Positive and negative density-dependence and boom-bust dynamics in enemy-victim populations: a mountain pine beetle case study

Abstract

Negative density-dependent population regulation in exploitative species is well studied. Positive density-dependence can arise if exploiters must cooperate to obtain access to well-defended resources. Most studies, however, focus on the first type of density-dependence at the expense of the other. Using a parasitoid-host model, we explored how positive density-dependence driven by host defenses in combination with negative density-dependence due to competition for resources impact transient population dynamics. Inspired by interactions between the mountain pine beetle and its pine hosts, we formulated a model of enemy-victim interactions in discrete-time in which the victim is capable of deadly self-defense against exploitation. We fitted the model to data and then analyzed its non-equilibrium dynamics to determine what conditions promote boom-bust dynamics. When present together, strong Allee effects and overcompensating competition for resources among exploiters can cause their populations to irrupt and then crash even though many exploitable resources remain. Accelerating population irruptions followed by precipitous collapse occur for realistic parameter values of our model of mountain pine beetle dynamics. Insect dynamics are often dominated by sudden irruptions and collapses on short time scales. Population crashes in exploitative species often happen enigmatically even when exploitable resources are not depleted. Herein, we argue that strong Allee effects in combination with overcompensation provide a plausible explanation for these boom-bust dynamics in some species.

Appendices

Appendix A: Negative binomial CDF

Here, we provide details that describe how the negative binomial cumulative distribution function can be written in terms of a regularized incomplete beta function (I(k,ϕ+1,a _t )). Pearson (1968) showed that Eq. 5 can be rewritten as

$$\begin{array}{@{}rcl@{}} F(\phi; a_{t}) &=& I(k, \phi + 1, q), \end{array} $$

(A.1a)

$$\begin{array}{@{}rcl@{}} q &=& k/(k + a_{t}), \end{array} $$

(A.1b)

where

$$ I(k, \phi + 1, q) = \frac{B(k,\phi+1, q)}{B(k,\phi+1,1)}. $$

(A.2)

The numerator in Eq. A.2 is the incomplete beta function

$$ B(k,\phi+1,q) = {{\int}_{0}^{q}}t^{k-1}(1-t)^{\phi}\text{dt}, $$

(A.3)

and the denominator is the beta function

$$ B(k,\phi+1,1) = {{\int}_{0}^{1}}t^{k-1}(1-t)^{\phi}\text{dt}. $$

(A.4)

Thus, the general host-parasitoid equation (3) can be rewritten as

$$\begin{array}{@{}rcl@{}} \text{P}_{t+1} &=& c\text{N}_{t}(1 - I(k, \phi + 1,q)), \end{array} $$

(A.5a)

$$\begin{array}{@{}rcl@{}} \text{N}_{t+1} &=& \lambda\text{N}_{t}I(k, \phi + 1,q), \end{array} $$

(A.5b)

$$\begin{array}{@{}rcl@{}} q &=& k/(k + a_{t}), \end{array} $$

(A.5c)

under the assumption that attacks are negative binomially distributed among hosts and an attack threshold of ϕ attacks must be exceeded for hosts to become exploitable.

Similarly, Eq 11 can be rewritten as

$$\begin{array}{@{}rcl@{}} \text{P}_{t+1} &=& c \alpha \text{N}_t \bigg(a_tz(1 + a_t/k(1-z))^{-(k+1)} - {\sum}_{i=0}^{\phi} if(i;a_t)z^i\bigg), \end{array} $$

(A.6a)

$$\begin{array}{@{}rcl@{}} \text{N}_{t+1} &=& (1-\alpha)\text{N}_t + \alpha\text{N}_tI(k, \phi + 1,q). \end{array} $$

(A.6b)

$$\begin{array}{@{}rcl@{}} q &=& k/(k + a_t), \end{array} $$

(A.6c)

$$\begin{array}{@{}rcl@{}} a_t &=& (1/\alpha)(2/3)\text{P}_t/\text{N}_t, \end{array} $$

(A.6d)

and the corresponding discrete time map in ratio state space (13) can be written

$$\begin{array}{@{}rcl@{}} \text{R}_{t+1} &=& \frac{ c \alpha (a_tz(1 + a_t/k(1-z))^{-(k+1)} - {\sum}_{i=0}^{\phi} if(i;a_t)z^i )}{(1-\alpha) + \alpha I(k, \phi + 1,q)}, \end{array} $$

(A.7a)

$$\begin{array}{@{}rcl@{}} q &=& k/(k + a_t), \end{array} $$

(A.7b)

$$\begin{array}{@{}rcl@{}} a_t &=& (1/\alpha)(2/3)\text{R}_t. \end{array} $$

(A.7c)

Although this notation makes the model slightly less readable, it makes it more amenable to fitting to data and simulation.

Appendix B: Data

Here we supply the data from (Klein et al. 1978) that we used to fit (11). We were unable to fit all five parameters (c, α, ϕ, k, and μ) in (11) at the same time with only five data-points and so we fixed the c parameter to 35 offspring (female)⁻¹, the estimate of the initial number of larvae produced per female with minimal competition (Goodsman et al. 2012). All other parameters were fitted to the (Klein et al. 1978) data.

The number of beetles present in study plots was estimated by (Klein et al. 1978) by counting emergence holes on successfully attacked trees and multiplying by the density of attacked trees. These data are expressed in terms of thousands of emergence holes (acre)⁻¹ in Fig. 4 of (Klein et al. 1978). Assuming that, on average, one beetle emerges from each emergence hole, we can estimate the number of beetles at large per acre in each year. These data were obtained from Fig. 4 (Klein et al. 1978) using WebPlotDigitizer software (http://arohatgi.info/WebPlotDigitizer/). Because the density of trees that were susceptible to mountain pine beetle attack per acre was also recorded ((Klein et al. 1978) Table 1), it was simple to compute the ratio (ratio in table below) of beetles (emerged in table below) to susceptible trees (hosts in table below). Note that the density of hosts per acre is recorded but the smallest diameter class (six inch diameter at breast height) listed in Table 1 of (Klein et al. 1978) was not included as six inch diameter at breast height trees were too small to be susceptible to mountain pine beetle attack.

Table 2 Data from Klein et al. (1978) that were used to fit (11)