Math 6410 • ODE

Three Species Predator-Prey Model. Consider the following Lotka-Volterra model for populations of three species \(x,y,z\ge 0\). \[ \begin {cases} \dot {x}=x(4-y-z),\\ \dot {y}=y(1-z),\\ \dot {z}=z(x+y-5);\\ x(0)=x_0,\quad y(0)=y_0,\quad z(0)=z_0. \end {cases} \] The only stationary point for this system with all positive coordinates is at \(P=(2,3,1)\). Show that there is an open \(U\subset \mathbb {R}^2\) such that \((2,3)\in U\), a function \(f(x,y)\in C^1(U)\) and \(\epsilon >0\) so that for initial points which are \(\epsilon \)-close to \(P\), that is for \(\lvert (x_0,y_0,z_0)-P\rvert <\epsilon \), if \(z_0=f(x_0,y_0)\) then the solution converges to \(P\) as \(t\to \infty \), but if \(z_0\ne f(x_0,y_0)\) then the solution eventually goes away from \(P\), that is for some \(t_0>0\), \(\lvert (x(t_0),y(t_0),z(t_0))-P\rvert \ge \epsilon \).