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admissible set Allee effect analysis AV(N compute constraint control variable costate equations denote density diagonal matrix difference equation differential equations drive the system ecology ecosystem model eigenvalues Example exists a positive feedback control policy finite fish population fishery follows function V(N global minimum globally stable harvesting policy Hence implies initial intraspecific interactions isoclines Liapunov function Liapunov-like function limit cycle linear logistic model Lotka–Volterra model maximum multispecies natural dynamics negative definite nonlinear nonnegative equilibrium optimal control problem optimal feedback optimal policy optimal size limit orthant parameters patch perturbations population models positive constants positive diagonal matrix positive equilibrium positive number positive orthant predator prey prey—predator system region of attraction region of ultimate satisfies sector stable single-species population singular control singular solution solution which begins species suitable Liapunov function Suppose Theorem theoretical ecology trajectory two-species ultimate confinement unstable values vector year-class