###### Modeling Density Dependence

Using (the wrong type of) density dependence

Not using density dependence

Bias in Rmax estimation

Underestimating maximum growth rate (Rmax)

Overestimating maximum growth rate (Rmax)

Incorrectly modeling impact under density dependence

Toxicity vs. harvest under density dependence

Using (the wrong type of) density dependence

Density dependence can have important effects on extinction risks (e.g., see Ginzburg et al. 1990), therefore it is necessary to use caution in selecting the type of density dependence, and specifying its parameters. For example, some types of density dependence assume that there is an equilibrium population abundance, K, and if the population size is below K, the population will have an average growth rate above 1 (will tend to grow). If this is not the case, in other words, if the population is subject to "systemic" pressure (Shaffer 1981) and declining deterministically, then assuming density-dependent regulation will cause an underestimation of the risks of decline and extinction.

Thus, if the population has a long-term declining trend, or the long-term average of survival rates and fecundities give a finite rate of population change (eigenvalue) of less than 1, then it may not be appropriate to use some types of density dependence (e.g., Scramble, Contest, Ricker, Beverton-Holt, logistic, etc.), depending on how exactly they are modeled and parameterized (however, there are some exceptions to this; see below). In such cases, a density-independent ("exponential") model, a Ceiling type of density dependence, or a user-defined density dependence model may be more appropriate.

Not using density dependence

Many species are threatened by habitat loss, which is most commonly (and perhaps most naturally) modeled as a decline in the carrying capacity of the population. Thus, models that do not include such a habitat-related parameter (e.g., most density-independent models) may not be appropriate for modeling threats such as habitat loss.

(Note that simulating habitat loss may require modifying Rmax or the stage matrix, in addition to the carrying capacity).

As discussed above, an observed population decline may indicate that the population is not under density-dependent regulation. However, there are some exceptions to this. In the following situations, using a density-dependent model may be appropriate, even though the population may be declining:

1. The population may be above the carrying capacity temporarily (for example due to recent environmental fluctuations or excess migration), and declining towards it. If the population was above the carrying capacity because the carrying capacity temporarily declined due to recent environmental fluctuations, then you should also use "Standard deviation of K" parameter to model stochasticity.

2. The population may be subject to Allee effects, and declining even though population is below the carrying capacity. In this case, you should specify a density-dependent model that includes Allee effects, or use a high extinction threshold.

3. The survival rates and fecundities (the stage matrix) may have been estimated during years at which the population happened to be declining due to environmental factors, and you have reason to believe that in the long run the population is stabilized by density-dependent factors.

4. The population may be declining because of habitat loss, or decreased habitat quality, and you have reason to believe that if habitat loss stopped, the population would be stabilized by density-dependent factors. In this case, you should use the "Temporal change in K" parameter to model habitat loss.

Conversely, if a population is observed to be increasing exponentially, this does not mean that a density-independent model is the most appropriate. The population may approach or reach its carrying capacity in the (future) simulated time horizon, even if currently it is far below the carrying capacity.

Not using density dependence may cause an underestimation of risks of decline or extinction (or an overestimation of risks of spread of an invasive species), if the population is under the influence of Allee effects ("positive density dependence"). Often there are not sufficient data at low population densities, so it is difficult to model Allee effects. In such cases, focusing on the risk of decline to a sufficiently high threshold abundance may be appropriate.

Bias in Rmax estimation

Fitting a density dependent model to a time series data (count of total population size over time) may introduce a bias. Suppose you have a time series of population size estimates, N(1), N(2), N(3), etc.; and for each time step t, you calculate the growth rate as R(t) = N(t+1) / N(t).

To estimate density dependence, you may want to perform a regression of R(t) [or r(t)=log(R(t))] on N(t), and use the y-intercept as the estimate of Rmax [or rmax], assuming it is a declining function.

However, note that N(t) appears both in the dependent and in the independent variable of the regression! This implies that if N(t) are measured with error (as they are in most cases), or are subject to other (stochastic) factors, then the estimate of the slope of the regression and, hence the estimate of Rmax, will be biased. In particular, if the data are from density-independent population fluctuations, you will often get an estimate of Rmax above 1.0, meaning that you will detect density dependence even though it does not exist.

For example, the left figure below shows a time series produced by a density-independent, age-structured model with environmental stochasticity in survival rates and fecundities. The figure on the right shows an attempt to fit a simple density dependence model to this data. The line shows linear regression of population growth rate, ln(R), on population size, N. The population growth rate is calculated as ln(R(t)) = ln( N(t+1) / N(t)). Even though the model that produced the data did not include density dependence, the negative slope of the regression line indicates density dependence, with an intercept of about 0.24, which corresponds to Rmax of about 1.3.

Detecting and parameterizing density dependence is a complicated problem. For a discussion of complexities inherent in detecting and estimating density dependence, see Ryu et al. (2016), Langton et al. (2002), Lande et al. (2002), Saether et al. (2002), as well as Hassell (1986), Hassell et al. (1989), Solow (1990), and Walters (1985).

Time series of population size (N), produced by a density independent, age-structured, stochastic model.

Fitting a density-dependent model to the time series on the left. The negative slope indicates density dependence, even though the time series was created by a density-independent model.

Underestimating maximum growth rate (Rmax)

Rmax is defined as the maximal growth rate in the absence of density effects, namely at low population sizes. Thus, the observed growth rates at higher population sizes, especially the population growth rate when the population is approaching its equilibrium (carrying capacity) often underestimates Rmax. For example, if the stage matrix is estimated from a population approaching carrying capacity, and the eigenvalue of the matrix is used as Rmax, then the risks of decline and extinction may be overestimated.

A similar mistake occurs when using Ceiling-type density dependence (which does not use Rmax). Ceiling-type density dependence assumes that the vital rates (the stage matrix) are not affected by the effects oif density until the population reaches the Ceiling level (K). If the vital rates are actually affected by density before the population reaches K, and the stage matrix is estimated from a population close to K, then this type of density dependence may overestimate the risks of extinction and decline.

Overestimating maximum growth rate (Rmax)

Although Rmax is defined as the maximal growth rate in the absence of density effects, merely observing a growth rate above 1.0 at low population densities does not justify using scramble- or contest-type density dependence (incl. logistic, Ricker, Beverton-Holt, theta logistic, etc.). You either have to fit an entire dataset as discussed above, or you need other evidence which shows that the population is regulated by these types of density dependence. This is because the observed growth rates are affected by factors other than density, such as stochasticity. Populations that are not regulated by scramble- or contest-type density dependence (for example those that are only subject to ceiling-type density dependence) will also frequently experience periods of positive growth, some of which will coincide with low population sizes. If you model such a population with a scramble- or contest-type density dependence, you may underestimate extinction risks, because of the stabilizing effect of these functions (see Ginzburg et al. 1990).

Similar cautions apply to the estimation of Rmax in cases where the use of scramble- or contest-type density dependence is justified. Simply using the maximum growth rate ever observed (or observed under laboratory conditions, etc.) may overestimate the strength of density dependence, because this observed value might have been the result of factors other than the low density of the population or the lack of competition. If you have census (count data), consider using the statistical methods referenced above.

Another way of overestimating Rmax is calculating its value for a long time step based on measurements at a short time step, especially if the long time step is longer than the generation time (or, even if it is shorter than generation time, but longer than period between reproduction events). For example, suppose that an insect species is observed to increase by two-fold in a generation (thus, Rmax is 2.0 per generation), and assume that there are 3 generations per year. Now, suppose you build a model with an annual time step. If the population can grow two-fold in one generation, it can grow 8-fold in three generations, if there is no density dependence. However, if you add density dependence to this model, it would be wrong to estimate Rmax as 8.0, because the effects of density would be felt in the population at a generation time scale, before the population grows by 8-fold. Thus, in this case it would be wrong to have a density dependent model with an annual time step. The correct model in this case would have a time step of 1/3 years and an Rmax of 2.0.

Incorrectly modeling impact under density dependence

When developing models for impact assessment, the interactions between density dependence and the simulated impact must be carefully considered; and the parameters of the model that need to be modified to simulate the impact must be carefully selected.

One common mistake is to modify only the stage matrix elements (survival rates or fecundity) under Scramble or Contest type of density dependence. If density dependence type is Scramble or Contest, the program modifies the stage matrix elements during a simulation, according to the population size at each time step and the parameters of the density dependence function (see the help file and manual for details). Because of this, when density dependence type is Scramble or Contest, changing only the stage matrix elements to model pollution (or any other impact that affects vital rates) is not adequate (it would not produce any substantial difference in dynamics). The solution is to change both stage matrix elements and the density dependence function.

Thus, modeling impacts on survival and fecundity under density dependence means changing either or both the maximum growth rate (Rmax) and carrying capacity (K). Which one should be modified to simulate impact depends on the interaction between the impact and the density dependence relationship. Changing only Rmaxor only K may not adequately describe what happens to the density dependence relationship under impact (see Figure 1).

For example, changing only Rmax assumes that there is no impact on the population if the population size (N) is close to carrying capacity (K), and that the growth rate under impact is actually higher if N>K (Figure 1A). Changing only K assumes that there is no impact on the population if the population size is small (see Figure 1B).

A. Impact (red): Rmax = 1.3; Baseline (blue): Rmax = 1.5

B. Impact (red): K = 60; Baseline (blue): K=100

Figure 1: The effect of changing only Rmax (A, left) and only K (B, right) to simulate impact under density dependence. In both graphs, the blue curve shows the baseline model, and the red curve shows the impact model. The baseline model has Rmax = 1.5, K = 100.

While the assumptions above may be valid in some situations, simulating impact in many cases requires modifying both the maximum growth rate (Rmax) and carrying capacity (K), as shown in Figure 2. The impact may reduce the growth rate of the population by the same amount regardless of population size, making the density dependence functions of baseline and impact models parallel (Figure 2A); or the effect may be stronger at high (Figure 2B) or low (Figure 2C) population sizes.

B. Impact (red): Rmax = 1.3, K = 35

A. Impact (red): Rmax = 1.3, K = 55

C. Impact (red): Rmax = 1.1, K = 35

Figure 2: The effect of changing both Rmax and K to simulate impact under density dependence.

In all three graphs, the blue curve shows the baseline model, and the red curve shows the impact model. The baseline model has Rmax=1.5; K=100.

Toxicity vs. harvest under density dependence

Another mistake is to model mortality due to toxicity as harvest. These two types of mortality are not equivalent; there is a fundamental difference between them under density dependence. When a population regulated by density dependence is harvested, the remaining individuals may be better off than before the impact (harvest) because there may be more resources per individual. This is called compensation. The situation under toxicity is entirely different: the remaining individuals may be few, and there may be more resources per individual, but they are still poisoned. Most likely, they have higher probability of mortality, lower reproduction, and slower growth; in other words, they are probably not better off than before the impact (pollution).

In most cases, harvest by itself does not change the density dependence relationship, and compensation effects are represented by the density dependence function. Toxicity, however, most likely shifts the density dependence curve downward, as in Figure 2 above.

Note that there are limits to compensation even with harvest. Depending on its rate and timing, harvest can drive a population to extinction, even if the population is regulated by density dependence. Harvest may also interact with other factors. For example, Allee effects and stochasticity may cause a population reduced by harvest to decline even further or increase its risk of extinction.