Zheng, Na , Wang, Zhen , Jin, Zhen , Li, Li
2026-06-01 MATHEMATICAL BIOSCIENCES 2026 396(卷), null(期), (null页)
The vegetation systems in arid regions often exhibit diverse spatial pattern structures, and their evolution with respect to key parameters (such as precipitation) is frequently accompanied by complex bifurcation behaviors and the emergence of multiple critical points. Although bifurcation theory has received extensive attention over the past half-century, systematic investigations into the dynamical behavior of the critical points themselves remain limited. In this study, we incorporate optimal control theory into the vegetation-water interaction model and treat human activities h(x, t) as a control variable, representing direct interventions on vegetation growth (such as planting, fertilization, grazing, etc.), and systematically explore the evolutionary tendencies of the system at four representative critical points. The central idea of this work lies in evaluating the intrinsic evolutionary preference of vegetation at each critical point by quantifying the difficulty of steering the system from the state at a critical point to different target steady states. To this end, we introduce four control tendency indicators-DKL, PSNR, Cost, and MRDR-to assess this transition difficulty. Specifically, the control cost (Cost) reflects the required intervention intensity; the DKL and PSNR measure the discrepancy between the controlled state and the target spatial pattern; and MRDR characterizes the convergence rate of control errors during the optimization process. By computing these indicators along different control trajectories and conducting quantitative comparisons, we reveal the system's evolutionary tendencies at each critical point. The study reveals that at the two Turing pattern-related critical points (pT1 and pT2), the vegetation system tends to evolve toward pattern states. Specifically, the Turing onset critical point pT1 exhibits a stronger preference for spot patterns, whereas the Turing terminal critical point pT2 favors the formation of gap patterns. In contrast, at the desertification critical point pdes and the transcritical bifurcation point pc, the system shows a tendency to evolve toward the bare soil state and the uniform vegetation state, respectively. This study is the first to characterize the evolutionary direction of the system at critical points from the perspective of optimal control, providing a novel theoretical framework and analytical tool for understanding the dynamical mechanisms of critical points in arid ecosystems. Moreover, existing aridity classification frameworks are primarily built on the correspondence between precipitation levels and the stable states that vegetation systems can sustain. By explicitly incorporating critical points and their associated evolutionary tendencies, this study systematically supplements and refines these frameworks, providing a dynamical and mechanism-based interpretation of transitions between aridity regimes.