Extinction in Random Environments

An important currency for individuals is their lifetime reproductive success (LRS), which is random simply due to demographic stochasticity. That randomness determines extinction probability. However, the distribution of LRS is also significantly affected by environmental variation. Previously we have shown (for a random environment that follows a Markov chain) how LRS is affected by an individual's birth environment. But our previous analysis severs the temporal linkage between a parent's birth environment and the environments into which its offspring are born. Here, we show how to compute the exact joint probability distribution of LRS for lineages spanning multiple environmental states (assuming a Markovian environment). From this joint distribution, we derive exact lineage extinction probabilities that fully incorporate demographic stochasticity, environmental frequency, and temporal autocorrelation. Applying our framework to Pacific Chinook salmon (semelparous with extreme early mortality) and European roe deer (iteroparous with delayed maturity), we demonstrate that initial birth states shape lineage fate. For salmon, the initial environment permanently separates trajectories; a poor birth state leads to near-certain extinction regardless of subsequent environmental shifts. For roe deer, we discover a counterintuitive dynamic where highly persistent poor conditions can rescue highly vulnerable individuals by expanding the right tail of reproduction. Furthermore, our exact calculations reveal that aggregated one-dimensional LRS distributions homogenize the reproductive landscape, overestimating extinction risks for lineages originating in poor environments and underestimating them for those in good environments. Accurately predicting evolutionary viability and the establishment of advantageous mutations requires preserving the environmental covariance. As global climate change amplifies environmental volatility, utilizing exact joint demographic models is critical for assessing true extinction risks.