Logistic-growth models measuring density feedback are sensitive to population declines, but not fluctuating carrying capacity

Abstract

Analysis of long-term trends in abundance of animal populations provides insights into population dynamics. Population growth rates are the emergent interplay of inter alia fertility, survival, and dispersal. However, the density feedbacks operating on some vital rates ("component feedback") can be decoupled from density feedbacks on population growth rates estimated using abundance time series ("ensemble feedback"). Many of the mechanisms responsible for this decoupling are poorly understood, thereby questioning the validity of using logistic-growth models versus vital rates to infer long-term population trends. To examine which conditions lead to decoupling, we simulated age-structured populations of long-lived vertebrates experiencing component density feedbacks on survival. We then quantified how imposed stochasticity in survival rates, density-independent mortality (catastrophes, harvest-like removal of individuals) and variation in carrying capacity modified the ensemble feedback in abundance time series simulated from age-structured populations. The statistical detection of ensemble density feedback from census data was largely unaffected by density-independent processes. Long-term population decline caused from density-independent mortality was the main mechanism decoupling the strength of component versus ensemble density feedbacks. Our study supports the use of simple logistic-growth models to capture long-term population trends, mediated by changes in population abundance, when survival rates are stochastic, carrying capacity fluctuates, and populations experience moderate catastrophic mortality over time.

Keywords: Australia; compensation; demographic rate; density dependence; megafauna; population size; stationarity; time series.

FIGURE 1

Scheme of the main elements of how density feedback operates in population dynamics (see Table 1 for a full glossary of terms indicated in italicized boldface). (i) a component density feedback can operate on survival probability (shown here as compensation ) where survival declines as population size increases. (ii) Another common component density feedback operates on fertility, where the number of offspring per female decreases with increasing population size. (iii) A time series of abundance estimates (“census data”) for a population captures an ensemble density feedback on the per capita rate of exponential population change (r) resulting from all component density feedbacks. In systems demonstrating stationarity , the underlying mechanisms (e.g., carrying capacity K) driving change in population size do not themselves shift over time. (iv) Plotting the rate of population change (r = N _t + 1/N _t) against population size (N _t) provides a way to measure the evidence for, and strength of, ensemble density feedback. In this representation, a Ricker logistic model estimates the linear slope between r and N _t (a negative slope here indicates compensation , but a positive slope would indicate depensation ). Where r = 0 intersects the linear Ricker logistic fit, the long‐term mean carrying capacity (K) can be estimated if not trending upward or downward. The black arrows indicate that, under compensatory dynamics, a population tends to grow towards K when r > 0 (i.e., low N) and to decline from K when r < 0 (i.e., high N). (v) In this example, the system is in a state of nonstationarity because the K is declining over time.

FIGURE 2

Strength of ensemble compensatory density feedback across demographic scenarios. Bootstrapped (10,000 uniform resamples between 95% confidence limits) across 21 test species (detailed in Table 2) of the strength of ensemble compensatory density feedback (Gompertz‐β) among scenarios (detailed in Table 3). Midpoints indicate means, and error bars are the interquartile ranges. Demographic scenarios include carrying capacity K fixed (K _fixed; Scenario ii), a pulse disturbance of 90% mortality at 20 generations (20G; Scenario iii), weakly declining ($\overline{r}$ ≅ −0.001; Scenario iv) and strongly declining ($\overline{r}$ ≅ −0.01; Scenario v) populations, K varying stochastically (K _stoch) around a constant mean with a constant variance (Scenario vi), K varying stochastically with a constant mean and increasing variance (K _stoch↑Var; Scenario vii), and K varying stochastically with a declining mean and a constant variance (↓K _stoch; Scenario viii).

FIGURE 3

Decoupling of ensemble and component density feedbacks in demographic scenarios with and without catastrophic mortality. Relationship between strength of ensemble (slope coefficient β of the Gompertz‐logistic model × [−1] in the time series) and component (1 – the modifier S _red on survival in the Leslie transition matrix) density feedback for: Scenario i (pink; stochastic mortality, no catastrophic mortality, stable K) and Scenario ii (grey: stochastic mortality, catastrophic mortality, stable K). Fitted curves across species are exponential plateau models of the form y = y _max − (y _max − y ₀)e^−kx . Shaded regions represent the 95% prediction intervals for each scenario. Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21  species; Table 2). Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker‐ and Gompertz‐logistic models across time series (ΣwAIC _c ‐density feedback) relative to the weights of two density‐independent models (random and exponential).

FIGURE 4

Decoupling of ensemble and component density feedbacks in demographic scenarios with catastrophic mortality and with catastrophic mortality + pulsed mortality and harvesting (see Figure 6). Relationship between strength of ensemble (slope coefficient β of the Gompertz‐logistic model × [−1]) and component (1 – the modifier S _red on survival) density feedback for: Scenario iii (green: pulse disturbance of 90% mortality at 20 generations); Scenario iv (red: weakly declining population at r ≅ −0.001); and Scenario iv (blue: strongly declining population at r ≅ −0.01). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species; Table 2). Fitted curves across species are exponential plateau models of the form y = y _max − (y _max − y ₀)e^−kx . Shaded regions represent the 95% prediction intervals for each scenario. Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker‐ and Gompertz‐logistic models across time series (ΣwAIC _c ‐density feedback) relative to the weights of two density‐independent models (random and exponential).

FIGURE 5

Decoupling of ensemble and component density feedbacks in demographic scenarios with catastrophic mortality and fluctuating carrying capacity. Relationship between strength of ensemble (slope coefficient β of the Gompertz‐logistic model × [−1]) and component (1 – the modifier S _red on survival) density feedback for: Scenario vi (purple: carrying capacity varying stochastically with a constant mean and an increasing variance); Scenario vii (green: carrying capacity varying stochastically with a constant mean and an increasing variance); and Scenario viii (red: carrying capacity K varying stochastically with a declining mean and a constant variance). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 test species, Table 2). Fitted curves across species are exponential plateau models of the form y = y _max − (y _max − y ₀)e^−kx . Shaded regions represent the 95% prediction intervals for each scenario. Also shown are the mean probabilities of median density feedback (Pr(density feedback): sum of the Akaike's information criterion weights for the Ricker‐ and Gompertz‐logistic models across time series (ΣwAIC _c ‐density feedback) relative to the weights of two density‐independent models (random and exponential).

FIGURE 6

Strength of ensemble density feedback and generation length for 21 vertebrate species for demographic scenarios with and without a component density feedback on mortality. Relationship between strength of ensemble (slope coefficient β × [−1] of the Gompertz‐logistic model) and generation length across the 21 species for: Scenario ii (black: with compensatory density feedback; see also Figure 2) and Scenario ix (grey: without compensatory density feedback). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 test species, Table 2). Probabilities of density feedback (Pr(density feedback) = sum of the Akaike's information criterion weights for the Ricker and Gompertz models relative to the weights of two density‐independent models (random and exponential)) calculated across simulations gave median Pr(density feedback) = 0.994 and 0.322 for the two stable scenarios with (Scenario ii) and without (Scenario ix) component feedback on survival, respectively.

FIGURE 7

Strength of ensemble density feedback in demographic scenarios with catastrophic mortality, catastrophic mortality with pulsed mortality, and two types of harvesting. Relationship between strength of ensemble density feedback (slope coefficient β × [−1] of the Gompertz‐logistic model) and the stationarity index ${\overline{T}}_{\mathrm{R}}/\mathrm{Var}\left(T_{\mathrm{R}}\right)$ across 21 test species over 40 generations for four demographic scenarios: (a) Scenario ii: carrying capacity (K) fixed, (b) Scenario iii: a pulse disturbance of 90% mortality at 20 generations, (c) Scenario iv: weakly declining population at r ≅ −0.001, and (d) Scenario v: strongly declining population at r ≅ −0.01. Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species, Table 2). Fitted curves across species exponential plateau models of the form y = y _max − (y _max − y ₀)e^−kx . Shaded regions represent the 95% prediction intervals for each type. ρ _med are the median Spearman's ρ correlation coefficients for the relationship between the ensemble strength and stationarity index across species (resampled 10,000 times; see Figure S8 for full uncertainty range of ρ in each scenario).

FIGURE 8

Strength of ensemble density feedback in demographic scenarios with catastrophic mortality and fixed carrying capacity versus three types of fluctuating carrying capacity and no catastrophic mortality. Relationship between strength of ensemble density feedback (slope coefficient β × [−1] of the Gompertz‐logistic model) and the stationarity index ${\overline{T}}_{\mathrm{R}}/\mathrm{Var}\left(T_{\mathrm{R}}\right)$ across 21 test species over 40 generations for four demographic scenarios: (a) Scenario ii: carrying capacity (K) fixed, (b) Scenario vi: K varying stochastically (K _stoch) around a constant mean with a constant variance, (c) Scenario vii: K varying stochastically with a constant mean and increasing variance (K _stoch ↑Var), and (d) Scenario viii: K varying stochastically with a declining mean and a constant variance (↓K _stoch). Each scenario includes 21,000 simulated time series of abundance (10,000 for each of 21 species. Fitted curves across species exponential plateau models of the form y = y _max − (y _max − y ₀)e^−kx . Shaded regions represent the 95% prediction intervals for each type. ρ _med are the median Spearman's ρ correlation coefficients for the relationship between the ensemble strength and stationarity index across species (resampled 10,000 times; see Figure S8 for full uncertainty range under each scenario).