<p>The logistic growth models serve as a framework to explore the biomechanics of systems in biology. These models help in understanding the population dynamics, cellular and tissue growth, systems physiology and cardiovascular mechanics. The fractional logistic growth models are extensions of these models with advantages of memory effects and non-local dynamics. This makes them more suitable for reflecting the complexities of biological systems. In this article, numerical algorithms are used to find approximate solutions of fractional logistic growth models. The Runge–Kutta methods are extended and analyzed for the case of fractional differential equations. Specifically, the second order and fourth order Runge–Kutta methods, referred as FRK2 and FRK4, are implemented and explored. The proposed algorithms are then implemented on few test problems and then on the fractional logistic population growth models. These logistic models include the nonlinear logistic model, the cubic logistic model and the quadratic logistic model. The convergence of proposed methods is discussed and comparison of approximate solutions is observed. The numerical computations are performed using the computing tool MATLAB. These methods are highly accurate, stable and computationally efficient. The numerical approximations reveal the accuracy and efficiency of our algorithms. The convergence and accuracy of fourth order Runge-Kutta method was found better. The proposed methods are efficient enough to predict the solution of fractional differential equations that do not possess exact solution. The accuracy of Runge Kutta methods increases with increase in fractional order.</p>