A Controlled Dynamical Systems Framework for Academic Career Evolution under Mentorship–Funding Interactions

We propose a deterministic compartmental model of academic career progression that couples behavioural mentorship dynamics with competition for a finite, logistically growing funding resource. The academic population is partitioned into early-career, experienced, redundant, and successful researchers, with transitions governed by bilinear mentorship interactions, Holling type-II funding-mediated progressions, and a Matthew-effect cumulative advantage term. Positivity, and boundedness of solutions are established via the comparison principles. Five equilibria are identified and characterised: trivial, funding-only, human-only, boundary, and interior coexistence equilibria. Local asymptotic stability is analysed through explicit Routh–Hurwitz conditions, including the full trace and determinant conditions for the boundary equilibrium and a necessary trace condition for the interior equilibrium. Global asymptotic stability of the trivial, funding-only, and interior equilibria is established. The model is then extended to an optimal control framework with four time-dependent policy instruments: mentorship quality enhancement, behavioural contagion mitigation, funding allocation efficiency, and redundancy rehabilitation. Necessary optimality conditions are derived via Pontryagins Maximum Principle to yield a coupled state–adjoint system solved numerically by a forward–backward fourth-order Runge–Kutta sweep coded in Python software. Simulations show that funding exhaustion is the central systemic constraint, that is, all strategies eventually deplete the shared resource pool, with funding efficiency control u_3 alone producing a severe boom-and-bust redundancy surge. Single controls yield only modest improvements, each optimising one objective at the cost of others, while two-control combinations reveal distinct equity efficiency trade-offs but still fail to simultaneously stabilise all three objectives. Under the fully integrated four-control strategy, the rehabilitation control dominates with a near-bang-bang profile, early-career researchers stabilise after an initial transient rise, redundancy is suppressed, and the funding pool is preserved substantially longer than under any partial strategy. These results establish that coordinated institutional policies are structurally necessary, not merely preferable, for sustainable academic career development.