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A thorough analysis is performed to find traveling waves in a qualitativereaction-diffusion system inspired by a predator-prey model. We providerigorous results coming from a standard local stability analysis, numericalbifurcation analysis, and relevant computations of invariant manifolds toexhibit homoclinic and heteroclinic connections, and periodic orbits in theassociated traveling wave system with four components. In so doing, we presentand describe a zoo of different traveling wave solutions. In addition,homoclinic chaos is manifested via both saddle-focus and focus-focusbifurcations as well as a Belyakov point. An actual computation of globalinvariant manifolds near a focus-focus homoclinic bifurcation is also presentedto unravel a multiplicity of wave solutions in the model.

A case study of multiple wave solutions in a reaction-diffusion system using invariant manifolds and global bifurcations