From disturbances to nonlinear fitness and back

Abstract

AbstractA fundamental goal of Ecology is to predict how natural populations respond to disturbances. Accordingly, the last decades have witnessed key theoretical developments in stochastic demography and transient dynamics. However, both areas, have to date been largely disconnected. Here, we introduce an expression for the second derivatives of population growth rate with respect to demographic rates (e.g. survival-dependent state transitions and reproduction) with direct links to transient dynamics. We use this connection to develop a new mathematical framework showing how transient responses to pulse disturbances lead to a quantitative description of press disturbances. Second-derivatives of population growth rate with respect to said demographic rates are valuable as they quantify the degree of nonlinear selection acting on demographic rates and how environments shape the long-term performance of populations. Whilst valuable, previous methods to quantify second-order derivatives have heavily relied on vector calculus-potentially obscuring important demographic processes connected to second-order derivatives. Here we offer an intuitive method using perturbation theory and our approach is valid for any discrete-time, st(age)-based structured population model. Importantly, our new method implicates an intimate relationship between the nonlinear selection pressures acting on demographic rates with the emergent transient dynamics of populations over time. We showcase these relationships through mathematical proofs, connecting to Cohen’s cumulative distance, and identifying a strong relationship between generation time across 439 unique plant and animal species (2690 population models). As such, this new method represents a valuable tool for population ecologists, comparative demographers, and conservation biologists to understand and protect structured populations in a changing world.