Catalyst-induced growth with limited catalyst lifespan and competition

Spatially extended catalyst-induced growth processes are studied. This type of processes exists in all domains of biology, ranging from ecology (nutrients and growth), through immunology (antigens and lymphocytes) to molecular biology (signaling molecules initiating signaling cascades). The extinction-proliferation transition is considered for a system containing discrete catalysts (A) that induces the proliferation of a discrete reactant (B). The realization of this model on an infinite capacity d-dimensional discrete lattice for immortal catalysts has been previously considered (the AB model). It was shown that the adaptation of the reactants to the diffusive noise induced by stochastic fluctuations of catalyst density yields proliferation even if the average environmental conditions lead to extinction. This model is extended here to include more realistic situations, like finite lifespan of the catalysts and finite carrying capacity of the reactants. By using a combination of Monte Carlo simulation, percolation-theory-based estimations and an analytic perturbative analysis, the asymptotic behavior of these systems is studied. In both cases studied, it turns out that the overall survival of the reactant population at the long run is based on the size and shape of a typical single colony, related to the localized proliferation around spatio-temporal catalyst density fluctuations. If the density of these colonies (based on the lifetime of the spatial fluctuation and the carrying capacity of the medium) is large enough, i.e. above the percolation threshold, the reactant population survives even in (on average) hostile environment. This model provides a new insight on the population dynamics in chemical, biological and ecological systems.