification. I've posted another question, which is related to this one but interesting in it's own: mathoverflow.net/questions/127654/… –  Jack Thomas Apr 15 '13 at 20:43