Density-dependent growth emerges from Bayesian adaptation of phenotype
Classical models often describe early tumor expansion as exponential growth, yet experimental and clinical evidence shows that tumor populations can deviate systematically from this behavior, exhibiting density dependent proliferation, cooperative low-density growth, intermediate growth optima, and finite upper growth bounds before resource limitation or spatial crowding dominate. These observations raise a common question: why should the per capita growth rate depend on population size? Here, we propose that sensing mismatch provides a mesoscopic link between environmental change and density dependent proliferation. We model the cell as a Bayesian adaptive agent whose coarse grained phenotype evolves on an intrinsic regulatory landscape, while environmental sensing reweights phenotypic states according to how well they account for the extracellular signal statistics generated by the population. In the weak phenotype signal correlation regime, the stationary phenotype distribution is Gaussian, with its mean displaced from the proliferative optimum by a population size-dependent baseline information mismatch. This displacement produces a quadratic penalty in the per capita growth rate. Coupling the framework to a receptor ligand decoding model, we show that basal readout error and nonlinear receptor saturation make the mismatch nonmonotonic in population size. This single structure gives rise to an intermediate proliferation optimum, an Allee survival threshold, a tissue specific capacity, and superlinear scaling at low density. A phase diagram in the phenotype signal coupling and readout-error plane partitions growth into regulated, uncontrolled, and arrested regimes. Thus, density dependent proliferation need not be imposed phenomenologically, but can emerge from cellular sensing and inference.