The Fisher–Kolmogorov (FK) model is a well-known nonlinear reaction–diffusion equation widely used to describe the interplay between diffusion and reaction and find important applications across disciplines, e.g., population dynamics, neurodegenerative diseases, cancer growth, epidemiology, and invasive species expansion. From a mathematical viewpoint, it can be proved that the solution of the FK equation with homogeneous Neumann boundary conditions remains nonnegative and bounded, provided that the initial conditions are nonnegative and the reaction term satisfies appropriate biological constraints, corresponding to the absence and saturation of the population or pathological agent, respectively. The inherent positivity-preserving mechanism of the FK dynamics is crucial from a modeling perspective. Indeed, in biological and physical applications, the solution to the FK equation typically represents a density or a concentration, for which negative values make no sense. At the numerical level, preserving positivity is particularly challenging, since numerical schemes do not automatically inherit this property. Classical Finite Element discretizations may lead to negative values of the discrete solution that are not only unphysical but can also negatively influence the overall stability of the schemes. In this article, we first propose a modified FK model that remains consistent with the original FK model, and we introduce its discretization, which is based on a high-order Discontinuous Galerkin method on general polygonal and polyhedral grids for spatial discretization, combined with an implicit second-order time integration scheme. Thanks to the proposed modified FK model, the corresponding semidiscrete formulation is automatically positivity preserving. The numerical experiments demonstrate that the formulation approach exhibits stability and optimal accuracy under both mesh refinement and high-order polynomial approximations. The proposed numerical method can find important applications in modeling neurodegenerative diseases, where the FK equation is used to simulate pathological protein propagation typical of proteinopathies like Alzheimer's or Parkinson's diseases.