Existence and Uniqueness of the Cauchy Problem for General Competition Systems? I have the original system
\begin{equation}
    \begin{array}{lcl}
      \dot u_{1} & = & A_{1}u_{1}(1 - u_{1} - a_{12}u_{2} - \dots - a_{1n}u_{n})  \\
        \dot u_{2} & = & A_{2}u_{2}(1 - a_{21}u_{1} - u_{2} - \dots - a_{2n}u_{n}) \\ 
        & \vdots & \\
        \dot u_{n} & = & A_{n}u_{n}(1 - a_{n1}u_{1} - \dots - a_{n(n-1)}u_{n-1} - u_{n}),
    \end{array}
\end{equation}
which is a general n-dimensional competition system. We use the fact that $A_{i} = a_{ii}$ to write this system as
\begin{equation}
    \dot u_{i} = \Phi_{i}(t; u_{1}, u_{2}, \dots, u_{n}),
\end{equation}
where
\begin{equation*}
\begin{array}{lcl}
    \Phi_{1} & = & u_{1}(a_{11} - a_{11}u_{1} - a_{11}a_{12}u_{2} - \dots - a_{11}a_{1n}u_{n}) \\
    \Phi_{2} & = & u_{2}(a_{22} - a_{22}a_{21}u_{1} - a_{22}u_{2} - \dots - a_{22}a_{2n}u_{n}) \\
    & \vdots & \\
    \Phi_{n} & = & u_{n}(a_{nn} - a_{nn}a_{n1}u_{1} - \dots - a_{nn}a_{n(n-1)}u_{n-1} - a_{nn}u_{n}).
\end{array}
\end{equation*}
Here, the $a_{ij}$'s are smooth, positive 1-periodic functions of t defined over $\mathbb{R}$. Before I properly consider the periodic case, I need to consider properties of the time-averaged system (which, in this case, will be autonomous). We define this as
\begin{equation}
    \dot w_{i} = \phi_{i}(w_{1}, w_{2}, \dots, w_{n}),
\end{equation}
where
\begin{equation*}
    \phi_{i}(w_{1}, w_{2}, \dots, w_{n}) = \int_{0}^{1} \Phi_{i}(w_{1}, w_{2}, \dots, w_{n}) \ dt.
\end{equation*}
It follows that the system (7) can be written as
\begin{equation}
    \dot w_{i} = w_{i}\bigg(\bar{a_{ii}}(1 - w_{i}) - \sum_{k \neq i} \bar{a_{ik}}w_{k}\bigg), \ \ \ i = 1, 2, \dots, n,
\end{equation}
where
\begin{equation*}
    \bar{a_{ij}} = \int_{0}^{1} a_{ij}(t) \ dt.
\end{equation*}
Now, my goal is to prove that the Cauchy problem for this system $\dot w_{i}$ has a unique global solution whenever the initial data $w_{0} = (w_{0_{1}}, w_{0_{2}}, \dots, w_{0_{n}})$ satisfy $w_{0_{i}} \in \mathbb{R}_{+0} = \mathbb{R}_{+} \cup \{0\} \forall i \in [1,n].$
Unfortunately, I have not been able to find much information on conditions for the existence and uniqueness of nonlinear systems of ODEs, or at least for general n-dimensional competition systems like this one.
How should I approach a proof for this? I'm a bit stuck.
 A: I think the hardest part here is to prove that solution exists for all positive times. Let me show this. Since $a_{ij}(t) > 0$, then $\overline{a}_{ij} > 0$. But that allows us to say that $\dot{w}_i \leqslant 0$ as long as $w_i \geqslant 1$ and other variables are non-negative. So, for any initial condition $(w_1^{(0)}, w_2^{(0)}, \dots, w_n^{(0)})$ this leads to the following statement: the trajectory doesn't leave a hyper-parallelepiped $\bigcap\limits_{i=1}^{n} \lbrace 0 \leqslant w_i \leqslant w_i^{(0)} \rbrace $. It is due to Bony-Brezis theorem: when $w_i = 0$, then $\dot{w}_i = 0$, and $\dot{w}_i < 0$ when $w_i = w_i^{(0)}$, which means that vector field is pointing inward at the boundary of this hyper-parallelepiped (when $w_i = w_i^{(0)}$) or is being tangent when $w_i = 0$. If trajectory does not leave a compact subset of phase space for $ t > 0$, it exists for all positive times (see, for example, theorem here, p. 19). For each starting point outside of $\bigcap\limits_{i=1}^{n} \lbrace 0 \leqslant w_i \leqslant 1 \rbrace $ we've shown that trajectory enters a compact set $\bigcap\limits_{i=1}^{n} \lbrace 0 \leqslant w_i \leqslant w_i^{(0)} \rbrace $ and does not leave it. The hyper-parallelepiped $\bigcap\limits_{i=1}^{n} \lbrace 0 \leqslant w_i \leqslant 1\rbrace $ is also invariant, so we have covered other points as well.
