Let $$\Delta=\{x_i\in\mathbb{R}^k_+: \sum_{i=1}^kx_i=1\}$$ where k is a positive integer. I've read in a book the following:
The simplex $\Delta \subset \mathbb{R}^k$ can be shown to be invariant in this dynamics.
I would like to know: What does it mean an invariant simplex (math definition)? and, given this dynamics, how can we prove that a simplex is invariant (an example will answer my question :))?
Well, the invariant simplex means that (in this case) standard simplex is invariant with respect to some dynamical system. The standard simplex is an object given by $$\Delta=\{x_i\in\mathbb{R}^k_+: \sum_{i=1}^kx_i=1\}. $$
It is invariant with respect to some dynamical system if all trajectories with initial condition at $\Delta$ stay at $\Delta$ for all time. If you have a vector field given by system of ODEs $\dot{\mathbf{x}} = V(\mathbf{x})$, then standard simplex is invariant iff for all $\mathbf{x} \in \Delta$ holds $$ (V(\mathbf{x}), \mathbf{1}) = 0, $$ where $\mathbf{1}$ denotes vector $(1, 1, \dots, 1)$. This condition is what we get if we apply Bony-Brezis theorem to $\Delta$ and system $\dot{\mathbf{x}} = V(\mathbf{x})$.