Changing environments causing time delays in population dynamics Erik Blystad Solbu a,⇑ , Steinar Engen a , Ola Håvard Diserud b,a a Centre for Biodiversity Dynamics, Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway b Norwegian Institute for Nature Research, NO-7485 Trondheim, Norway article info Article history: Received 25 January 2013 Received in revised form 29 May 2013 Accepted 3 June 2013 Available online 13 June 2013 Keywords: Diffusion process Non-stationary Density dependence Time series Detrending Herring abstract We use a linear diffusion process to approximate a stochastic density regulated population model where parameters can change through time. Contrary to stationary models, there is a difference between the expected value and the carrying capacity of a population at any given time. This time delay can be considerable and depends on the vital rates of the population and the magnitude of the change. We emphasize the importance of acknowledging this difference when assessing viability of populations. As an illustration, we consider the population of Norwegian spring spawning herring and its collapse in the 1960s. Based on our analysis, the stock was already at a critical level a decade before the collapse was observed. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction Population dynamics is most commonly analyzed by stationary processes, assuming no catastrophic effects or continuous trends in the environment. A major problem in population viability analysis (PVA) is then the exploration of mean time to extinction [1], or der- ivation of prediction intervals for the time to extinction [2]. These analyzes are important in relation to the precautionary principle [3], that is, how to avoid unacceptable small population sizes [4], and estimating the probability of extinction [5]. However, human activities often affect populations in ways that make the stationary assumption unrealistic. By introducing dynamic parameters that change through time, discrete or contin- uous changes in the environment, caused by human activities, can be dealt with. Examples of sudden changes are construction of dams for hydro-electrical power production, forest clearance for roads or to acquire land for agriculture [6], and accidents such as oil spills or the release of other pollutants. A recent paper by Fukaya et al. [7] studies the effect of habitat fluctuations on the population dynamics of a marine copepod. Such changes in habitat can be modeled as temporal variation in the growth rate of a spe- cies. Gradual changes may occur due to climate change, leading to trends in e.g. temperature and climate indices such as the North Atlantic Oscillation index [8], or changes in stochastic variability. A study on climate effects on Eurasian oystercatcher by van de Pol et al. [9] used a stage-structured model, and showed that the time to extinction increased as the mean average temperature in- creased, while an increase in the standard deviation of average temperature reduced the time to extinction. Renwick et al. [10] used generalized linear models to study changes in species abun- dance under climate change, in addition to other explanatory vari- ables such as land, habitat and rainfall. Under different scenarios for increasing mean global temperature, they predicted an increase in population size for Eurasian nuthatch and green woodpecker, and a decrease in abundance for Eurasian curlew and meadow pi- pit. Climate effects have also been studied for mammals, e.g. seal [11], red deer [12] and soay sheep [13], and for fish species such as the Atlantic cod [14]. Another anthropogenic effect is overexploitation which has dri- ven many populations extinct, or close to extinction [15–20]. With increasingly effective technology, harvesting has introduced dra- matic effects on the population dynamics; effects that may be rep- resented by temporal changes in the parameters describing their dynamics. For instance, the collapse of the Norwegian spring spawning herring population at the end of the 1960s was followed by a period of about 30 years when the population could not be harvested at all [15]. The dynamic response of a given species to changes in the envi- ronment, for example caused by the above mentioned human activities, depends in general on how the mean vital rates of sur- vival and fecundity are affected. The dynamics of populations change in different ways depending on which parameters or char- acteristics of the species that respond to the environmental changes, as well as the values of other parameters. Here, we study such responses, analyzing in particular how changes in population 0025-5564/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2013.06.003 ⇑ Corresponding author. Tel.: +47 97584012. E-mail address: erik.solbu@math.ntnu.no (E.B. Solbu). Mathematical Biosciences 244 (2013) 213–223 Contents lists available at SciVerse ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs