# Bird and mammal population dynamics

The life history models (LHM) of the species in Bird & Mammal Populations (BMP) are age-structured population dynamic models at population dynamic equilibrium with zero growth. By adding density regulation (γ parameter) and regulation by population dynamic feed-back selection (γ_{ι} parameter) to the birth rate, the life history models extend into selection-regulated population dynamic models.

Each species with a body mass estimate has a population dynamic model in BMP, and it may also have several population specific models. These models are obtained by maximum likelihood fitting to 3,847 time-series of abundance estimates. The time-series are among others obtained from the Living Planet Index, the North American Breeding Bird Survey, the PanEuropean Common Bird Monitoring Scheme, the Netwerk Ecologische Monitoring, Sovon, the Swiss Breeding Bird Index, the British Trust for Ornithology, the Danish Ornithological Society, and Svensk Fågeltaxering.

Population dynamic studies have traditionally assumed density-regulated growth. By allowing also for regulation by population dynamic feed-back selection, the population models of BMP cover a broader range of dynamics (Fig. 1).

An essential question is whether natural populations of birds and mammals are predominantly density-regulated (the γ parameter) or regulated by natural selection (the γ_{ι} parameter). Figure 2 shows the distribution of the importance of selection regulation relative to total regulation [i.e., γ_{ι}/(γ+γ_{ι})] across all the time-series where the best population dynamic model explains more than 50% of the variance in the data. With median estimates of 0.60 (se:0.01) for birds and 0.69 (se:0.02) for mammals, selection regulation is found to be more important than density regulation in most populations. While the distribution of estimates covers the range from almost pure selection regulation to almost pure density regulation, only 6% of the bird and 5% of the mammal populations have selection regulation below 10% of total regulation by density and selection. There is, in other words, no support for the traditional view that natural populations are exclusively density-regulated.

_{ι}/(γ+γ_{ι})] among time-series where the best population dynamic model explains more than 50% of the population dynamic variance of the data.

The widespread importance of selection regulation has some profound population dynamics implications. Where density-regulated growth returns monotonically to the carrying capacity with a damping ratio around unity (Fig. 1, left), selection-regulated population dynamics show damped to stable population cycles (Fig. 1, right), with damping ratios that decline to zero for stable cycles. Some populations may even have exploding cycles with negative damping ratios during smaller time periods, although time-series with negative damping ratio estimates may reflect uncertainty in our estimation of regulation.

The distributions of the estimated damping ratios are shown in Figure 3. With median damping ratios around 0.13 (se:0.01) and 0.08 (se:0.02) the general population dynamics of birds and mammals is best characterised as strongly cyclic. 85% of the bird populations, and 83% of the mammals, have damping ratios that are estimates to be smaller than 0.5. Strongly damped density-regulation-like growth with damping ratios above 0.9 is estimated for 5% of the bird populations, and 6% of mammals.

The period of the population cycles is best measured in time steps of generations, allowing for a comparison across species with different lifespans. The distributions of these cycle periods are shown in Figure 4. The estimated periods are nearly always above five generations. Although the distributions have long tails toward very long periods, they are highly peaked in the lower range with 52% of all birds, and 74% of all mammals, having periods below ten generation. Median estimates are 9.5 (se:60) generations for birds and 6.7 (se:160) for mammals, but the period is longer in populations with more damped dynamics. Thus, the median period increases from 7.8 (se:0.6) and 6.1 (se:1.5) generations for birds and mammals with stable dynamics (damping ratios around zero), to 38 (se:8) and 18 (se:4) generations for birds and mammals with damping ratios around 0.8.

## Estimation

The population dynamic models are fitted to the time-series of abundance estimates following the procedure in Witting (2021). This implies maximum likelihood estimates for the two population regulation parameters γ and γ_{ι} with the underlying age-structure being the LHM of the relevant species. Model selection by the Akaike (1973) information criterion allows for a linear trend in the equilibrium abundance, and in rare cases also for catastrophic mortality in a single year. A few time-series have long records of substantial harvest and shorter records of abundance estimates, and these are maximum likelihood fitted to the abundance data following the back-calculation method in Witting (2013).

Caution should be taken when interpretating population models with poor fits to abundance data. Yet, BMP lists all fitted models to show cases that are well-explained, as well as cases that are not. Only models that explain more than 50% of the variance in the population dynamic data are used to summaries population dynamic results and to extrapolate species level models. On average, 66% of the population dynamic variance is explained across all population dynamic models, while on average 80% of the variance is explained among models that explain at least 50%.

For species with a single time-series of abundance data and a model that explains at least 50% of the variance, the fitted population model is used also as a general population model for that species. For species with more than a single time-series and at least one model that explains 50% of the variance, the γ and γ_{ι} parameters at the species level are the explained variance weighted average across the population models of that species. The species level models of the remaining species are estimated by inter-specific extrapolation, based on an allometric correlation between the reproductive period and the maximum likelihood estimated regulation parameters γ and γ_{ι}.

Click here to read how selection regulation changes our interpretation of population dynamic time-series.

## References

- Akaike, H. 1973. Information theory as an extension of the maximum likelihood principle. pp. 267--281, In: B. N. Petrov and F. Csaki (eds.) Second International Symposium on Information Theory. Akademiai Kiado.
- Witting, L. 2013. Selection-delayed population dynamics in baleen whales and beyond. Population Ecology 55:377--401, https://dx.doi.org/10.1007/s10144--013--0370--9.
- Witting, L. 2021. Selection-regulated population dynamic in birds and mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/2021.11.27.470201.