**The behavior of the solutions as a function of the tumor parameters ***r*_{11 }and *r*_{22}. System of equations (7): The red surface is the plot of Φ =

*β*_{1}(

*g*_{11}(1 +

*r*_{11}) 1) +

*β*_{2}(

*g*_{22 }-

*r*_{22 }- 1) as a function of

*r*_{11 }and

*r*_{22 }(as in (11)). If the point on the surface corresponding to (

*r*_{11},

*r*_{22}) is negative, then the solutions have decreasing amplitude oscillations converging to the nontrivial steady state; if positive, then the solutions have increasing amplitude and unstable oscillations. The values

*r*_{11 }= .005 and

*r*_{22 }= 0.2 in Fig. 4 and Fig. 5 correspond to -.00145 on the red surface, and the solutions converge slowly to the nontrivial steady state

= 5.0,

= 316.0. The values

*r*_{11 }= .02 and

*r*_{22 }= 0.2 in Fig. 6 and Fig. 7 correspond to .0002 on the red surface, and the solutions are unstable. The other parameters are

*r*_{12 }= 0,

*r*_{21 }= 0,

*α*_{1 }= 3.0,

*α*_{2 }= 4.0,

*β*_{1 }= 0.2,

*β*_{2 }= .02,

*g*_{11 }= 1.1,

*g*_{22 }= 0.0,

*g*_{12 }= 1.0,

*g*_{21 }= -0.5, γ

_{T }= .005,

*L*_{T }= 100.

## PubMed Commons