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 Kα=k(1−κ1/r)$K_{\alpha }=k(1-\kappa _1/r)$ . Under the positivity condition Dα(t)=κ0+μ(t)(r−κ1)>0$\mathcal {D}_{\alpha }(t)=\kappa _0+\mu (t)(r-\kappa _1)>0$ , positive solutions are global and forward invariant. According to the relation between r$r$ and κ1$\kappa _1$ , 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 α∗=1−r$\alpha ^{*}=1-r$ . 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.