Abstract

Wildlife poaching threatens biodiversity across sub-Saharan Africa, and is especially acute for critically endangered species such as the black rhinoceros (Diceros bicornis). Etosha National Park, Namibia (22,935 square kilometers), is patrolled by approximately 295 anti-poaching rangers, posing two interlinked operational questions: where should a limited workforce be placed to maximise conservation return, and how many rangers constitute a dynamically viable minimum below which the prey population tips toward long-run decline? We present a three-layer mathematical pipeline that addresses both questions jointly, with three analytical extensions beyond sequential layer application. Layer 1 generates a Wildlife Protection Potential (WPP) field combining nonlinear Multi-Attribute Utility Theory (MAUT) utilities for endangered (exponential) and abundant (logarithmic) species, a waterhole attraction term, and a non-homogeneous Poisson process (NHPP) threat intensity field. Layer 2 solves a Stackelberg-structured patrol allocation as a nonlinear constrained optimisation (SLSQP) over the WPP field, incorporating a season-adaptive terrain-friction matrix. Layer 3 analyses a three-population prey-predator-poacher ODE system; Jacobian eigenvalue analysis identifies the critical ranger threshold T*. Three extensions advance beyond the base pipeline. First, a self-consistent feedback loop iterates the three layers to a fixed-point equilibrium in which the deployment is optimal given the threat landscape and the threat landscape is consistent with the population dynamics induced by that deployment. Second, a multi-objective Pareto analysis maps the trade-off between capture efficiency and patrol equity, identifying the knee point of maximum marginal return. Third, real Diceros bicornis occurrence records from GBIF (n=103 within the Etosha bounding box) are integrated as a spatial prior, anchoring the WPP field to observed animal distributions. Applied to Etosha, the framework yields a patrol allocation (Gini index approximately 0.77) anchored on waterhole arcs, a stability threshold of approximately 240-260 rangers, and a feedback-loop objective gain of approximately 33% over a single open-loop pass. Sensitivity analysis confirms that UAV and sensor uptime dominate over headcount adjustments.