We study the non-equilibrium phase transition between survival and extinction of spatially extended biological populations using an agent-based model. We especially focus on the effects of global temporal fluctuations of the environmental conditions, i.e., temporal disorder. Using large-scale Monte-Carlo simulations of up to $3times 10^7$ organisms and $10^5$ generations, we find the extinction transition in time-independent environments to be in the well-known directed percolation universality class. In contrast, temporal disorder leads to a highly unusual extinction transition characterized by logarithmically slow population decay and enormous fluctuations even for large populations. The simulations provide strong evidence for this transition to be of exotic infinite-noise type, as recently predicted by a renormalization group theory. The transition is accompanied by temporal Griffiths phases featuring a power-law dependence of the life time on the population size.
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