Modeling the 2026 Gaokao Essay Prompts: Growth, Resilience, and Optimal Control

Using the logistic model introduced by ecologist Verhulst in 1838, the author shows that real‑world growth follows an S‑shaped curve rather than pure exponential growth. When the population is far below the environmental carrying capacity, growth appears exponential; as it approaches the capacity, the growth rate slows and the curve converges to a limit. The inflection point marks the transition from accelerating to decelerating growth, illustrating that growth proceeds in stages with different evaluation criteria.

Applying the same dynamical‑systems framework to resilience, the article distinguishes stable and unstable equilibria. A stable equilibrium (a ball in a bowl) returns quickly after disturbance, while an unstable one (a ball on a hill) diverges. C.S. Holling’s 1973 definition of ecological resilience is presented: the magnitude of disturbance a system can absorb without shifting to a different, often worse, equilibrium. For gradient systems, a potential‑energy function visualizes these equilibria, where the depth of the potential well determines the system’s ability to recover.

Turning to personal development, the author maps planning and effort onto an optimal‑control problem. The system’s state is knowledge or ability, the control input is learning effort, and the objective is to reach a target state while minimizing a convex cost that grows faster with additional effort. Using Pontryagin’s maximum principle, the optimal effort distribution over time is shown to be non‑uniform, depending on the system’s parameters (learning efficiency and forgetting rate), which must be identified through sufficient practice.

The discussion then contrasts parameter tuning with structural change. Parameter tuning adjusts variables within a fixed feasible region (e.g., mixing paint colors), whereas structural change expands the feasible region itself (e.g., adding new edges to a water‑distribution network). The author explains that misidentifying the nature of a problem—treating a structurally unsolvable issue as a parameter‑tuning task—leads to wasted effort.

Finally, the article examines how technology reshapes the “possibility space” of imagination. Historical examples such as telescopes and the internet illustrate expansion, while recommendation algorithms create a feedback loop that narrows the space, forming a stable “filter bubble.” Generative AI simultaneously expands creative possibilities and risks shrinking the mental capacity to generate ideas from scratch. The author concludes that, despite these shifts, the invariant ability to pose new problems from nothing remains essential.