The Case 2 heteroclinic-orbit conjecture for the wound-healing angiogenesis system

Let $(u_{C_1},w_{C_1})$, $(u_{C_2},w_{C_2})$, and $(u_W,w_W)$ be the equilibria of system

in the positive quadrant, with the system parameters in Case 2, $(\alpha,\beta,c)=(2/5,5/2,\sqrt{2}/2)$. **Case 2 heteroclinic-orbit conjecture.** Under parameter regime Case 2, system

possesses heteroclinic orbits connecting $(u_{C_2},w_{C_2})$ to $(u_{C_1},w_{C_1})$ and $(u_{C_1},w_{C_1})$ to $(u_W,w_W)$. These connections are suggested by the phase plane but are not rigorously proved in the paper; the no-limit-cycles conjecture is used elsewhere to establish a different heteroclinic connection.