We study the spatially homogeneous time-dependent solutions and their bifurcations of the Gray-Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens-Bogdanov and a Bautin bifurcations, in the space of two parameters k-F . With the aid of numerical continuation of local bifurcation curves we give a global description of all the possible bifurcations.

Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non compact.

One of the simplest model of immune surveillance and neaoplasia was proposed by Delisi and Resigno [7]. Later Liu et al [9] proved the existence of non-degenerate Takens-Bogdanov bifurcations defining a surface in the whole set of five positive parameters. In this paper we prove the existence of Bautin bifurcations completing the scenario of possible codimension two bifurcations that occur in this model. We give an interpretation of our results in terms of the three phases immunoediting theory:elimination, equilibrium and escape.

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