When is cumulative dose response monotonic? Analysis of incoherent feedforward motifs

We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motifs (IFFM) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and nonmonotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. The structural conditions identify eight canonical IFFM models, four of which are analyzed in detail within the proposed framework. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially nonmonotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 exhibits nonmonotone cDR. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how system structure shapes monotonicity in cumulative input--output responses.