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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.

MSC:
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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