Deterministic and Probabilistic Modeling of the Logistic Growth

The logistic growth concept investigated by many researchers has wide applications in different fields. An exact solution to the logistic growth problem can always be obtained using the first order differential equation. However, this is not always possible when the fractional order of derivative is used. This work investigated the use of deterministic and probabilistic approaches for modeling the logistic growth models. The deterministic model was built using classical and fractional differential equations. Hadamard type fractional derivative and integral were used to prove the existence and uniqueness of the solution to the fractional logistic differential equation using theorems. Numerical methods were employed to approximate the solution in the fractional case since it has no analytic form. The probabilistic approach used by employing the Gaussian kernel smoothing. A comparison of deterministic and probabilistic methods performance in modeling the logistic growth concept, minimum error levels were achieved with the fractional method, and Gaussian kernel smoother method with bandwidth 22. Keywords: Gaussian kernel, optimal bandwidth, fractional differential equation, Hadamard derivative, Caputo-Fabrizio, Grünwald-Letnikov, generalized Euler method, carrying capacity.