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

• 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.

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 carbo  is 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)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.