Molding the dynamic system with memory-dependent derivative

As a new alternative of fractional-order derivative, we have brought forth the concept of “memory-dependent derivative”. It is more flexible than the fractional-order one in describing the memory effect. In fact, not only the time-delay but also the weighted function can be chosen freely to adapt the request of different dynamic processes. As an example of applications the Malthus population model in ecology is studied. It follows from numerical simulations that the memory-dependent differential equation type of model has more power in representing the variation of population evolution. Theoretical study reveals that to ensure the existence and uniqueness of solution the time delay should smaller than an upper bound determined by the weighted function. Till now, the study on the theory and applications of memory-dependent derivative is just on its prime stage, it needs further research.