Response of population size to changing vital rates in random environments

Background/Question/Methods

Population size and population growth rate respond to changes in vital rates like survival and fertility. For instance, in the absence of environmental variation, the elasticity of the long-term growth rate λ to a vital rate quantifies the proportional change in λ due to a small proportional change in a vital rate when the population is in its stable stage distribution (SSD). Given λ and the initial conditions, we can precisely determine the population size for any future point in time. As a consequence, most studies have focused on how the population growth rate, rather than population size, responds to changes in vital rates.  However, in random environments, population size at any time t is a random variable, so that a change in size as a result of perturbation obeys a probability distribution. So, the natural question is: Can we predict the effect of changing a vital rate on future population size based on its effect on the long-term stochastic growth rate alone? Further, we are interested in, how population size responds to changes in two vital rates with different elasticities of the long-term population growth rate. Answers to these issues have implications for understanding population dynamics and devising policies for management.

Results/Conclusions

We analytically show that the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ_s while the standard deviation scales as √t. Consequently, a vital rate with a larger elasticity of λ_s may produce a larger mean change in population size compared to one with a smaller elasticity of λ_s. But, a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. In conclusion, we show that the response of population size to vital rate perturbations depends not only on the change in the long-term population growth rate, but also on the variance in this response. Our results facilitate calculating the probability of achieving a specified percentage change in population size even for short time periods. We illustrate the results with demographic data from two plant species.