Connecting host physiology to host resistance in the conifer-bark beetle system

Abstract

Host defenses can generate Allee effects in pathogen populations when the ability of the pathogen to overwhelm the defense system is density-dependent. The host–pathogen interaction between conifer hosts and bark beetles is a good example of such a system. If the density of attacking beetles on a host tree is lower than a critical threshold, the host repels the attack and kills the beetles. If attack densities are above the threshold, then beetles kill the host tree and successfully reproduce. While the threshold has been found to correlate strongly with host growth, an explicit link between host physiology and host defense has not been established. In this article, we revisit published models for conifer-bark beetle interactions and demonstrate that the stability of the steady states is not consistent with empirical observations. Based on these results, we develop a new model that explicitly describes host damage caused by the pathogen and use the physiological characteristics of the host to relate host growth to defense. We parameterize the model for mountain pine beetles and compare model predictions with independent data on the threshold for successful attack. The agreement between model prediction and the observed threshold suggests the new model is an effective description of the host–pathogen interaction. As a result of the link between the host–pathogen interaction and the emergent Allee effect, our model can be used to better understand how the characteristics of different bark beetle and host species influence host–pathogen dynamics in this system.

Appendices

Appendix A

In the absence of recruitment, the beetle–host model presented by Stenseth (1989) is

$$ \frac{dA}{dt}=-b_{\!1}AR $$

(27)

$$ \frac{dR}{dt}=b_{\!2}-b_{\!3}A-(b_{\!4}-b_{\!5}A)R $$

(28)

where A(t) is the density of attacking beetles per tree and R(t) is the volume of resin per beetle gallery. Introducing the following dimensionless variables

$$ \tilde{A}=A \frac{b_{\!5}}{b_{\!4}}, \: \tilde{R}=R \frac{b_{\!5}}{b_{\!3}}, \: \tilde{t}=t b_{\!4} $$

and the dimensionless parameters

$$ \alpha=\frac{b_{\!1} b_{\!3}}{b_{\!4} b_{\!5}} , \: \beta=\frac{b_{\!5} b_{\!2}}{b_{\!4} b_{\!3}} $$

we can write the dimensionless version of the modified model as (after dropping the tildes)

$$ \frac{dA}{dt}=-\alpha A R $$

(29)

$$ \frac{dR}{dt}=\beta-A-R+AR $$

(30)

The model given by Eqs. 29 and 30 has two steady states. The first is (A ^*,R ^*) = (0,β), which is a steady state where the tree is alive and all attacking beetles are dead. The second is (A ^*,R ^*) = (β,0), which is a steady state where the beetles have successfully killed the host tree. The Jacobian of Eqs. 29 and 30 is given by

$$ \mbox{\bf{J}}_{(A^*,R^*)}= \left( \begin{array}{cc} -\alpha R^* & -\alpha A^* \\ R^*-1 & A^*-1 \end{array} \right) $$

(31)

At the steady state (A ^*,R ^*) = (0,β), both eigenvalues are negative, which means that the tree alive steady state is stable. At the steady state (A ^*,R ^*) = (β,0), one eigenvalue is negative and one is positive. Thus, the tree dead steady state is an unstable saddle.

Appendix B

Stability analysis of the Powell et al. (1996) model in the absence of beetle recruitment (Eqs. 9 and 10). The Jacobian is given by

$$ \mbox{\bf{J}}_{(H^*,R^*)}= \left( \begin{array}{ccc} -R^* && -H^* \\ -R^* && \beta-2\beta R^*-H^* \end{array} \right) $$

(32)

At the steady state (H ^*,R ^*) = (0,1), both eigenvalues are negative and the live host steady state is stable. At the steady state $(H^*,R^*)=(\bar{H},0)$, one of the eigenvalues is zero, which means that the eigenvalues are insufficient for characterizing stability. To investigate stability at this steady state, we study the nonlinear perturbation equations. Let H(t) = H ^* + h(t) and R(t) = R ^* + r(t), where h(t) and r(t) are small perturbations around the steady state (H ^*,R ^*). The perturbation equations from the steady state $(H^*,R^*)=(\bar{H},0)$ are

$$ \frac{dh}{dt}=-r(\bar{H}+h) $$

(33)

$$ \frac{dr}{dt}=r(\beta-\bar{H}-h)-\beta r^2 $$

(34)

We begin by considering the relationship between r and h at the dr/dh = 0 isocline (denoted by $\hat{r}$ and $\hat{h}$).

$$ \hat{r}=1-\frac{\bar{H}+\hat{h}}{\beta} $$

(35)

Because r ≥ 0 and dh/dt ≤ 0, the phase-plane trajectories always decrease along h. For values of h greater than the isocline, dr/dt is negative, and for values of h less than the isocline, the gradient dr/dt is positive. This can be seen by substituting the point $h=\hat{h}+\epsilon$ into Eq. 34, which yields

$$ \frac{dr}{dt}=-r\epsilon $$

(36)

If ε is negative, then r will increase, and if ε is positive, then r will decrease. The isocline given by Eq. 35 crosses the stable steady state of $\hat{r}=0$ at the point $(\hat{h},\hat{r})=(\beta-\bar{H},0)$. The critical trajectory can now be defined as the one that passes through the point $(\hat{h},\hat{r})=(\beta-\bar{H},0)$ because only trajectories with smaller r (or larger h) than this critical trajectory will be in the basin of attraction of r ^* = 0. The critical trajectory for the system given by Eqs. 9 and 10 can be solved analytically, which allows us to write the perturbation conditions exactly. By defining $\eta=\ln(\bar{H}+h)$ and $\mu=r \exp(-\beta \eta)$, we can rewrite Eqs. 33–34 as

$$ \frac{du}{d \eta}=e^{\eta (1-\beta)}-\beta e^{-\beta \eta} $$

(37)

If β ≠ 1, then the solution of Eq. 37 through the critical point $(\hat{h},\hat{r})=(\beta-\bar{H},0)$ is

$$ \hat{r}=\frac{1}{1-\beta}\left(1-\beta+\bar{H}+\hat{h}-\left(\frac{\bar{H}+\hat{h}}{\beta}\right)^\beta \right) $$

(38)

If β = 1, then the solution is

$$ \hat{r}=1+(\bar{H}+\hat{h})\left(\ln(\bar{H}+\hat{h})-1\right) $$

(39)

The stability criterion for $\bar{H}$ at the steady state $(H^*,R^*)=(\bar{H},0)$ can be determined numerically for arbitrary perturbations using Eqs. 38 and 39. As r →0 and h →0, the stability criterion is $\bar{H}=\beta$.

Appendix C

Stability analysis of the linear host–pathogen model given by Eqs. 16–18. The Jacobian is

$$ \mbox{\bf{J}}_{(A^*,S^*,R^*)}= \left( \begin{array}{ccc} -R^* & 0 & 0 \\ -\zeta S^* & -\zeta A^* & 0 \\ -R^* & \gamma (1-R^*) & -\gamma S^*-A^* \end{array} \right) $$

(40)

At each of the three steady states $(A^*,S^*,R^*)=(\bar{A},0,0)$, $(0,0,\bar{R})$, $(0,\bar{S},1)$, there is at least one zero eigenvalue, which means that a linear analysis around the steady state is not sufficient to assess stability. To determine stability of the steady states, we study the nonlinear perturbations through simulation. The full perturbation equations for all steady states are

$$ \frac{da}{dt}=-(A^*+a)(R^*+r) $$

(41)

$$ \frac{ds}{dt}=-\zeta (A^*+a)(S^*+s) $$

(42)

$$ \frac{dr}{dt}=\gamma (S^*+s)(1-R^*-r)-(A^*+a)(R^*+r) $$

(43)

where a = A − A ^*, s = S − S ^*, and r = R − R ^* are perturbations around the steady state (A ^*,S ^*,R ^*). The perturbations surrounding each steady state are obtained by setting $(A^*,S^*,R^*)=(\bar{A},0,0)$, $(0,0,\bar{R})$, or $(0,\bar{S},1)$. Because a and s can only decrease, stability for all steady states is assessed by whether or not r decays to zero. Unless otherwise noted, we explored the parameter space of ζ and γ from zero to 10¹⁰ (i.e., 0 ≤ ζ ≤ 10¹⁰ and 0 ≤ γ ≤ 10¹⁰).

Near the steady state $(A^*,S^*,R^*)=(0,\bar{S},1)$, r decays to zero from the initial conditions of $(a_o,s_o,r_o)=(10^{-8},10^{-8},-10^{-8})$ for all values of ζ explored, all values of $0 \leq \bar{S} \leq1$, and for values of γ > 0. Thus, we conclude that the steady state $(A^*,S^*,R^*)=(0,\bar{S},1)$ is stable.

Near the steady state $(A^*,S^*,R^*)=(0,0,\bar{R})$, r increases to $r^*=1-\bar{R}$ from the initial conditions of $(a_o,s_o,r_o)=(10^{-8},10^{-8},0)$ for all values of ζ explored, all values of $0.01 \leq \bar{R} \leq1$, and for values of γ > 0. Values of $\bar{R}<0.01$ yielded unreliable numerical simulations for small values of γ. Thus, for γ > 0 and $\bar{R} \geq 0.01$, the steady state $(A^*,S^*,R^*)=(0,0,\bar{R})$ is unstable.

The steady state $(A^*,S^*,R^*)=(\bar{A},0,0)$ is a little different from the others in that it is locally stable but not globally stable for $\bar{A}$ bounded away from zero. For a given set of parameters, a sufficiently small perturbation could be found such that r decayed to zero. Specifically, (a _o ,r _o ,s _o ) = (0,ε, ε), r decays to zero for ε sufficiently small, for 10^− 10 ≤ ζ ≤ 10¹⁰, 0 ≤ γ ≤ 10¹⁰, and $\bar{A} \geq 0.01$. We did not check 0 < ζ < 10^− 10, and values of $\bar{A}<0.01$ yielded unreliable numerical simulations. Thus, for slightly positive values of ζ and $\bar{A} \geq 0.01$, the steady state $(A^*,S^*,R^*)=(\bar{A},0,0)$ is stable.

Appendix D

• 1. Initial resin density, relative to R _m , is always low (e.g., Raffa and Smalley 1995). We assume R _o = 0.01R _m based on Wallin and Raffa (1999).

• 2. Raffa and Smalley (1995) report a maximum monoterpene concentration of 305 mg per gram of dried phloem. Assuming a resin density of 0.858 g ml^− 1 based on the largest component of resin α-pinene, a dried phloem density of 0.46 g cm^− 3 (Bouffier et al. 2003), and a monoterpene concentration in resin of 0.5, we estimate a maximum resin concentration of r _o  = 327 (l m^− 3).

• 3. Maximum resin volume can be estimated from R _m  = r _o x B, where x is phloem thickness and B is bark area. From Waring and Pitman (1985), the average bark area was B = 9.4 (m²). Assuming an average phloem thickness of x = 0.015 (m) (e.g. Zausen et al. 2005) gives an estimate of R _m  = 46.1 (l).

• 4. Raffa and Berryman (1983) report a 30% beetle mortality rate over a ~20-day period in host trees that are killed by mountain pine beetles. If we assume that resin volume was maximal (i.e., R _m ), then this gives a rough mortality rate estimate of h _o  = 0.0003869 (l^− 1 day^− 1).

• 5. From Raffa and Smalley (1995), we can get an estimate for the resin loss rate within the fungal/beetle activity zone (g _z ). Using an initial resin concentration of 250 mg per gram, and a final concentration of 210 mg per gram over a 15-day period, we estimate the loss rate of resin within the fungal zone as g _z  = 0.0116 (A^− 1 day^− 1). To convert this into a per-capita loss rate of resin over the entire tree from each attack, we use the sampled lesion size of 36 cm² from Raffa and Smalley (1995), and the average bark area of B = 9.4 (m²) from Waring and Pitman (1985), to estimate a resin loss rate of $g_o=4.4\times 10^{-6}$ (A^− 1 day^− 1).

• 6. The linear growth of the damaged area is roughly between 0.5 and 1 cm per day (Reid et al. 1967). Thus, we assume an area increment in the range of 0.196–0.785 cm² per day of damaged tissue. If we assume that sieve tube damage is best accounted for by the area of damage per area of bark, then, assuming an average bark area of B = 9.4 (m²) from Waring and Pitman (1985), the sieve tube damage rate is given by the range of $k_o=2.1\times 10^{-6}$ to $k_o=8.4\times 10^{-6}$.

• 7. Using the average DBH of 0.15 (m) from Waring and Pitman (1985), the sapwood area to DBH relationship from Bond-Lamberty et al. (2002), and the sapwood area to leaf area relationship from Callaway et al. (1994) for lodgepole pines, we estimate the average leaf area for the site as L = 20 (m²).

• 8. From Lavigne and Ryan (1997). Value used is averaged over locations and age classes and agrees well with the estimate for generic wood of m _w  = 0.25 (Penning de Vries 1975).

• 9. Czimezik et al. (2002).

• 10. From Gershenzon (1994), the metabolic cost of producing monoterpenes is 3.54 (g g^− 1) of glucose per monoterpene. Using the molar mass of monoterpenes (136.23 g Mol^− 1) and glucose (180.16 g Mol^− 1), the total carbon cost by mass is 3.2 g glucose per gram of resin. Converting this to a dimensionless proportion yields m _r  = 0.69 (g g^− 1).

• 11. From the molar mass of monoterpenes (136.23 g Mol^− 1), c _r  = 1.14 (g g^− 1).

• 12. Assuming a 180-day growing season.

• 13. Using a resin density of 0.858 g ml^− 1 for pinene, which is the most abundant component of resin.

• 14. We assume a typical value of x = 0.015 (m) (e.g., Zausen et al. 2005).

Appendix E

The model given by Eqs. 16–18 assumes that the time-scale of beetle aggregation to a host tree is sufficiently fast, relative to the time-scale of the attack dynamics, that the process of aggregation can be subsumed into the initial conditions of the model (i.e., A _o ). To assess the validity of this assumption, we can explicitly incorporate aggregation dynamics and compare this with the simplified model. The dimensional model with aggregation dynamics is given by

$$ \frac{dA}{dt}=A_o\Gamma(t,\alpha,\beta)-h_oAR $$

(44)

$$ \frac{dS}{dt}=-k_o AS $$

(45)

$$ \frac{dR}{dt}=f_o \left(1-\frac{R}{R_m}\right)S -g_oAR $$

(46)

where Γ(t,α,β) describes the proportion of the total attacking beetles (A _o ) that arrive at time t. To parameterize the aggregation distribution for an empirical example, we fit the distribution to the arrival data in Fig. 1 of Raffa and Berryman (1983). Fitting the function yields parameter estimates of α = {2.61, 5.21, 3.01} and β = {0.87, 0.47, 0.96} for the years 1977, 1978, and 1979 (Fig. 8). To demonstrate the impact of incorporating both time-scales, we use the mean parameter estimates of α = 3.61 and β = 0.77. Figure 9 shows predicted dynamics of the attack process under both models. The similarity of the dynamics demonstrates that the simplifying assumption of subsuming the aggregation process into an initial condition is a good approximation to the full model.

Fig. 8

Proportion of beetle attacks through time from Raffa and Berryman (1983). Circles are digitized data and lines are fit gamma distribution. Black shows dynamics in 1977, dark gray those in 1978, and light gray those in 1979

Fig. 9

Dynamics of the attack process for two levels of attack density (A _o  = 20 and A _o  = 50). Black dots show dynamics when aggregation is explicitly incorporated Eqs. 44–46, and gray lines show dynamics under the simplifying assumption that aggregation can be subsumed into an initial attack density A _o Eqs. 16–18. Circles denote initial conditions for both models, and arrows show the direction of time. Note that the gray lines overlay the black dots for much of the dynamics. Despite the differing initial conditions, time trajectories of the simplified model approach that of the full model with explicit aggregation, which suggests that the simplified model is a good approximation to the asymptotic dynamics of the full model well. All other parameter values are given in Table 2