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Diffusion models of stochastic population dynamics, including both demographic and environmental stochasticity, can be transformed to a scale with isotropic noise. On this scale, demographic stochasticity contributes a net downward component to population trajectories that is inversely proportional to population size, creating a type of Allee effect. In populations with appreciable demographic stochasticity and a small long-run growth rate, such as many threatened and endangered species, there may be a stochastically unstable equilibrium below which most population trajectories tend to decline toward extinction. Population sizes corresponding to stochastic equilibria on the transformed scale can be obtained directly, without transformation. An explicit transformation and its inverse are derived for illustration in a model with stochastic density-independent growth (including demographic and environmental stochasticity) and deterministic density-dependence.

Empirical studies have shown that animal populations from a wide array of taxa exhibit spatial patterns of correlation in fluctuating abundance. In the search for explanations for this phenomenon it has been proposed that subpopulation interactions in the form of spatial dispersal, or variability in external factors, such as weather, would be the crucial driving forces responsible for spatial synchrony. Nevertheless, dispersal and external factors have been shown to produce different patterns of synchrony. We show here that observed patterns in synchronous dynamics can be reproduced by using a spatially linked population model. Further, we analyse how local and global environmental stochasticity and dispersal influence the pattern of spatial synchrony. We contrast our theoretical results with data on long-term dynamics of North American game animals and emphasize that the data and our spatial population dynamics models are compatible.

A metapopulation model with stochastic local dynamics is developed assuming a small individual migration rate and many local populations. A diffusion approximation for local population dynamics is employed to derive how rates of local extinction and colonization depend on individual migration rate and habitat occupancy in the metapopulation. For a given migration rate, increasing habitat occupancy increases numbers of migrants and the average size of local populations, which together can substantially decrease the rate of local extinction (rescue effect) and increase the rate of colonization (establishment effect). Coupling with local dynamics influences metapopulation dynamics both qualitatively and quantitatively. For some parameters, multiple equilibria may exist for habitat occupancy, with an unstable equilibrium at low habitat occupancy (a type of Allee effect at the metapopulation level) as suggested by Hanski and Gyllenberg. Decreasing local extinction rate and increasing colonization rate with increasing habitat occupancy both increase the mean time to metapopulation extinction. Large underestimates in metapopulation persistence times can result from neglecting rescue and establishment effects.

I analyse the importance and influence of noise with positive autocorrelation ('red noise') to extinction risk in spatially structured populations using a set of simple population dynamical simulation models. Noise colour has a major influence on both local and global extinction risk, level of synchrony in population dynamics, and spatial patterns of the synchrony. Generally, the stronger the autocorrelation in the noise is, the higher is the global extinction risk - even though local extinction risk may decrease. Populations with high intrinsic growth rate are more prone to global extinction than populations with low growth rate. The influence of autocorrelated noise on population synchrony depends strongly on the way in which environmental noise is introduced into the model.

For a spatial population assemblage, extinction risk should be greatly affected by features of local population dynamics and interpatch migration patterns. In a variable environment, the magnitude of environmental correlation between local population patches may have great impact on local dynamics and thereby global extinction risk. We examined the effect of correlated environmental variation on global extinction risk in a coupled lattice model consisting of local populations governed by density dependent population growth and density independent interpatch migration. We let each local population experience a stochastic environment expressed as a variation in maximum birth rate and let this environmental variation be correlated among local populations. We simulated global population growth under different magnitudes of environmental variability, correlation of environmental variability, emigration rate and migration survival, in order to evaluate the magnitude of their effect on local population dynamics and global extinction risk. The risk of global extinction increases with increasing magnitude of environmental correlation and environmental variability. The major determinant of global extinction risk is the balance between local population variability and the synchrony in local population fluctuations. A low rate of successful interpatch migration connects the local populations to each other, exposing them to less extinction risk than when they are isolated. High levels of interpatch migration are often negative for population persistence. The reason for this is that increased migration survival causes an increased risk of population crashes, due to overcompensatory population growth. This effect is amplified by a high emigration rate. Thus, local dynamics are affected by temporal and spatial variability in birth rates as well as interpatch migration levels. An assemblage of local populations in a variable environment will suffer least risk of global extinction when environmental correlation is low and interpatch migration is moderate.

Based on data from radio-collared individuals, we present an analysis of the viability of two small populations of the Scandinavian brown bear, Ursus arctos. The northern and southern populations had different demographic characteristics, even though the population growth rate r and the demographic variance sd2 were high in both populations (r = 0.13 and sd2=0.180 in the north, and r = 0.15 and sd2=0.155 in the south). In the northern population the environmental variance se2 was not significantly different from 0, whereas in the south se2=0.003. In the south, this was related to high environmental stochasticity in the survival rate of the youngest animals, which resulted in an increase in survival with age in this population. In contrast, in the north, the probability of survival showed a slight decrease with age. Uncertainties were obtained from the joint distribution of bootstrap replications of r, sd2 and se2. Although the uncertainty in these estimates is quite large, it is unlikely that even relatively small populations (> 10 females ≥ 1 year old) will decline to size less than 1 after 100 years. Analysis of the distribution of the critical population size (i.e. the population size where the population's logarithmic growth rate is zero) shows that these brown bear populations must be larger than 3-4 females 1 year or older to secure a positive growth rate. Similarly, if we define a viable population as the population size where the chance of survival is greater than 90% during a period of 100 years, 8 females ≥ 1 year old must be present in the north and 6 females in the south. This high viability of even small brown bear populations is due to high reproductive and survival rates. A relatively small increase in the mortality rate will strongly reduce the viability of even relatively large brown bear populations.

This paper explores the correspondence between the parameters of an extinction model analysed by Lande, Foley and Middleton et al. and the parameters of the incidence function model of metapopulation dynamics. The parameters of the extinction model, the intrinsic rate of population increase (r), its variance (v) and the population ceiling (K), can be mapped to the parameters of the incidence function model describing the scaling of the probability of local extinction (E) with patch area (A), E=e/Ax, via the equations s = x and r=eDs/s, where s = 2r/v, D is population density and K = DA. I explore this correspondence with two empirical examples, a mainland- island metapopulation of the European common shrew (Sorex araneus) on islands in lakes and a classical metapopulation of the American pika (Ochotona princeps). The most robust result is the correspondence x = 2r/v, which value decreases with increasing strength of environmental stochasticity. Thus the impact of environmental stochasticity on population dynamics can, in principle, be inferred from the pattern of habitat patch occupancy in a metapopulation.

Here we review the use of stochastic models in population biology. We provide precise definitions of demographic and environmental variance, and present a new concept of demographic covariance. We suggest that population fluctuations in many cases can be approximated by a diffusion process. Many interesting properties of the process can be expressed by the Green function, including the expected time to extinction which is of great interest in population viability analysis. We then apply this approach on density-dependent models without age structure and give a summary of some results obtained for age-structured models without density regulation. We suggest that these results may be very useful in practical management because they also give us the possibility of estimating the uncertainties in our estimates of for instance the probability of extinction of small populations. Their application will, however, depend on the availability of long-term data sets, where we can estimate the order of magnitude of essential parameters and evaluate their predictive power.

Many natural resource stocks have collapsed under exploitation. There is a need for management methods that are less prone to failure. Here I compare various management strategies for stocks with complicated stochastic dynamics. The complications include \"critical depensation\" where the stock tends to collapse if brought below a critical size, catastrophes where the stock suddenly decreases in size, and heavy-tailed distribution of fluctuations in net reproduction. The results indicate that these complications are all influential in determining the probability of early collapse and in determining the expected sum of discounted harvests of the stock. A strategy involving abrupt changes in harvest size in response to fluctuations in the stock abundance produces a lower probability of early collapse and a higher expected sum of discounted harvests than other strategies.