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$u_{t}= -\alpha_{k}(-1)^{k} \nabla^{2k}u+\sum_{j=1}^{k-1}\alpha_{j} (-1)^{j}\nabla^{2j}u+\nabla^{2}(u^{m})+u(1-u^{2p}), $

with periodic boundary conditions on a one-dimensional spatial domain. The equation generalises nonlinear diffusion model for the case when higher-order differential operators are present. Furthermore, we establish the asymptotic positivity as well as the positivity of solutions for all times under certain restrictions on the initial data. The above class of equations reduces for some particular values of the parameters to classical models such as the KPP-Fisher equation which appear in various contexts in population dynamics.

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