Reaction–diffusion models: on the conservation laws and integrability of the time-fractional, density-dependent cases

The Fisher and family of Fisher type equations, viz., Kolmogorov, density-dependent diffusion–reaction model equation, inter alia, do not admit any conservation laws. However, under certain conditions of some underlying parameters in the models, the reduction of such systems may be variational. In particular, traveling wave reduction leads to ordinary differential equations that generate Lagrangian and first integrals via Noether’s Theorem. Here, a variational symmetry approach leads to the properties and integrability of Painlevé. A similar situation occurs for the Fisher–Kolomorov and Nagumo type equations. For classes of these equations, we will investigate the existence of Conservation Laws for the fractional cases even though these do not exist for the integral derivative cases, viz., for the fractional Fisher and family of Fisher type equations.