Proefschrift

APPENDIX B. SUPPLEMENT TO CHAPTER 6 B.1 Adjusting Competition Matrices While the initial assumptions of the model required the competition coefficients to be within [0 , 1], evolutionary competition studies have shown that competing cancer cells may result in competition coefficients that are far greater than one (Freischel et al., 2021; Kaznatcheev et al., 2019). In this way, we relax the constraint on the competition coefficients having values less than one and identify which competition coefficients are most pertinent to stability for the three responder patient competition matrices. To identify stable points that exist within the patient viability constraint that include T + and T P cells, we set ˙ x T − ≤ 0. Rearranging the state equation for x T − in Equation 3.1 gives the following. ˙ x T − = x T − r T − 1 − α 31 x T + + α 32 x T P + x T − K T − Setting ˙ x T − ≤ 0. x T − r T − 1 − α 31 x T + + α 32 x T P + x T − K T − ≤ 0 1 − α 31 x T + + α 32 x T P + x T − K T − ≤ 0 1 ≤ α 31 x T + + α 32 x T P + x T − K T − K T − ≤ α 31 x T + + α 32 x T P + x T − K T − − x T − ≤ α 31 x T + + α 32 x T P Under the patient viability constraint where i ∈T x i ≤ 9000 (B.1) we see that the competition coefficients α 31 and α 32 are the key parameters affecting stability for all values of x T − . Specifically, α 31 and/or α 32 must be greater than one. α 31 and α 32 are the competition of T + and T P cells on T − cells, respectively. To see how the increase of α 32 affects the stable points we show a sensitivity analysis of increasing α 32 for Matrix 2 (the matrix explored in detail within the main text). Figure B.1 graphically displays this stability analysis for α 32 = 1 . 0, α 32 = 1 . 1, α 32 = 2 . 0, and α 32 = 5 . 0. 138