m.r.Life ι**=7/3ψ

Population dynamic causes

Biological indicators should reflect changes in the Earth’s living systems. Programs based on population dynamic indicator species, and indicator indices from multiple species, aim to understand how the environmental drivers of an expanding human population affect the ecosystem of the indicator species, or affect the biological diversity of the joint set of ecosystems covered by all the species in the multi-species indicator index. The most ambitious indicator index is probably the Living Planet Index, which attempts to measure the overall state of the world’s biological diversity based on thousands of population trends for vertebrate species from terrestrial, freshwater, and marine habitats.

The conceptual foundation of population dynamic indicators is density-regulated growth where, quite generally, i) populations decline only when their habitats deteriorate and equilibrium abundancies decline, and ii) stable populations start to increase when habitats improve, and equilibrium abundancies increase. The problem, however, is that density regulation is weaker than selection regulation in most populations (Witting, 2021).

Fig. 1 Three examples of selection-regulated dynamics with a stable (left), declining (middle), and increasing (right) equilibrium abundance, likely reflecting a stable, deteriorating, and improving habitat. Data from the Swiss Breeding Bird Index and the British Trust for Ornithology.

With selection regulation added the population dynamics is cyclic. This means, as illustrated in the left plot of Fig. 1, that populations decline about 50% of the time when the equilibrium abundance and habitat is stable. Even more detrimental to the concept of population dynamic indicators are examples like i) the middle plot where the equilibrium abundance declines but the population increases about 50% of the time, and ii) the right plot where the equilibrium increases but the population declines about 50% of the time.

Populations will eventually decline if the habitat continues to deteriorate, but at least for long lived species the indicator inconsistencies in Fig. 1 may persist for decades. Across the population dynamic models in Bird & Mammal Populations (BMP) that explain more than 50% of the variance in the data, 75% of the population dynamic declines are not associated with an estimated decline in the equilibrium abundance, and 76% of the population dynamic increases are not associated with an equilibrium increase. In fact, 24% of the population declines have increasing equilibria, and 29% of the population increases have declining equilibria. No population dynamic indicators are constructed by BMP because of this lack of a direct connection from population growth to the underlying environment.

While we may continue to use population dynamic indicators as red flags for species and habitats that require conservation attention, BMP provides population dynamic models that allow for a broader understanding of the interactions between populations, their habitats, resources, and history.

History is not important for density-regulated growth, where the current environment and density define the growth rate. But it is essential in selection-regulated dynamics where initial conditions are just as important for current growth as the density dependent environment: The selection in BMP models changes the birth rate by altering a competitive quality parameter between cohorts of parents and offspring. The initial growth of a population at the population dynamic equilibrium abundance is zero only for an initial quality of unity. Populations at equilibrium abundancies will instead increase if their quality is less than one and decline if quality is greater than one.

North American birds

The BMP models for North American birds provide an example of model-based population dynamic insight. By analysing the dynamics over the past 50 years, Witting (2021) found that most waterfowl and raptors were increasing across the continent, with more populations having increasing than declining equilibria. Yet, the growth rates of most waterfowl were decelerating while those of most raptors were accelerating. This means that waterfowl were more often above than below their population dynamic equilibria, while raptors were more often below. Several waterfowl that are currently increasing may thus decline soon, while most raptors should continue to increase for a while.

The dynamics of shorebirds differed from that of waterfowl and raptors. Shorebirds were mainly increasing in the middle of the continent and declining in areas to the west and east. Yet, their equilibrium abundancies were more often declining than increasing over the whole continent, indicating a possible deterioration or fragmentation of shorebird habitats. And with most of the population dynamic growth rates decelerating, shorebirds were more often above than below the equilibrium. This indicates that we may expect a future decline even in the central areas where shorebirds have mainly increased up till now.

The BMP population dynamic simulator

All population models in BMP are selection-regulated models that are run by a population dynamic simulator in the user-browser. For the simple case with non-overlapping generations, the selection-regulated model reduces to the following form

nt+1 = nt [nt / n*] / qtwithqt = qt-1 [nt-1 / n*]γι

where n is population density, n* the equilibrium density, q the competitive quality, λ=1/q the growth rate, γ regulation by density, and γι regulation by population dynamic feed-back selection. More generally, the model incorporates the age-structure of the life history, with regulation on annual reproduction.

You can use the BMP simulator to construct your own simulations by altering initial conditions, adding a trend in the equilibrium abundance, including catastrophic mortality in a chose year, adding random variation to the birth rate, or altering the length of the simulation. The only parameters that are fixed at their best estimates are the density and selection regulation parameters γ and γι, and the life history of the species that determines the age-structure of the dynamics.

You run and control the appearance of each simulation by four buttons:

plots the natural selection induced acceleration/deceleration of the population dynamic growth rate r = log λ.

All population specific models are determined by model selection allowing for changes in the population dynamic equilibrium. These changes will often reflect a fragmentation or expansion of the original habitat, with each BMP simulation having a habitat button that shows the estimated density trajectory in the original habitat in addition to the trajectory of the whole population [assuming that changes in the overall equilibrium occurs from an expansion or fragmentation of suitable habitats].

simulates from chosen parameters and initial conditions.

resets the simulation.

Click here to continue to the population dynamic simulator with species and population specific models.


  • Witting, L. 2021. Selection-regulated population dynamic in birds and mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/2021.11.27.470201.