Revisited Fisher’s equation in a new outlook: A fractional derivative approach

• Published in: Physica A 2015-11, Vol.438, p.81-93
• Main Authors: Alquran, Marwan, Al-Khaled, Kamel, Sardar, Tridip, Chattopadhyay, Joydev
• Format: Article
• Language: English
• Subjects: Approximate solutions; Fisher’s equation; Fractional differential equation; Generalized Taylor series; Residual power series; Sinc method
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2015.06.036

The well-known Fisher equation with fractional derivative is considered to provide some characteristics of memory embedded into the system. The modified model is analyzed both analytically and numerically. A comparatively new technique residual power series method is used for finding approximate solutions of the modified Fisher model. A new technique combining Sinc-collocation and finite difference method is used for numerical study. The abundance of the bird species Phalacrocorax carbois considered as a test bed to validate the model outcome using estimated parameters. We conjecture non-diffusive and diffusive fractional Fisher equation represents the same dynamics in the interval (memory index, α∈(0.8384,0.9986)). We also observe that when the value of memory index is close to zero, the solutions bifurcate and produce a wave-like pattern. We conclude that the survivability of the species increases for long range memory index. These findings are similar to Fisher observation and act in a similar fashion that advantageous genes do. •A modified fractional Fisher is proposed to demonstrate the embedded memory index.•We performed new analytical and numerical treatments to solve Fisher equation.•Phalacrocorax carbodata set is considered as a test bed to validate our findings.•Non-diffusive and diffusive systems are similar in a certain interval of memory index.•The wave-like pattern observed in our findings coincides with the thought of Fisher.