| Abstract | We analyze the population dynamics of blowout penstemon (Penstemon haydenii). In Nebraska this endangered plant naturally occurs in "blowouts", which are sparsely vegetated depressions in active sand dunes created by wind erosion. Our goal is to identify factors limiting the population size and to make recommendations for its management. We use discrete time integral projection models to predict plant population dynamics, and to determine how the dynamics depend upon the life history parameters and initial conditions. The kernel for the integral operator is obtained by estimating the size-dependent survival, growth, seed production, and recruitment probability (the probability that a seed will become a seedling in the following year) from a large data set spanning 13 blowout sites in western Nebraska. This model is density-dependent, since the recruitment probability is dependent upon the number of seeds produced. We do a numerical analysis of the transient and asymptotic dynamics, and a mathematical analysis of the asymptotic dynamics. The novel aspect of the mathematical proofs involves writing the projection operator as a closed-loop feedback operator (where the recruitment probability is a feedback of the population) and using control theoretic small-gain techniques to obtain convergence. We find that there is an asymptotic population and stage structure, which is independent of the nonzero initial population and stage structure. This is observed numerically and proved mathematically. We give formulas for the asymptotic population density and quantity in terms of the parameters, and provide global convergence proofs. We also analyze the transient dynamics that are predicted if the population deviates from the stable stage distribution. Our model predicts that in the early phase of blowout colonization population density drops to very small numbers before increasing to the asymptotic population size. This suggests a very small colonization success of this plant since small populations have a high extinction risk because of demographic and environmental stochasticity and Allee effects. We use robustness analysis to evaluate different management strategies. |
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