۱- Introduction

۲- Preliminary results

۳- Existence and uniqueness theorems

۴- Stability analysis for the logistic models

۵- Numerical discussion

۶- Conclusion

References

Abstract

In this article, we study different types of fractional-order logistic models in the frame of Caputo type fractional operators generated by conformable derivatives (Caputo CFDs). We present the existence and uniqueness theorems to solutions of these models and discuss their stability by perturbing the equilibrium points. Finally, we furniture our results by illustrative numerical examples for the studied models.

Introduction

Fractional calculus is a branch of mathematical analysis that takes into consideration the integration and differentiation of real or complex order. In spite of the fact that this kind of calculus is old, it gained popularity and started to catch the interest of scientists only in the last 20 or 30 years because important results were reported when fractional derivatives and integrals were applied to describe many real world phenomena [1–۱۰]. A big virtue of the fractional calculus is that there are many different fractional derivatives or integrals. This virtue gives the opportunity to choose the most appropriate derivative or integral in order to describe complex systems of real world problems eligibly. Nevertheless, in order to have better mathematical models of real world problems, scientists started to disclose some new types of fractional integrals and consequently fractional derivatives using two main methods. The first method is the traditional method based on iterating to find the nth order integral and then replacing n by any number α. Hadamard, generalized fractional operators and the fractional operators generated from conformable derivatives, can be considered as examples of fractional operators obtained using this approach [11–۱۷]. These operators usually have singular kernels. The second method is subject to the limiting process and using some properties of the Dirac-Delta functions.