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Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response. (English) Zbl 1132.65092
The authors study two semi-implicit fully discrete, piecewise linear finite element methods for numerical approximation of the solutions of a class of nonlinear reaction-diffusion systems modeling predator-prey interactions, where the local growth of the prey is logistic and the predator displays the Holling type II functional response. The finite element scheme uses a semi-implicit discretisation in time. A rigorous analysis is made to establish a priori estimates and error bounds for the finite element approximations.
Numerical simulations are carried out for 1-D and 2-D problems. It is also shown that a modest change in one of the parameters can lead to dramatic changes in the qualitative dynamics of solutions; namely stationary, smooth oscillatory, intermittent chaos and chaos. Some more numerical simulations are needed for estimating the error bounds with time before the finite element scheme is implemented.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
92D25 Population dynamics (general)
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
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