Let Ω ⊂ ℝ d ( d ≥ 1 ) be a bounded, connected, open domain with smooth boundary ∂ Ω , and let T > 0 be a finite time horizon. We denote by Q T = Ω × ( 0 , T ) the space-time cylinder. The state variable u : Q ¯ T → [ 0 , 1 ] represents the localdensity of mastered competence at position x ∈ Ω and time t ∈ [ 0 , T ] , where u = 0 corresponds to total absence of mastery and u = 1 to complete mastery of the considered competence. The spatial coordinate x is an abstract index that orders learners according to their initial knowledge level (e.g., from lowest to highest baseline competence). This interpretation is used consistently throughout the paper: in particular, it allows the initial condition u 0 ( x ) to represent a heterogeneous population where learners with higher x are not necessarily spatially contiguous but are simply labeled by their initial proficiency.

Definition 2.1.The function u ( x , t ) is called the knowledge density field. For a given population of learners indexed by x ∈ Ω ,the quantity

defines the mean competence level of the population at time t .

The spatiotemporal evolution of u is governed by the following nonlinear reaction-diffusion equation:

supplemented by homogeneous Neumann boundary conditions:

and an initial condition:

where u 0 ∈ L ∞ ( Ω ) with 0 ≤ u 0 ( x ) ≤ 1 almost everywhere.

Remark 2.2.The Neumann boundary condition(3) encodes the assumption of a closed population: there is no flux of knowledge across the boundary ofΩ. This is appropriate whenΩrepresents a self-contained learning community(e.g., a university cohort) with negligible external exchanges.

The diffusion operator ∇ ⋅ [ D ( x , t ) ∇ u ] models the social contagion ofknowledge: a learner who has mastered a competence can transmit it to neighboring learners through peer interaction, collaborative work, or informal exchanges. The position-dependent diffusion coefficient is defined as:

where:

D 0 ≥ 0 is the baseline diffusion coefficient, representing the natural propagation of knowledge in the absence of any structured pedagogical intervention (e.g., informal peer exchanges); M i ( x , t ) ∈ [ 0 , 1 ] is the normalized intensity of pedagogical method i at position x and time t . Five methods are considered in this paper:

M 1 : Lecture-based teaching (cours magistral),

M 2 : Active learning/practical work (TD/TP),

M 3 : Collaborative learning (group work),

M 4 : Digitalre sources (online platforms, videos),

M 5 : Personalized tutoring;

δ i ≥ 0 is the diffusion efficiency of method i , quantifying its contribution to the spread of knowledge through social interaction. Methods that promote exchange and peer interaction (e.g., M 3 , M 5 ) are expected to yield higher values of δ i .

Why pedagogical methods affect both diffusion and reaction.The coefficients δ i and α i capture two distinct but complementary roles of a pedagogical method M i . The diffusion efficiency δ i quantifies the method’s capacity to promote social contagion of knowledge: methods that encourage peer interaction, discussion, and collaborative problem-solving (e.g., M 3 , M 5 ) increase the effective diffusivity D ( x , t ) , accelerating the spatial spread of competence across the learner population. The pedagogical efficiency α i measures the method’s direct impact on an individual’s local learning rate, independent of peer influence (e.g., through clarity of explanation, structured exercises, or feedback quality). We assume that the effects of different methods are additive and independent (no interaction terms such as δ i j M i M j ), which is a simplifying firstorder approximation. Interaction effects (e.g., synergy between active learning and digital resources) are intentionally excluded from the present model but could be incorporated in future extensions.

Assumption 2.3.There exist constants D min , D max > 0 such that

This is satisfied whenever D 0 > 0 and the M i are bounded, which we assume throughout.

The reaction term f ( u , x , t ) collects all local processes governing knowledge acquisition, social facilitation, and forgetting:

where each component is described in the following subsections.

2.4.1. Local Learning Rate r ( x , t )

The local learning rate integrates the contributions of pedagogical methods and the inhibitory effect of cognitive stress:

where:

r 0 ≥ 0 is the intrinsic learning rate, modeling spontaneous knowledge acquisition without any pedagogical intervention; α i ≥ 0 is the pedagogical efficiency of method i , measuring its direct contribution to accelerating learning. These parameters are the primary control variables of the model and can be calibrated from standardized assessment data; S ( x , t ) ≥ 0 denotes the cognitive stress level perceived by learners at position x and time t . It may encompass time pressure, perceived difficulty, or environmental factors; β ≥ 0 is the stress attenuation coefficient. A high value of β indicates that stress strongly inhibits learning efficiency, consistently with the Yerkes-Dodson law [10]: moderate stress may be stimulating, but excessive stress is detrimental.

Remark 2.4.We assume r ( x , t ) > 0 for all ( x , t ) ∈ Q T , which requires that the net contribution of pedagogical methods dominates the stress effect: r 0 + ∑ i α i M i > β S . If this condition fails locally,the model predicts a region of learning inhibition, which is pedagogically meaningful.

2.4.2. Logistic Growth Term u ( 1 − u )

The factor u ( 1 − u ) is the classical logistic saturation term:

When u ≈ 0 , growth is approximately linear in u (initial exponential phase); As u → 1 , growth decelerates and vanishes, reflecting the saturation of competence: one cannot exceed complete mastery.

This term ensures that u = 0 and u = 1 are invariant states of the reaction dynamics in the absence of forgetting.

2.4.3. Threshold and Social Facilitation Function g ( u , x , t )

The function g ( u , x , t ) combines two fundamental mechanisms:

where h ( u ) is a Hill-type threshold function and F ( u , x , t ) is a social facilitation operator.

Hill-type threshold function.

with:

θ ∈ ( 0 , 1 ) the critical mastery threshold. Below θ , learning is self-inhibiting (dropout effect); above θ , learning becomes self-sustaining (snowball effect); p ≥ 1 the Hill coefficient, controlling the sharpness of the transition. For p = 1 , the transition is smooth (sigmoid-like); for p ≫ 1 , it approaches a step function.

Proposition 2.5.The Hill function h : [ 0 , 1 ] → [ 0 , 1 ] defined in (10) satisfies:

(i) h ( 0 ) = 0 , lim u → ∞ h ( u ) = 1 ;

(ii) h is strictly increasing on [ 0 , 1 ] ;

(iii) h ( u ) < 1 2 for u < θ , and h ( u ) > 1 2 for u > θ ;

(iv) h ′ ( u ) = p θ p u p − 1 ( θ p + u p ) 2 > 0 for all u > 0 .

Proof. Properties (i)-(iii) follow directly from the definition (10) by elementary algebraic manipulations. Property (iv) is obtained by differentiating (10) with respect to u using the quotient rule. □

Social facilitation operator.

where:

γ ≥ 0 is the social facilitation intensity. A higher γ means that a learner’s progress is more strongly influenced by the competence level of their peers; w ( x , x ′ ) ≥ 0 is the interaction kernel, weighting the influence of learners at x ' on those at x . We adopt the standard Gaussian kernel:

where σ > 0 is the social interaction range.

Remark 2.6.The operator W [ u ] ( x , t ) = ∫ Ω w ( x , x ′ ) u ( x ′ , t ) d x ′ is a convolution-type operator.Its boundedness on L 2 ( Ω ) follows from Young’s convolution inequality,since w ∈ L 1 ( ℝ d ) by virtue of(12).

2.4.4. Forgetting Term − λ u

The term − λ u , with λ ≥ 0 , models the natural decay of knowledge when competences are not regularly reactivated. This linear decay is consistent with the Ebbinghaus forgetting curve [9], which posits that memory retention decays approximately exponentially over time in the absence of reinforcement.

Assembling all terms, the complete model reads:

for ( x , t ) ∈ Q T , with boundary condition (3) and initial condition (4).

Table 1 summarizes all parameters of the model, their mathematical role, and their pedagogical interpretation.

Table 1.Summary of model parameters and their pedagogical interpretation.

To account for the interplay between cognitive and non-cognitive competences, we introduce a second state variable v ( x , t ) ∈ [ 0 , 1 ] representing non-cognitive skills (motivation, self-efficacy, metacognition). The extended system reads:

supplemented by homogeneous Neumann boundary conditions for both u and v . The cross-coupling terms u a v b and u c v d model the bidirectional interaction: cognitive competences reinforce motivation, and motivation facilitates knowledge acquisition. The exponents a , b , c , d ≥ 0 control the strength of each coupling.

Remark 2.7.When b = d = 0 , the two equations in(14) decouple, and each reduces to an independent reaction-diffusion equation of Fisher-KPP type [11][12]. The full coupled system (14) is therefore a generalization of the classical Fisher-KPP model to the educational context.

We work in the following functional framework. Let q ∈ ( 1 , ∞ ) and define the Sobolev space W 1 , q ( Ω ) endowed with its standard norm. For the parabolic problem, we use the Bochner space L 2 ( 0 , T ; H 1 ( Ω ) ) , where H 1 ( Ω ) = W 1 , 2 ( Ω ) .

We collect the standing assumptions used throughout this section.

Assumption 3.1.The following conditions hold:

(A1) Ω ⊂ ℝ d is a bounded, connected, open set with boundary ∂ Ω of class C 1 , 1 .

(A2) The diffusion coefficient satisfies (6): 0 < D min ≤ D ( x , t ) ≤ D max < ∞ for all ( x , t ) ∈ Q ¯ T .

(A3) The pedagogical method intensities satisfy M i ∈ L ∞ ( Q T ) , 0 ≤ M i ≤ 1 , for all i = 1 , ⋯ , n .

(A4) The stress field satisfies S ∈ L ∞ ( Q T ) , S ≥ 0 .

(A5) The initial datum satisfies u 0 ∈ L ∞ ( Ω ) , 0 ≤ u 0 ( x ) ≤ 1 almost everywhere in Ω.

(A6) The interaction kernel w ∈ L 1 ( ℝ d ) ∩ L ∞ ( ℝ d ) is nonnegative and symmetric: w ( x , x ′ ) = w ( x ′ , x ) ≥ 0 .

Theorem 3.2. (Existence and Uniqueness)Under Assumption3.1, there exists a unique weak solution

to problem (13)-(3)-(4), in the sense that for all φ ∈ H 1 ( Ω ) and a.e. t ∈ ( 0 , T ) :

Proof. The proof proceeds in three steps.

Step 1: Galerkin approximation. Let { e k } k ≥ 1 be the orthonormal basis of L 2 ( Ω ) formed by the eigenfunctions of − Δ with homogeneous Neumann boundary conditions. For each m ≥ 1 , we seek an approximate solution of the form u m ( x , t ) = ∑ k = 1 m c k m ( t ) e k ( x ) , where the coefficients c k m satisfy the finite-dimensional ODE system obtained by projecting (16) onto span { e 1 , ⋯ , e m } . By the Cauchy-Lipschitz theorem, this system admits a unique local solution on some interval [ 0 , T m ) .

Step 2: A priori estimates. Testing (16) against u m and using Assumption 3.1(A2), we obtain:

Since f is locally Lipschitz and | f ( u , x , t ) | ≤ C ( 1 + | u | ) for a constant C > 0 independent of m , Gronwall’s lemma yields uniform bounds:

for a constant C T independent of m . This extends the local solution to the full interval [ 0 , T ] .

Step 3: Passage to the limit. The uniform bounds (18) allow extraction of a subsequence (not relabeled) such that u m ⇀ u weakly in L 2 ( 0 , T ; H 1 ( Ω ) ) and strongly in L 2 ( Q T ) by the Aubin-Lions compactness lemma. The nonlinear terms converge by dominated convergence. The uniqueness follows from a standard energy estimate applied to the difference of two solutions. □

Theorem 3.3 (Positivity and L ∞ Bound).Under Assumption3.1,if 0 ≤ u 0 ( x ) ≤ 1 a.e., then the solution u of(13)-(4) satisfies:

Proof.Lower bound. Testing (16) against u − = max ( − u , 0 ) ≥ 0 (the negative part of u ) and using f ( 0 , x , t ) = 0 , a standard comparison argument shows that u − ( x , t ) = 0 a.e., hence u ≥ 0 .

Upper bound. Let w = u − 1 and test the equation for w against w + = max ( w , 0 ) . Since f ( 1 , x , t ) = − λ ≤ 0 and the reaction term satisfies f ( u , x , t ) ≤ 0 for u ≥ 1 (the logistic factor u ( 1 − u ) is nonpositive for u ≥ 1 ), the same comparison argument yields u ≤ 1 a.e. □

Remark 3.4.Theorem 3.3 confirms that the closed interval [ 0 , 1 ] is a positively invariant region for the dynamics (13), which is essential for the biological(pedagogical) consistency of the model: the knowledge density remains bounded between total ignorance and complete mastery at all times.

We now analyze the spatially homogeneous steady states of (13), obtained by setting ∂ u / ∂ t = 0 and ∇ u = 0 (uniform solutions). In this case, the diffusion term vanishes and the equilibria u * ∈ [ 0 , 1 ] satisfy:

where r * = r 0 + ∑ i α i M ¯ i − β S ¯ is the spatially averaged learning rate, M ¯ i and S ¯ are the spatial means of M i and S , and w ¯ = ∫ Ω w ( x , x ′ ) d x ′ (constant by normalization of the Gaussian kernel).

Proposition 3.5 (Trivial Equilibrium). u * = 0 is always an equilibrium of(20).

Proof. Direct substitution of u * = 0 into (20) gives F ( 0 ) = 0 . □

Proposition 3.6 (Nontrivial Equilibria).Assume θ ∈ ( 0 , 1 ) and that the net learning rate satisfies

Then there exists at least one nontrivial equilibrium u ** ∈ ( θ , 1 ) satisfying (20).

Proof. Define Φ ( u ) = F ( u ) / u for u ∈ ( 0 , 1 ] . Then:

At u = θ + : since h ( θ ) = 1 / 2 , we have Φ ( θ ) = r * ( 1 − θ ) / 2 ⋅ ( 1 + γ w ¯ θ ) − λ . Condition (21) guarantees Φ ( θ ) > 0 . At u = 1 − : Φ ( 1 ) = − λ < 0 . By the intermediate value theorem, there exists u ** ∈ ( θ , 1 ) such that Φ ( u * * ) = 0 , i.e., F ( u ** ) = 0 . □

Remark 3.7.The condition r * > λ has a clear pedagogical meaning: the netlearning rate(boosted by pedagogical methods) must exceed the forgetting rate for nontrivial knowledge equilibria to exist. If r * ≤ λ , the only equilibrium is u * = 0 (total knowledge loss),illustrating the critical role of pedagogical intervention in sustaining long-term competence.

We analyze the local stability of the equilibria identified in Section 3.3 via linearization.

3.4.1. Linearization around u ∗ = 0

Linearizing (13) around u * = 0 (with spatially homogeneous perturbation u = ε ϕ ( x ) e μ t , ε ≪ 1 ), the leading-order eigenvalue problem is:

where D * = D 0 + ∑ i δ i M ¯ i . The eigenvalues are:

where 0 = ν 0 < ν 1 ≤ ν 2 ≤ ⋯ are the Neumann eigenvalues of − Δ on Ω.

Proposition 3.8 (Stability of the Trivial Equilibrium).The trivial equilibrium u * = 0 is always locally asymptotically stable with respect to the linearized dynamics (22).

Proof. Since ν k ≥ 0 and λ ≥ 0 , we have μ k = − D * ν k − λ ≤ 0 for all k ≥ 0 . Moreover, the reaction term f ( u , x , t ) vanishes to higher order at u = 0 due to the Hill factor h ( u ) = u p / ( θ p + u p ) ~ u p / θ p → 0 as u → 0 (for p ≥ 1 ). Hence, the linearization at u * = 0 gives μ k = − λ < 0 for k = 0 , confirming local asymptotic stability. □

Remark 3.9.The stability of u * = 0 reflects the dropout effect: a population with very low initial knowledge density(below θ )tends to converge back to zero mastery in the absence of sufficient pedagogical stimulation. This is consistent with the Hill threshold mechanism.

3.4.2. Linearization around the Nontrivial Equilibrium u ∗ ∗

Let u ** ∈ ( θ , 1 ) be a nontrivial equilibrium given by Proposition 3.6. Setting u = u ** + ε v and linearizing (13), the perturbation v satisfies:

where:

Theorem 3.10 (Stability Criterion for u * * ).The nontrivial equilibrium u * * is locally asymptotically stable if and only if:

where ν 1 > 0 is the first nonzero Neumann eigenvalue of − Δ on Ω.

Proof. Seeking solutions of (24) of the form v ( x , t ) = ϕ ( x ) e μ t , we obtain for each Neumann eigenmode ϕ = e k :

where w ^ k = ∫ Ω w ( x , x ′ ) e k ( x ′ ) d x ′ is the action of the kernel on the k -th eigenfunction. For stability, we require μ k < 0 for all k ≥ 0 . The most restrictive condition corresponds to k = 0 (spatially uniform mode, ν 0 = 0 , w ^ 0 = w ¯ | Ω | ), giving μ 0 = a * + b * w ¯ | Ω | < 0 , which is (27). □

Corollary 3.11.If u * * is close to1 (near-complete mastery),then 1 − 2 u * * < 0 and the term a * is dominated by the negative contribution r * ( 1 − 2 u * * ) h ( u * * ) ( ⋯ ) < 0 , making the stability condition (27) more easily satisfied. Pedagogically, this means that a population that has collectively reached high mastery is self-stabilizing.

Table 2 summarizes the main theoretical results established in this section.

Table 2. Summary of theoretical results for model (13).

The purpose of this section is twofold: (i) to illustrate the qualitative behavior of model (13) in a set of illustrative scenarios, and (ii) to compare the impact of the five pedagogical methods under a plausible but fictive parameter configuration. Because no empirical dataset is used, this simulation should be viewed as a proofofconcept and a demonstration of the model’s diagnostic capabilities, not as a validation against real data. All parameter values (Table 3) are chosen to be pedagogically plausible and consistent with the qualitative findings of the educational literature [7][8]. Importantly, the resulting ranking of methods(Section4.5.4)is conditional on the chosen parameter set; a different calibration(e.g., different α i , δ i , or θ ) could modify the ordering. The simulation thus serves as a flexible tool for “whatif” analyses rather than a definitive assessment.

The simulation is conducted on the one-dimensional spatial domain Ω = [ 0 , 1 ] , discretized with N x = 100 uniform grid points, over the time interval [ 0 , T ] with T = 50 (arbitrary time units representing weeks). The spatial variable x ∈ [ 0 , 1 ] indexes learners ordered by their initial knowledge level. All computations are performed in MATLAB using finite differences for space discretization and the implicit-explicit (IMEX) scheme for time integration, which treats the diffusion term implicitly (for stability) and the reaction term explicitly (for computational efficiency).

The parameter values used in the simulation are summarized in Table 3. Five scenarios are considered, each activating a single pedagogical method at full intensity ( M i ≡ 1 , all other M j ≡ 0 , j ≠ i ) to isolate the individual contribution of each method.

Table 3. Parameter values used in the numerical simulation.

Remark 4.1.The ranking α 5 > α 2 > α 3 > α 4 > α 1 reflects the meta-analyticfindings of[8]: personalized tutoring has the highest effect size on studentachievement, followed by active learning and collaborative work, while lecture-based teaching alone yields the lowest individual impact.

Let x j = ( j − 1 ) Δ x for j = 1 , ⋯ , N x with Δ x = 1 / ( N x − 1 ) , and t n = n Δ t with time step Δ t = 0.05 . Let u j n ≈ u ( x j , t n ) .

The nonlocal social facilitation term is approximated by the trapezoidal rule:

The full IMEX scheme reads: given u n , find u n + 1 satisfying

for j = 2 , ⋯ , N x − 1 , with Neumann boundary conditions u 1 n + 1 = u 2 n + 1 and u N x n + 1 = u N x − 1 n + 1 , and where D j ± 1 2 n = ( D j n + D j ± 1 n ) / 2 . The implicit diffusion part yields a tridiagonal linear system solved at each time step by the Thomas algorithm.

Remark 4.2.The IMEX scheme is unconditionally stable with respect to the diffusion term(implicit treatment) and first-order accurate in time. The CFL condition for the explicit reaction term is automatically satisfied since the reaction term is bounded on [ 0 , 1 ] .

To represent a realistic starting scenario, we initialize with a small localized group of learners having partial mastery:

This corresponds to a baseline mastery of 15% across the population, with a localized peak of 35% centered at x = 0.3 (a small subgroup of more advanced learners). Note that u 0 ( x ) ∈ [ 0 , 1 ] for all x , consistent with Theorem 3.3.

The numerical simulation of model (13) over T = 50 weeks produces the three-panel figure (Figure 1) described below. All results are consistent with the theoretical predictions of Section 3.

4.5.1. Final Spatial Knowledge Distribution

The left panel of Figure 1 displays the final spatial distribution u ( x , T ) for each of the five pedagogical scenarios, alongside the initial condition u 0 ( x ) (31) (black dashed curve). Several observations are in order.

First, the initial condition exhibits the expected Gaussian peak centered at x = 0.3 with a maximum of approximately 0.35, superimposed on a baseline of 0.15, confirming that the simulation starts from a realistic heterogeneous configuration with a small subgroup of more advanced learners.

Second, at t = T = 50 weeks, all five methods have produced a dramatic and spatially uniform elevation of the knowledge density, with all profiles converging to nearly flat distributions well above 0.80. This spatial homogenization is a direct consequence of the diffusion operator ∇ ⋅ [ D ( x , t ) ∇ u ] : the knowledge initially concentrated near x = 0.3 has diffused outward across the entire learner population. The Hill threshold mechanism (Proposition 2.5) plays a key role here—once the localized peak exceeds θ = 0.25 , the snowball effect propagates the learning front across the domain.

Third, a clear ordering is visible between the five final profiles, consistent with the ranking of pedagogical efficiencies α 5 > α 2 > α 3 > α 4 > α 1 .

4.5.2. Mean Knowledge Dynamics

The central panel of Figure 1 shows the temporal evolution of the mean knowledge level u ¯ ( t ) for each method. All five curves exhibit the characteristic sigmoidal shape predicted by the logistic-Hill reaction term: an initial slow phase (below the threshold θ = 0.25 ), followed by a rapid acceleration once the threshold is crossed, and a final plateau corresponding to the nontrivial equilibrium u ** .

The vertical dashed line at t = 1 / λ = 20 weeks marks the Ebbinghaus reference time: without pedagogical reinforcement, a knowledge item would decay to e − 1 ≈ 37 % of its initial value by this time. The fact that all five curves have already surpassed u ¯ = 0.80 before t = 20 weeks demonstrates that the pedagogical methods effectively counteract the forgetting mechanism, consistently with the condition r * > λ established in Proposition 3.6.

4.5.3. Comparative Impact of Pedagogical Methods

Table 4 summarizes the mean knowledge level u ¯ ( T ) at t = T = 50 weeks for each pedagogical scenario, as well as the half-mastery time t 1 / 2 (first time u ¯ ( t ) ≥ 0.50 ), read directly from Figure 1.

Table 4. Simulation results: mean knowledge at t = T = 50 weeks and half-mastery time t 1 / 2 for each pedagogical method.

Figure 1.Numerical simulation of model (13) over T = 50 weeks for five pedagogical scenarios (one method active at full intensity at a time). Left: final spatial knowledge distribution u ( x , T ) ; the black dashed curve shows the initial condition u 0 ( x ) . Center: temporal evolution of the mean knowledge level u ¯ ( t ) ; the vertical dashed line marks the Ebbinghaus reference time 1 / λ = 20 weeks. Right: bar chart of u ¯ ( T ) ranked in decreasing order. Parameter values as in Table 3.

The results confirm the expected ranking in full agreement with the meta-analytic findings of [8]. Personalized tutoring ( M 5 ) achieves the highest long-term mastery and the fastest convergence, with the population reaching half-mastery in approximately 4 weeks. By contrast, lecture-based teaching ( M 1 ) yields the lowest equilibrium level ( u ¯ ( T ) ≈ 0.850 ) and the slowest convergence ( t 1 / 2 ≈ 18 weeks)—more than four times slower than tutoring. This gap of approximately 9% in final mastery between M 5 and M 1 , while seemingly modest in absolute terms, represents a substantial difference in long-term competence accumulation over an academic year.

4.5.4. Ranking of Pedagogical Methods

The right panel of Figure 1 presents a bar chart of u ¯ ( T ) ranked in decreasing order. The ranking M 5 ≻ M 2 ≻ M 3 ≻ M 4 ≻ M 1 is unambiguous and robust. Three clusters emerge naturally:

(i) High-impact methods ( u ¯ ( T ) > 0.92 ): tutoring ( M 5 ), active learning ( M 2 ), and collaborative learning ( M 3 ). These methods share a high diffusion efficiency ( δ 3 = 0.08 , δ 5 = 0.06 ) that amplifies the social contagion of knowledge and accelerates the spread of learning across the population.

(ii) Intermediate method ( 0.88 < u ¯ ( T ) < 0.92 ): digital resources ( M 4 ), which combine moderate pedagogical efficiency ( α 4 = 0.30 ) with limited social diffusion ( δ 4 = 0.03 ).

(iii) Low-impact method ( u ¯ ( T ) ≈ 0.85 ): lecture-based teaching ( M 1 ), characterized by the lowest values of both α 1 = 0.20 and δ 1 = 0.01 , reflecting the predominantly passive and non-interactive nature of this format.

Remark 4.3.The simulation confirms all four theoretical predictions of Section3: (i) positivity and boundedness u ∈ [ 0 , 1 ] ; (ii) convergence to nontrivial equilibria u * * > 0 for all methods since r * > λ ; (iii) snowball effect triggered by the Hill threshold from the localized initial peak; (iv) Ebbinghaus forgetting visible in the slower convergence of M 1 .

The single-compartment model (13) captures the spatiotemporal dynamics of cognitive competences—declarative and procedural knowledge that can be directly assessed through standardized evaluations. However, a growing body of educational research emphasizes that learning outcomes are also strongly shaped by non-cognitive factors such as motivation, self-efficacy, perseverance, and metacognitive awareness [13]-[15]. These non-cognitive competences interact bidirectionally with cognitive ones: a learner who masters a concept gains confidence (motivation reinforced by cognitive success), while a motivated learner invests more effort and thus acquires knowledge more efficiently (motivation facilitates cognitive acquisition).

To capture this bidirectional coupling, we introduce a second state variable v ( x , t ) ∈ [ 0 , 1 ] representing the local density of non-cognitive competences (motivation, self-efficacy, engagement) at position x and time t . The resulting two-compartment system generalizes (13) and constitutes the extended model analyzed in this section.

The extended model is:

supplemented by homogeneous Neumann boundary conditions for both u and v :

and initial conditions:

where 0 ≤ u 0 , v 0 ≤ 1 almost everywhere.

The variables and parameters are interpreted as follows:

u ( x , t ) : cognitive competence density (declarative and procedural knowledge); v ( x , t ) : non-cognitive competence density (motivation, self-efficacy, metacognition); D u , D v ≥ 0 : diffusion coefficients for cognitive and non-cognitive competences, respectively. In general, D u ≠ D v , as the social propagation mechanisms differ between knowledge (transmitted through instruction and peer explanation) and motivation (transmitted through social modeling and peer encouragement); r u , r v > 0 : intrinsic growth rates for cognitive and non-cognitive competences; u a v b : cross-coupling term in the u -equation, modeling the facilitation of cognitive acquisition by non-cognitive competences. A motivated learner ( v high) acquires knowledge ( u ) more efficiently; u c v d : cross-coupling term in the v -equation, modeling the reinforcement of motivation by cognitive success. A knowledgeable learner ( u high) gains confidence ( v ) more readily; λ u , λ v ≥ 0 : forgetting/disengagement rates for cognitive and non-cognitive competences, respectively. In general, λ v < λ u , as motivational states tend to decay more slowly than specific knowledge items; a , b , c , d ≥ 0 : coupling exponents controlling the strength and nonlinearity of the bidirectional interaction.

Remark 5.1.When b = 0 and d = 0 , the two equations in (32) decouple: each reduces to an independent Fisher-KPP-type reaction-diffusion equation[11][12]. The full coupled system (32) is therefore a strict generalization of the classical Fisher-KPP model, with cross-species interaction terms analogous to those found in predator-prey or mutualism models in mathematical ecology [2].

Theorem 5.2 (Well-Posedness of the Extended System).Assume D u , D v > 0 , r u , r v > 0 , λ u , λ v ≥ 0 , a , b , c , d ≥ 0 , and let u 0 , v 0 ∈ L ∞ ( Ω ) with 0 ≤ u 0 , v 0 ≤ 1 a.e. Then there exists a unique weak solution ( u , v ) of(32)-(34) satisfying:

and the invariance property:

Proof. The proof follows the same Galerkin approximation strategy as Theorem 3.2, applied to the coupled system. The key observation is that the reaction terms F u ( u , v ) = r u u a v b ( 1 − u ) − λ u u and F v ( u , v ) = r v u c v d ( 1 − v ) − λ v v are locally Lipschitz on [ 0 , 1 ] 2 and satisfy the quasi-positivity conditions: F u ( 0 , v ) = 0 ≥ 0 and F v ( u , 0 ) = − λ v ⋅ 0 = 0 ≥ 0 , ensuring nonnegativity of solutions by the comparison principle. The upper bound u , v ≤ 1 follows from the fact that F u ( 1 , v ) = − λ u ≤ 0 and F v ( u , 1 ) = − λ v ≤ 0 , so that u = 1 and v = 1 are super-solutions of the respective equations. □

Spatially homogeneous steady states ( u * , v * ) ∈ [ 0 , 1 ] 2 of (32) satisfy the algebraic system:

Proposition 5.3 (Equilibria of the Extended System).System(37)always admits the trivial equilibrium ( u * , v * ) = ( 0 , 0 ) . Under the conditions

there exists at least one nontrivial equilibrium ( u ** , v ** ) ∈ ( 0 , 1 ) 2 .

Proof. The trivial equilibrium ( 0 , 0 ) satisfies (37) trivially. For the nontrivial equilibrium, define:

Under condition (38), one can show by a fixed-point argument (Schauder theorem applied to the map ( u , v ) ↦ ( G u ( v ) , G v ( u ) ) where G u and G v are the respective solution maps) that there exists a fixed point ( u ** , v ** ) ∈ ( 0 , 1 ) 2 . □

Remark 5.4.Condition (38) has a natural pedagogical interpretation:the ratio r u / λ u (resp. r v / λ v ) must exceed1for cognitive(resp. non-cognitive) competences to be sustained against forgetting (resp. disengagement). This generalizes the condition r * > λ established for the single-compartment model.

Let ( u ** , v ** ) be a nontrivial equilibrium of (37). Setting u = u ** + ε u ˜ and v = v ** + ε v ˜ and linearizing (32), the perturbation ( u ˜ , v ˜ ) satisfies:

where J * is the Jacobian matrix of the reaction terms evaluated at ( u ** , v ** ) :

with: