Coupling Mountain Pine Beetle and Forest Population Dynamics Predicts Transient Outbreaks that are Likely to Increase in Number with Climate Change

Abstract

Mountain pine beetle (MPB) in Canada have spread well beyond their historical range. Accurate modelling of the long-term dynamics of MPB is critical for assessing the risk of further expansion and informing management strategies, particularly in the context of climate change and variable forest resilience. Most previous models have focused on capturing a single outbreak without tree replacement. While these models are useful for understanding MPB biology and outbreak dynamics, they cannot accurately model long-term forest dynamics. Past models that incorporate forest growth tend to simplify beetle dynamics. We present a new model that couples forest growth to MPB population dynamics and accurately captures key aspects of MPB biology, including a threshold for the number of beetles needed to overcome tree defenses and beetle aggregation that facilitates mass attacks. These mechanisms lead to a demographic Allee effect, which is known to be important in beetle population dynamics. We show that as forest resilience decreases, a fold bifurcation emerges and there is a stable fixed point with a non-zero MPB population. We derive conditions for the existence of this equilibrium. We then simulate biologically relevant scenarios and show that the beetle population approaches this equilibrium with transient boom and bust cycles with period related to the time of forest recovery. As forest resilience decreases, the Allee threshold also decreases. Thus, if host resilience decreases under climate change, for example under increased stress from drought, then the lower Allee threshold makes transient outbreaks more likely to occur in the future.

Appendices

Appendix A: Small k Approximation for F

For aggregation levels that are very high, $k\ll 1$, we can expand the survival function (9) around $k=0$ to obtain the following approximation

$$\begin{aligned} 1-F(m;k,\varphi ) \approx {\left\{ \begin{array}{ll} 0 &{} m < m_0 \\ - k \left( \log \left( \frac{k \varphi }{m}\right) + \gamma \right) &{} m>m_0, \end{array}\right. } \end{aligned}$$

(85)

where $\gamma$ is the Euler gamma number, and we can determine $m_0$ by solving $\log \left( \frac{k \varphi }{m}\right) + \gamma =0$ to obtain $m_0 = k \varphi e^\gamma$.

The analysis that follows is similar to that using the Hill function approximation, although note that in this case the equation for the fixed points (32) can be solved directly in terms of the Lambert W function. In other words, with this approximation, $m/c = 1-F(m;k, \varphi )$ can be solved for m to obtain

$$\begin{aligned} m = -c k W_{\{0,-1\}} \left( \frac{-\varphi }{c}e^\gamma \right) \end{aligned}$$

(86)

where the $W_0$ branch is the intermediate (unstable) solution and the $W_{-1}$ branch is the larger (predominately stable) solution. As $\varphi /c \rightarrow e^{-\gamma -1}$, the two solutions converge to a single unique value $m^*_k=c k$, as $W_0(1/e)=W_{-1}(1/e)=-1$. Additionally, for $\varphi /c \le e^{-\gamma -1}\approx 0.205$ there is no solution (because W(x) does not have real solutions for $x<-1/e$). This means that the threshold must be less than about 20% of the number of beetles emerging per tree for the existence of the outbreak equilibrium. The stability analysis of these fixed points follows similarly.

Fig. 9

The spectral radius of the upper branch near the bifurcation point with $\kappa =5$. The vertical lines indicate the values of $\theta$ chosen for the dynamical simulation in Fig. 11. The subplots show different ranges of $\theta$, where the second subplot is very close to the critical point. Note that the two vertical lines in the second subplot overlap in the first subplot

Fig. 10

The real and imaginary parts of the dominant eigenvalue of the upper branch near the bifurcation point with $\kappa =5$. The subplots show different ranges of $\theta$, where the second subplot is very close to the critical point

Fig. 11

The mean number of beetles from dynamical simulations near the bifurcation point with $\kappa =5$. We set the initial conditions to the critical values for all variables. The value of $\theta$ is given in each subplot and is also indicated in Fig. 9. The horizontal line is the critical value of $m=m^*=(\kappa -1)/\kappa$

Appendix B: Stability Near the Bifurcation Point

Analytically, using the Hill function approximation, we find that there is a fold bifurcation at $m=m^*$ (44), and that the upper branch of the solution very near the bifurcation point must be stable. However, in Fig. 4 the upper branch appears unstable numerically near the critical $m^*$. Here we show numerically that very near the bifurcation point the upper branch is indeed stable, but that it very quickly loses stability before regaining it at a slightly smaller value of $\theta$. The dynamical simulations near equilibrium for values of $\theta$ in this regime show that the mean number of beetles m oscillates rapidly before eventually stabilizing or losing stability. However, even in the case where stability is lost, the simulations often show positive numbers of beetles over times much longer than those relevant biologically.

Figure 9 shows the spectral radius of the upper branch very near the bifurcation point. This is similar to Fig. 5 in the main text, but over a much smaller range of $\theta$ and only for the upper branch. The second subplot shows that the spectral radius is less than one only very near the critical point, before quickly increasing to greater than one. We additionally plot the real and imaginary parts of the dominant eigenvalue separately in Fig. 10, which are both positive over this range of $\theta$. As neither are zero when the spectral radius crosses one, we suspect the system may undergo consecutive Neimark-Sacker bifurcations wherein the oscillations around the fixed points either spiral towards or away from the fixed point.

Fig. 12

The time to extinction for the beetle population with small variations in the initial mean number of beetles at $\theta =0.60242$. The vertical line shows the critical point $m^*$. Note that when the initial mean number of beetles is small, the time to extinction is very short, between 6 and 13 years for the range considered here

We simulate the system for four values of $\theta$ to test what the dynamics of the system look like very near the bifurcation point in Fig. 11. We set the first of these values of $\theta$ to be near enough to the critical point to be in the stable region (halfway between where the spectral radius crosses one and the critical value of $\theta ^*$). We set the next values of $\theta$ to be at the maximum value of the spectral radius, halfway between the values of $\theta$ where the spectral radius crosses one, and very near the final point where the spectral radius crosses one. These values of $\theta$ are shown as vertical lines in Fig. 9. We set the initial values of all parameters and variables to their critical values and simulate forward in time for thousands of years. Note that unlike the simulations in the main text, there are no additional immigrating beetles in this case as we start the simulation close to equilibrium.

The first simulation very near the bifurcation point does indeed settle to the stable mean number of beetles in a damped oscillation. For the value of $\theta$ where the spectral radius is at its maximum, the number of beetles remains non-zero for around 200 time steps (years) before collapsing. For the next two values of $\theta$, even though the spectral radius is greater than 1, the mean number of beetles oscillates consistently for thousands of years. Thus, even though this fixed point is technically unstable, it is nearly stable for biologically relevant time scales.

For the second largest value of $\theta$, we additionally test how this time to extinction depends on the initial mean number of beetles and plot the results in Fig. 12. We find that small variations in the initial conditions can lead to thousands of time steps difference for the time to extinction. However, as long as there are enough beetles initially, the time to extinction is long enough that the beetle population is effectively stable for biologically relevant time scales.