A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment

A numerical method is developed for a general structured population model coupled with the environment dynamics over a bounded domain where the individual growth rate changes sign. Sign changes notably exhibit nonlocal dependence on the population density and environmental factors (e.g., resource availability and other habitat variables). This leads to a highly nonlinear PDE describing the time-evolution of the population density coupled with a nonlinear-nonlocal system of ODEs describing the environmental time-dynamics. Stability of the finite-difference numerical scheme and its convergence to the unique weak solution are proved. Numerical experiments are provided to demonstrate the performance of the finite difference scheme and to illustrate a range of biologically relevant potential applications.