Mathematical models using differential equations with integer-order have proved valuable in understanding the dynamics of biological systems. However, most biological, physical, and engineering systems have long-range temporal memory [3, 114, 309, 363] and/or long-range space interactions [204, 220, 408]. Modeling such systems using fractional-order differential equations is more advantageous than classical integer-order mathematical modeling, in which the effects of existence of time memory or long-range space interactions are neglected. Moreover, the fractional-order derivative is related to the whole space for a physical process, whereas the integer-order derivative describes the local properties of a certain position. Accordingly, the subject of fractional calculus (i.e., calculus of integral and derivatives of arbitrary order) has gained popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields of science and engineering. It has been successfully applied to system biology [86, 117, 309, 311, 399], physics [100, 116, 158, 407], chemistry and biochemistry [406], hydrology [211, 351], engineering [230, 231], medicine [16, 129, 139], and finance [83]. Examples of fractional-order systems in modeling and control can be found in [78, 228, 229].

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Fractional-Order Delay Differential Equations with Predator-Prey Systems

  • Fathalla A. Rihan

摘要

Mathematical models using differential equations with integer-order have proved valuable in understanding the dynamics of biological systems. However, most biological, physical, and engineering systems have long-range temporal memory [3, 114, 309, 363] and/or long-range space interactions [204, 220, 408]. Modeling such systems using fractional-order differential equations is more advantageous than classical integer-order mathematical modeling, in which the effects of existence of time memory or long-range space interactions are neglected. Moreover, the fractional-order derivative is related to the whole space for a physical process, whereas the integer-order derivative describes the local properties of a certain position. Accordingly, the subject of fractional calculus (i.e., calculus of integral and derivatives of arbitrary order) has gained popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields of science and engineering. It has been successfully applied to system biology [86, 117, 309, 311, 399], physics [100, 116, 158, 407], chemistry and biochemistry [406], hydrology [211, 351], engineering [230, 231], medicine [16, 129, 139], and finance [83]. Examples of fractional-order systems in modeling and control can be found in [78, 228, 229].