<p>This study develops and analyzes a nonlinear dynamical model describing the co-evolution of consumer trust, perceived reliability, and advertising effort in brand adoption. The model comprises a system of ordinary differential equations incorporating saturation effects and interdependent growth rates. Analytical results establish boundedness, positivity, and the existence of equilibria governed by a basic influence number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{R}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">R</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. Detailed stability analysis shows that the boundary equilibrium is globally asymptotically stable when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{R}_1<\textbf{R}_{1,\textrm{SN}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">R</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi mathvariant="bold">R</mi> <mrow> <mn>1</mn> <mo>,</mo> <mtext>SN</mtext> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, while the interior equilibrium becomes globally stable for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{R}_1>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">R</mi> <mn>1</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Global stability is proven using classical results from the theory of monotone dynamical systems and order-preserving semiflow properties of the system. Rigorous bifurcation analyses reveal the existence of transcritical and saddle-node bifurcations, highlighting scenarios where consumer dynamics exhibit bistability. The absence of Hopf bifurcation is demonstrated using monotone systems theory and Perron–Frobenius spectral arguments. Numerical simulations illustrate forward and backward bifurcations, bistability, and global sensitivities using Partial Rank Correlation Coefficients. These findings extend classical diffusion models by showing how early advertising and credibility signals can determine long-term brand adoption, offering a novel, mathematically grounded framework for nonlinear adoption dynamics.</p>

Bifurcation and stability analysis of a nonlinear brand adoption model with advertising, trust, and reliability interactions

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发表日期：

2026年3月16日

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Skand Dwivedi,
Ram Naresh Tripathi,
Shyam Sundar,
Ahana Verma,
Maninder Singh Arora

摘要

This study develops and analyzes a nonlinear dynamical model describing the co-evolution of consumer trust, perceived reliability, and advertising effort in brand adoption. The model comprises a system of ordinary differential equations incorporating saturation effects and interdependent growth rates. Analytical results establish boundedness, positivity, and the existence of equilibria governed by a basic influence number $\textbf{R}_1$ $R_{1}$ . Detailed stability analysis shows that the boundary equilibrium is globally asymptotically stable when $\textbf{R}_1<\textbf{R}_{1,\textrm{SN}}$ $R_{1} < R_{1, SN}$ , while the interior equilibrium becomes globally stable for $\textbf{R}_1>1$ $R_{1} > 1$ . Global stability is proven using classical results from the theory of monotone dynamical systems and order-preserving semiflow properties of the system. Rigorous bifurcation analyses reveal the existence of transcritical and saddle-node bifurcations, highlighting scenarios where consumer dynamics exhibit bistability. The absence of Hopf bifurcation is demonstrated using monotone systems theory and Perron–Frobenius spectral arguments. Numerical simulations illustrate forward and backward bifurcations, bistability, and global sensitivities using Partial Rank Correlation Coefficients. These findings extend classical diffusion models by showing how early advertising and credibility signals can determine long-term brand adoption, offering a novel, mathematically grounded framework for nonlinear adoption dynamics.