A response surface framework for persistence-aware lime requirement

Hamza Jouichat , Lotfi Khiari

Conventional lime-requirement (LR) methods estimate the dose needed to attain a target soil pH, but do not indicate how long the correction remains acceptable during re-acidification. We tested whether repeated dose–response measurements could provide a persistence-aware LR. Single applications of calcium carbonate (CaCO{\! }_{3}) and calcium oxide (CaO) were evaluated in clay, sandy loam, and organic soils over five 12-week cropping cycles. Each cycle included ryegrass growth, fertilisation, and three harvests; pH was measured at the end of each cycle. Response surfaces described how the pH response changed during continued cropping. We defined the peak lime requirement, LR_{\mathrm{peak}}, as the smallest dose whose highest fitted pH across C1–C5 attained the target; peak therefore denotes the maximum fitted response, not the duration of the pH correction. We defined the persistence-aware lime requirement, LR_{\mathrm{freq}}(N), as the dose that remained within selected pH bounds and closest to the target over N cycles. We also defined the persistence ratio, r(N), as LR_{\mathrm{freq}}(N) divided by LR_{\mathrm{peak}}; it expresses the persistence-aware rate relative to the peak target-attainment rate. Under CaCO{\! }_{3}, pH at LR_{\mathrm{peak}} fell below the lower agronomic bound in the second cycle in all three soils. For a proposed persistence horizon, the surface first tests whether any dose remains within both bounds through every cycle; when feasible, LR_{\mathrm{freq}}(N) selects the rate closest to the target. Under CaCO{\! }_{3}, LR_{\mathrm{freq}}(5) was feasible for clay and organic soil. Sandy loam had no five-cycle strict-bound solution; its longest feasible horizon was two cycles, for which LR_{\mathrm{freq}}(2) was selected. At each soil’s longest feasible horizon, LR_{\mathrm{freq}} (t ha{\! }^{-1}) was 5.00 for clay, 4.15 for sandy loam, and 7.72 for organic soil; corresponding r(N) values were 1.66, 1.14, and 1.36. Over the corresponding horizons, predictions at LR_{\mathrm{freq}} had mean squared deviations from target pH 84.0%, 32.4%, and 75.7% lower than predictions at LR_{\mathrm{peak}} for clay, sandy loam, and organic soil, respectively. The response surfaces therefore show how lime rate, the selected pH limits, and the intended number of cycles interact. When one dose can remain within both limits, LR_{\mathrm{freq}} identifies the rate that stays closest to the target.