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I'm a little bit confused with the theorem of existence and uniqueness for ODEs (the Cauchy-Lipschitz or Picard-Lindelof theorem if I'm right). Let assume we can apply the theorem on the ODEs $X'=F(t,X)$ in $\mathbb{R}^n$. Let consider that initial conditions are in the positive quadrant $X(0) \in \mathbb{R}_+^n$ (I mean $X(0)>0$.

Is it true to say that if there is the trivial equilibrium $F(t,0^n)=0$, then all solution starting in the positive quadrant $\mathbb{R}_+^n$ stay in $\mathbb{R}_+^n$ ?

The point is that I found a paper in ecology where authors study a system of ODEs on 2 dimensions (in $\mathbb{R}^2$) with $(0,0)$ an equilibrium, and they assume $x_1(0)>0$ and $x_2(0)>0$. Then, they write that theorem of existence and uniqueness ensures positives solutions and the axis cannot intersect and therefore: $x_1(t) >0$ and $x_2(t)$. I don't understand why the trajectory cannot turn around the point $(0,0)$ and so intersect the axis.

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The result is false in general. Counterexample: $$ x'=y,\quad y'=-x. $$ Trajectories are circles centered at $0$.

However, it might be true for the specific differential equation modeling the ecological system in the paper. That will depend on the properties of the function $F$. If for instance $$ x'=f(x,y),\quad y'=g(x,y) $$ with $f,g\ge0$, both $x$ and $y$ will be increasing.

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  • $\begingroup$ Thank you. I actually missed the information that $f(0,y)=0$ and $g(x,0)=0$, and so axis bound the system. $\endgroup$ – Leon-Alph Dec 4 '14 at 16:34

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