Conformable Logistic Dynamic Equations on Timescales

Abstract

In this paper, we study an autonomous conformable logistic dynamic equation in proportional conformable calculus on timescales. A reciprocal transformation reduces the nonlinear problem to a mixed first-order conformable equation and yields an exact representation, including the constant-coefficient solution with effective carrying capacity $K_{\alpha }=k(1-{\kappa }_1/r)$. Under the positivity condition $D_{\alpha }\left(t\right)={\kappa }_0+\mu \left(t\right)(r-{\kappa }_1)>0$, positive solutions are global and forward invariant. According to the relation between $r$ and ${\kappa }_1$, solutions converge to a positive equilibrium, exhibit algebraic extinction in the critical case, or decay exponentially. For the standard proportional choice, the critical order is ${\alpha }^{\ast }=1-r$. We also establish conformable-exponential comparison and conditional Hyers–Ulam stability results. In the standard autonomous case, the equation can be rewritten as a delta logistic model with effective growth and removal coefficients, clarifying how the conformable order controls their balance. The critical order is proved to be a transcritical bifurcation value for constant graininess and for the associated period map on periodic or forward-periodic timescales. The conditional Hyers–Ulam stability result is formulated through explicit initial-state and defect thresholds without requiring bounded graininess. The theory is further illustrated on several discrete, hybrid, and Cantor-type timescales.