An analytically tractable non-autonomous logistic population model with autonomously evolving coefficients

<p>In this paper, we extend the classical logistic law by incorporating <span>autonomously evolving, time-dependent coefficients</span> that allow both the intrinsic growth rate <span class="math inline">\(\gamma(t)\)</span> and the carrying capacity <span class="math inline">\(K(t)\)</span> to vary over time according to logistic modulated dynamics. In particular, the carrying capacity is modeled as a logistic process with intrinsic growth rate <span class="math inline">\(\alpha\)</span> and saturation parameter <span class="math inline">\(\beta\)</span>, yielding an asymptotic level of <span class="math inline">\(\frac{\alpha}{\beta}\)</span>. The objective is to investigate how temporal variability in the governing coefficients influences both transient and asymptotic regimes of the population dynamics and to assess the extent to which the system behavior can be controlled through a reduced set of key parameters. <span>Analytical results are derived in closed form, expressed in terms of hypergeometric functions, and compared with numerical integrations for validation purposes.</span> It is shown that the model admits a long-term equilibrium determined by the ratio <span class="math inline">\(\frac{\alpha}{\beta}\)</span>, independently of the initial population size <span class="math inline">\(S_0\)</span>, while short- and medium-term dynamics are strongly shaped by the interplay between <span class="math inline">\(S_0\)</span> and the <span>non-autonomous logistic evolution</span> of the carrying capacity <span class="math inline">\(K(t)\)</span>. These results illustrate how analytically tractable <span>non-autonomous logistic models with internally generated coefficient trajectories</span> can enhance the qualitative understanding of population dynamics and provide reliable benchmarks for numerical simulations, with potential applications in sustainable resource management, aquaculture, and ecological modeling.</p>