Stable Relationships

The paper defines the same 2D linear update A(t+1)=(1−αγ)A(t)+αγB(t), B(t+1)=βγA(t)+(1−βγ)B(t), computes eigenvalues λ1=1 and λ2=1−γ(α+β), and proves: (i) equilibria are exactly A*=B*; (ii) the system is “marginally stable” iff −α ≤ β ≤ 2/γ − α; (iii) when |λ2|<1, A(t),B(t) converge to (βA(0)+αB(0))/(α+β); (iv) when λ2=−1 there is a 2-cycle with the stated P2; and (v) when λ2=1 the matrix is defective and trajectories grow linearly via a Jordan block. These are all aligned with the candidate solution’s steps and conclusions, which use the same spectral decomposition (and an invariant βA+αB). The only issue we found is a minor typo in the paper’s Jordan-block section: M^t and B(t) are missing a +1 term in the (2,2) entry and corresponding +B(0) term; however the qualitative conclusion (linear divergence unless A(0)=B(0)) remains correct. Model and paper therefore agree in substance and method. Citations supporting the above: model setup and eigenstructure, equilibria, and stability region (Propositions 1–2) ; convergence for |λ2|<1 and the 2-cycle at λ2=−1 with explicit P2 ; the Jordan-block case λ2=1 (with the noted typo) .