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We consider a reaction-diffusion-chemotaxis model for the spatial spread of bark beetle infestations. (Our specific focus is on the southern pine beetle, which destroys tens of thousands of acres of pine forests annually.) Standard Keller-Siegel-type chemotaxis terms model the pheromonal attraction and repulsion of beetles to and from colonized trees. The hyperbolic character of these nonlinear chemotaxis terms typically results in very steep gradients and induces an amazing variety of complex behaviors, including traveling waves and pulses.
The mixed parabolic-hyperbolic character of the model makes numerical simulation a highly delicate problem. Following the work of LeVeque and others, we apply a fractional-step method in which different methods specially suited for reaction, diffusion, and chemotaxis, respectively, are combined without sacrificing second-order accuracy. We will discuss the implementation of this technique in Mathematica and present a number of generated animations.
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