Figure 8

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 = β₁(g₁₁(1 + r₁₁) 1) + β₂(g₂₂ - r₂₂ - 1) as a function of r₁₁ and r₂₂ (as in (11)). If the point on the surface corresponding to (r₁₁, r₂₂) 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 r₁₁ = .005 and r₂₂ = 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 r₁₁ = .02 and r₂₂ = 0.2 in Fig. 6 and Fig. 7 correspond to .0002 on the red surface, and the solutions are unstable. The other parameters are r₁₂ = 0, r₂₁ = 0, α₁ = 3.0, α₂ = 4.0, β₁ = 0.2, β₂ = .02, g₁₁ = 1.1, g₂₂ = 0.0, g₁₂ = 1.0, g₂₁ = -0.5, γ _T = .005, L _T = 100.