The collaborative learning model is based on the following assumptions:

1) The system consists of two interacting learners with knowledge levels y i ( t ) , i = 1 , 2 .

2) Both learners share the same total knowledge target M .

3) Knowledge evolves in continuous time and is governed by a deterministic system of ordinary differential equations (ODEs).

4) Each learner is characterized by a learning rate A i > 0 and a forgetting rate L i > 0 , i = 1 , 2 .

5) The interaction is symmetric, i.e., the influence of learner 1 on learner 2 is equal to the influence of learner 2 on learner 1, and is governed by a common coupling coefficient k .

6) The interaction depends only on the knowledge difference between the learners.

The classical learning-with-decaying memory model is described by the ordinary differential equation:

with initial condition

where:

M , denotes the total amount of knowledge to be acquired, y ( t ) , represents the knowledge level at timet, ε , is the proportion of initially known material, A > 0 , is the learning parameter, L > 0 , the decaying memory rate, reflecting the individual’s ability to retain acquired knowledge over time.

The long-term equilibrium of the system is given by

which represents the maximum attainable knowledge level under constant learning and forgetting conditions.

The model expresses a balance between knowledge acquisition and decay. The term A ( M − y ) represents the remaining learning part of M , while the term L y captures the loss of acquired knowledge over time. This structure ensures boundedness of solutions and convergence to a finite equilibrium.

We extend the model to two interacting persons with knowledge levels y 1 ( t ) at time t and y 2 ( t ) . The initial conditions are given by

The admissible region

is positively invariant under the dynamics of system (7), provided that A i > 0 , L i > 0 , k > 0 , and the initial conditions satisfy 0 ≤ y i ( 0 ) ≤ M , i = 1 , 2 .

Indeed, on the boundary y i = 0 , the corresponding derivative satisfies

where j ∈ { 1 , 2 } and j ≠ i . Similarly, on the boundary y i = M ,

Therefore, trajectories cannot leave the region Ω, and the bounds

are preserved for all t ≥ 0 .

Interaction between persons is modeled through knowledge exchange proportional to their differences. The coupling terms are defined as

This interaction tends to reduce knowledge differences between learners, driving the system toward cognitive equalization. It should be emphasized that the interaction term is not intended to represent a literal transfer of knowledge from one learner to another. Rather, it provides an idealized consensus-type mechanism that models the tendency of interaction to reduce cognitive disparities between learners.

In this interpretation, the conservation property refers only to the interaction contribution and should not be interpreted as implying that knowledge is literally transferred or depleted through collaboration. Knowledge acquisition continues to occur through the external learning terms, while the interaction acts as a mathematical equalization mechanism.

The interaction satisfies the conservation property

which ensures that knowledge is redistributed within the system without being artificially created or destroyed.

This diffusive coupling structure reflects peer-learning mechanisms and depends only on relative knowledge differences.

Combining individual learning dynamics with interaction, we obtain the system

Equivalently, the system can be written as,

In matrix form, we write

where

The system (9) is a non-homogeneous linear dynamical system admitting a unique equilibrium. Its structure allows a complete analytical treatment, including explicit computation of equilibrium states and stability properties. In the next section, we exploit this formulation to derive equilibrium solutions, analyze eigenvalues, and characterize convergence dynamics in terms of the model parameters.

We begin by determining the steady state of the system (9).

Theorem 3.1 (Existence and uniqueness of equilibrium)The system (9) admits a unique equilibrium point Y * = ( y 1 * , y 2 * ) given by

where A and b are defined in (10).

Proof. The equilibrium is obtained by setting d Y ( t ) d t = 0 in the system (9). Thus, the equilibrium satisfies

which gives

provided that A is invertible. The determinant of A is

Since all parameters are positive, it follows that det ( A ) > 0 , hence A is invertible. Therefore, the equilibrium exists and is unique. □

The equilibrium represents the maximum attainable knowledge levels of the two persons under the combined influence of learning, decaying memory, and interaction.

We examine the stability of the equilibrium point of system (9).

Theorem 3.2 (Global asymptotic stability)The equilibriumpoint Y * = ( y 1 * , y 2 * ) of system (9) is globally asymptotically stable.

Proof. The eigenvalues λ 1 , λ 2 of matrix A are

where

Since tr ( A ) < 0 and det ( A ) > 0 , both eigenvalues are strictly negative. □

Therefore, the system converges monotonically to equilibrium without oscillations, forming a stable node.

The explicit form of the eigenvalues provides additional insight into the convergence dynamics. In particular, the eigenvalues are always real and negative, implying that the system converges monotonically toward equilibrium without oscillations. Geometrically, this behavior corresponds to a stable node in the phase plane. All trajectories are directed toward the equilibrium point and no spiraling occurs. This behavior is illustrated in Figure 1, where all trajectories converge toward Y * and the vector field points inward.

The solution of system (9) can be expressed in terms of the eigenvalues and eigenvectors of matrix A . In particular, the general solution takes the form

where λ 1 , λ 2 are the eigenvalues of matrix A , v 1 , v 2 the corresponding eigenvectors, and c 1 , c 2 constants determined by the initial conditions. Since both eigenvalues are real and negative, the system exhibits monotone convergence toward equilibrium without oscillations.

Fast and slow dynamics

The solution naturally decomposes into two distinct modes associated with the eigenvalues:

Figure 1. Phase portrait of the linear system. Trajectories converge monotonically toward the equilibrium point, forming a stable node.

A fast mode, corresponding to the eigenvalue with larger absolute value,A slow mode, corresponding to the eigenvalue closest to zero.

The fast mode decays rapidly and governs the initial transient behavior, while the slow mode determines the long-term convergence toward equilibrium.

Characteristic time scale

The dominant eigenvalue (i.e., the one closest to zero) determines the characteristic convergence time of the system, defined as

A larger absolute value of the dominant eigenvalue corresponds to faster convergence.

Geometric interpretation

This behavior admits a clear geometric interpretation in the phase plane. Trajectories initially align with the direction of the fast eigenvector, corresponding to rapid decay of one component. Subsequently, they evolve along the direction of the slow eigenvector, which governs the final approach to equilibrium. This two-stage convergence mechanism is reflected in the trajectories shown in Figure 1, which exhibit an initial rapid motion followed by a slower asymptotic approach. This separation of time scales will be further analyzed in Section 3.5, where the role of the coupling parameter is examined in detail.

The eigenvectors of matrix A determine the directions along which trajectories approach the equilibrium in the phase plane. In particular, the eigenvectors associated with the eigenvalues λ 1 and λ 2 , given by (13), define the principal directions of convergence. The corresponding eigenvectors are given by

where d is defined in (14). The eigenvector corresponding to the dominant eigenvalue (i.e., the one closest to zero) determines the asymptotic direction of convergence. Denoting by m the slope of this eigenvector, we obtain

This quantity provides a direct measure of the relative variation of the two knowledge levels during convergence. If m > 1 , then y 2 exhibits a stronger variation than y 1 , indicating that the second person undergoes a larger adjustment. Conversely, if m < 1 , the first person adjusts more significantly. In this way, the interaction induces a stronger response in the dynamically weaker person.

To illustrate this behavior, consider, for example m = 3 . Along this direction, an increase of 10 units in y 1 , corresponds to an increase of approximately 30 units in y 2 , indicating a significantly stronger response of the second person.

For large values of the coupling parameter k , the slope satisfies m → 1 , implying that the two variables evolve symmetrically. In this regime, the system approaches a state of near-complete equalization of knowledge.

The coupling parameter k plays a central role in shaping the qualitative behavior of the system. When the coupling parameter k = 0, the system decouples and the two persons evolve independently according to their individual learning decaying memory dynamics. In this case, the eigenvectors coincide with the coordinate axes y 1 and y 2 (Figure 2), and their slopes take the limiting values 0 and + ∞ . This indicates that changes in the knowledge of one person have no influence on the other, and each person converges to its individual equilibrium at a rate determined by the parameter ( A i + L i ) , i = 1 , 2 .

As k increases, the interaction between persons becomes progressively stronger, and the system transitions from independent evolution to cooperative dynamics. This transition is reflected in the geometry of the phase plane: the eigenvectors rotate away from the coordinate axes, and the trajectories bend toward a common direction. The slope m of the dominant eigenvector depends explicitly on both k and the parameter difference d = ( A 1 + L 1 ) − ( A 2 + L 2 ) , which represents the difference in the intrinsic dynamical responses of the two persons. When this difference is significant, the slope becomes highly sensitive to variations in k, indicating that interaction strongly amplifies the adjustment of the dynamically weaker person. As the coupling strength increases ( k → + ∞ ), the influence of the individual parameters A i and L i on the system dynamics becomes progressively weaker. In this regime, the coupling parameter k dominates the evolution, effectively bending the individual learning trajectories and forcing them to converge toward a common direction. As a result, individual learning behavior is transformed into collective stability. Overall, the coupling parameter k acts as a synchronization mechanism that converts individual evolution into collective dynamics, driving the system toward a common trajectory and reducing knowledge disparities between the persons.

Figure 2. Effect of the coupling parameter k on the phase-plane geometry. As k increases, trajectories bend toward a common direction, reflecting the transition from independent to cooperative dynamics.

In the individual learning model, the equilibrium knowledge level is given by

which represents the maximum level of knowledge that a person can attain under constant learning and decaying memory conditions. In the collaborative model, the equilibrium point ( Y * ) can be interpreted as a weighted compromise between the corresponding individual equilibria, resulting from the interaction between the two persons through the coupling term. A key role in determining the final knowledge levels is played by the ratio A i L i , i = 1 , 2 , which can be interpreted as a measure of the cognitive strength of each person. This ratio reflects the ability of a person to retain acquired knowledge relative to the rate of forgetting, and thus determines the level toward which knowledge converges in the long term. In particular, if

then person 1 attains a higher equilibrium knowledge level than person 2, even in the presence of strong interaction. This highlights an important structural property of the model: while the coupling parameter (k) governs the rate of convergence and promotes the equalization of knowledge levels, it does not fully eliminate the influence of intrinsic individual characteristics. Consequently, although collaboration reduces cognitive differences between persons, the final distribution of knowledge remains strongly determined by the ratio A i L i , i = 1 , 2 , which encodes the long-term cognitive capacity of each person. Geometrically, this effect can be interpreted as a shift of the equilibrium point in the phase space. Persons with higher values of the ratio A i L i , i = 1 , 2 , exhibit stronger knowledge retention, leading to higher equilibrium levels. As a result, the equilibrium point is displaced toward the person with greater cognitive strength, as illustrated in Figure 3.

To gain deeper insight into the structure of the system, we introduce the transformation

where S ( t ) represents the mean knowledge of the system and D ( t ) the cognitive difference between the two persons.

3.7.1. Dynamical Decomposition

Using the transformation (22), the system can be rewritten in terms of S ( t ) and D ( t ) . The initial conditions (4) become

where D ( 0 ) quantifies the initial difference in knowledge between the two persons. By adding the two equations of system (9), we obtain

Figure 3. Influence of the cognitive strength ratio A_i/L_i on the equilibrium point. Higher ratios lead to higher equilibrium knowledge levels, shifting the steady state accordingly.

Similarly, subtracting the two equations gives

A key structural property follows directly from Equations (24)-(25): The coupling parameter k does not appear in the evolution of S ( t ) , while it enters explicitly in the decay of D ( t ) . This implies that collaboration does not increase the total knowledge of the system, but instead redistributes it by reducing cognitive differences between the learners. It should be emphasized that this statement refers only to the interaction terms. The total knowledge of the system may still increase through the external learning inputs A 1 M and A 2 M , which represent exogenous knowledge acquisition. The role of collaboration is therefore redistributive rather than generative.

3.7.2. Dissipation of Knowledge Disparity

To obtain an explicit closed-form expression for the difference variable, we consider the symmetric case

Under this assumption, Equation (25) reduces to

Equation (25) admits the explicit solution

This expression shows that the knowledge difference decays exponentially over time, with an effective convergence rate given by

This expression shows that the coupling parameter k enhances the decay rate linearly, acting as a dissipative mechanism that accelerates the reduction of cognitive differences.

3.7.3. Critical Coupling Threshold

The threshold interpretation introduced below is based on the symmetric case considered in Section 3.7.2, namely

The expression (27) suggests the definition of a critical coupling value

This threshold separates two qualitatively different regimes:

For k ≪ k c , convergence is dominated by the intrinsic learning and decaying memory dynamics.For k ≥ k c , the interaction becomes dominant, leading to rapid equalization of knowledge.

Thus, k c defines a transition scale between individual and collective learning behavior.

3.7.4. Phase-Space Interpretation

It should be emphasized that the complete equalization condition D * = 0 is obtained only in the symmetric case A 1 = A 2 and L 1 = L 2 . For asymmetric learners, characterized by different cognitive strengths A 1 L 1 ≠ A 2 L 2 , the system generally converges to a nonzero equilibrium disparity D * ≠ 0 . In this case, collaboration still reduces cognitive differences and accelerates convergence, but it does not completely eliminate the steady-state disparity induced by intrinsic differences between the learners. The dynamics in the (S, D) plane are illustrated in Figure 4.

For small k , trajectories approach the equilibrium gradually, with slow decay of D ( t ) .Near the critical value k c , trajectories exhibit noticeable curvature, reflecting the increasing influence of interaction.For large k , trajectories exhibit rapid reduction of D ( t ) , approaching D * = 0 in the symmetric case and D * ≠ 0 in the asymmetric case.

This representation clearly separates two fundamental processes:

knowledge accumulation, governed by S ( t ) ,knowledge equalization, governed by D ( t ) .

Overall, the mean-difference formulation reveals that collaborative learning is governed by two distinct mechanisms: the accumulation of knowledge and the dissipation of cognitive differences, with the coupling parameter controlling only the latter.

Figure 4. Trajectories in the (S, D) phase plane for different values of the coupling parameter k: (a) weak coupling, (b) near-critical coupling, and (c) strong coupling. Increasing k accelerates the decay of knowledge disparity, leading to stronger synchronization and faster convergence toward equilibrium.

3.7.5. Visualization and Trajectory Interpretation

The dynamical behavior of the system is further illustrated in Figure 4, where trajectories are represented in the ( S , D ) phase plane for different values of the coupling parameter k . The trajectories originate from the initial condition ( S ( 0 ) , D ( 0 ) ) and converge toward the equilibrium point.

Three characteristic regimes can be distinguished:

Weak coupling ( k ≪ k c ): The trajectory approaches equilibrium gradually (Figure 4(a)). The cognitive difference D decays slowly, indicating that the dynamics are dominated by the intrinsic learning and forgetting rates of the persons. Interaction plays a minor role in this regime.Critical coupling ( k = k c ): The trajectory exhibits noticeable curvature, reflecting the increasing influence of interaction (Figure 4(b)). The decay of D ( t ) becomes comparable to the evolution of S ( t ) , marking the transition between individual and collective dynamics.Strong coupling ( k > k c ): A rapid initial reduction of the cognitive difference D ( t ) is observed, corresponding to strong synchronization between the learners. (Figure 4(c)). This is followed by a slower evolution along the S direction toward equilibrium. In this regime, collaboration dominates the dynamics.

These trajectories reveal a two-stage convergence mechanism:

A fast horizontal (or vertical, depending on representation) motion associated with the decay of cognitive difference,Followed by a slower motion toward the final equilibrium governed by the evolution of the mean knowledge.

Thus, the phase-plane representation (Figure 4) provides a clear geometric interpretation of the learning process, separating the roles of knowledge accumulation and knowledge equalization.

We replace the constant coupling parameter k with a function k ( D ) . A natural and analytically convenient choice is the Gaussian form

where the coefficient k 0 > 0 represents the maximum interaction strength and the parameter a controls the sensitivity of the interaction to the knowledge difference. This choice captures the essential feature of the ZPD: interaction is strongest when the persons have comparable knowledge levels and decreases rapidly as the cognitive difference increases.

The introduction of state-dependent coupling leads to qualitatively richer dynamics. If the initial cognitive difference D ( 0 ) is sufficiently small, then the coupling function remains close to its maximum value k ( D ) ≈ k 0 , and the system behaves similarly to the strongly coupled linear case. In this regime, the cognitive difference decays rapidly, leading to a fast convergence toward a common knowledge level. This corresponds to an effective collaborative learning process.

In contrast, when the initial cognitive difference is large, the coupling function becomes negligible k ( D ) ≈ 0 , and the system effectively decouples. The persons then evolve almost independently, and the reduction of cognitive difference occurs at a much slower rate. This behavior naturally leads to the notion of a convergence zone, defined as the set of initial conditions for which the interaction remains sufficiently strong to ensure rapid equalization of knowledge.

The nonlinear formulation introduces the concept of a critical cognitive difference scale, beyond which interaction becomes ineffective. More specifically:

For small values of | D | , the coupling remains strong and the system operates in a collaborative regime.For large values of | D | , the coupling weakens significantly and the system enters a decoupling regime.

Thus, the system exhibits two qualitatively distinct modes of behavior:

A collaborative regime, characterized by rapid convergence toward equal knowledge.A decoupling regime, in which persons evolve quasi-independently.

Importantly, this transition is not externally imposed, but emerges intrinsically from the state-dependent structure of the interaction.

The qualitative behavior of the nonlinear system is illustrated in Figure 5, which depicts representative trajectories in the ( S , D ) phase plane. The green trajectory corresponds to an initial condition within the convergence zone. In this case, the coupling remains strong and the trajectory rapidly collapses toward the line D = 0 , indicating efficient knowledge equalization. In contrast, the red trajectory corresponds to a large initial cognitive difference. Here, the coupling function is weak, and the trajectory evolves slowly without significant reduction of disparity,

Figure 5. Nonlinear dynamics with state-dependent coupling. The green trajectory corresponds to an initial condition within the convergence zone, leading to rapid equalization of knowledge, while the red trajectory illustrates a decoupling regime where large initial disparity suppresses interaction.

illustrating the decoupling regime.

The phase space can therefore be partitioned into two regions:

A convergence zone, where interaction is effective and learning is collaborative.A decoupling zone, where interaction is suppressed due to excessive disparity.

Overall, the nonlinear extension reveals that collaboration is not uniformly effective, but depends critically on the initial cognitive distance between persons. The model provides a dynamical interpretation of the Zone of Proximal Development, demonstrating that successful interaction requires a sufficient level of cognitive proximity.