A compartmental dynamical system model for concept retention, forgetting, and relearning in calculus education

This study develops a four-compartment dynamical system for examining concept acquisition, retention, forgetting, and relearning in undergraduate calculus. The model extends compartmental state-transition logic to four educational states: susceptible learners S(t), actively learning students I(t), retained knowledge R(t), and concept decay S₂(t). The linear ordinary differential equation system is analyzed for positivity, conservation, feasibility, equilibrium structure, and local stability on the invariant population manifold. Recalculation of the full Jacobian shows one eigenvalue −α, one zero eigenvalue associated with conservation of the total learner population, and two non-zero eigenvalues determined by λ 2 + (β + γ + δ)λ + (βγ + γδ + βδ) = 0. Hence, for α, β, γ, and δ > 0, the equilibrium for a fixed total population is locally asymptotically stable; stability does not require γ > δ. The quantity 𝓛 = δ/γ is therefore interpreted as a retention–relearning ratio rather than a stability threshold. At equilibrium, 𝓛 = R*/S₂*, so it describes whether retained knowledge or concept decay has the larger equilibrium share. Numerical simulations illustrate transient redistribution among the four learning states and show that increasing the forgetting rate lowers the retained-knowledge level, whereas increasing the relearning rate improves recovery and raises the retained-knowledge equilibrium. The model is not proposed as a new numerical algorithm and does not claim computational superiority over stochastic, delayed, fuzzy, or non-standard finite-difference epidemic models; its novelty lies in the educational interpretation of a closed learning–mastery–forgetting–relearning cycle and in the explicit equilibrium implications for calculus retention.