Theoretical Ecology

, Volume 1, Issue 3, pp 163–177

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

Authors

    • Department of Biological SciencesUniversity of Alberta
    • Department of BiologyQueen’s University
  • Mark A. Lewis
    • Department of Mathematical and Statistical SciencesUniversity of Alberta
Original Paper

DOI: 10.1007/s12080-008-0017-1

Cite this article as:
Nelson, W.A. & Lewis, M.A. Theor Ecol (2008) 1: 163. doi:10.1007/s12080-008-0017-1
  • 107 Views

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.

Keywords

Host–pathogen modelsAttack thresholdAllee effectBark beetlesResin defensesMountain pine beetlesCarbon budget model

Introduction

Hosts can generate Allee effects (Allee 1931) in a pathogen population when host defense depends on pathogen densities (Courchamp et al. 1999; Ogden et al. 2002; Uma Devi and Uma Maheswara Rao 2006). Bark beetles are a classic example of a pathogen that suffers an Allee effect from host resistance (Berryman 1979). Bark beetles are a common and destructive pathogen of pine forests (Logan and Powell 2001) that must kill host tissue to ensure successful survival and reproduction (Christiansen et al. 1987; Raffa et al. 2005). As a result of host defenses, many individual beetles are required to successfully overwhelm and kill a single host tree (e.g., Christiansen et al. 1987; Fig. 1). Most species of bark beetles have evolved a pheromone communication system that helps aggregate the beetle population during flight (Raffa 2001). If insufficient beetles are available to attack a host, then host defenses kill the attacking beetles. If the attack density is above a critical threshold, then attacking beetles kill the host tree and successfully reproduce. Because beetle survival is tightly coupled to host defense, the threshold for successful attack on an individual tree produces a population-level Allee effect in bark beetles (Berryman and Stenseth 1989). Owing to the close host–pathogen interaction between conifer trees and bark beetles, as well as the economic impact of this pest species, a good deal of empirical and theoretical research has been carried out.
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig1_HTML.gif
Fig. 1

Empirical threshold for successful attack of mountain pine beetles as a function of host vigor (data from Waring and Pitman 1985). Vigor is defined as the mass of wood added to a host tree per year divided by the leaf area. Each symbol represents an individual tree. Solid circles indicate trees killed by beetles, open circles are living trees that resisted beetle attack, and gray circles are living trees where a portion of the bark area was killed by beetles. The gray line is the empirically estimated threshold for host mortality

A number of models have been developed to describe the host–pathogen interaction in bark beetles (Berryman et al. 1989; Stenseth 1989; Powell et al. 1996). However, these models blend two distinct processes: the host–pathogen interaction within the tree and the immigration of new beetles to the host tree. Biologically, these processes can be separated because they occur over different time scales. Beetle immigration is mediated by aggregation pheromones that are generated by beetles in the process of attacking a host tree. These pheromones can attract additional beetles to the host tree from those flying in the local vicinity (Raffa 2001). Such pheromone-mediated immigration occurs at the start of an attack and lasts less than a week (e.g., Raffa and Berryman 1983). In contrast, host death from high attack densities occurs between 4 and 6 weeks after the attack has started (Kirisits and Offenthaler 2002). The difference is time scales suggest that the host–pathogen interaction operates at a slower rate than beetle migration. This is also supported by empirical observations of the attack threshold (e.g., Fig. 1), that are well described by the beetle attack density after all migration has occurred. Thus, from the perspective of the host–pathogen interaction, the effect of pheromone-mediated immigration can be thought of as influencing initial attack density.

In this article, we analyze existing bark beetle models in the absence of recruitment from flying beetles to study the core host–pathogen interaction. Our results reveal that these modified models predict unrealistic steady state characteristics, suggesting that current descriptions of the beetle–host interaction need further development. Based on these results, we develop a new model of the beetle–host interaction that explicitly describes host damage and demonstrate that it has biologically realistic steady state characteristics. Using what is known about tree physiology, we then derive a link between host growth and resin production under the assumption that growth in carbohydrate is limited. In doing so, we provide the first model to integrate physiologically based host resin defense with process-based attacking beetle dynamics. We parameterize the model for mountain pine beetles using literature data and compare model predictions against independent empirical data on the attack threshold in mountain pine beetles. The success of our new model at predicting the empirical threshold suggests that it reflects a more accurate description of the beetle–host interaction that can be used to understand how the characteristics of the host–pathogen interaction influence beetle population dynamics.

Reanalysis of published models

To begin, we reanalyze previous models of the host–pathogen interaction in the absence of recruitment from flying beetles.

Berryman et al. (1989) and Stenseth (1989)

One of the earlier models of the host–pathogen interaction was developed by Berryman et al. (1989). A similar model is presented in Stenseth (1989), which is analyzed in Appendix A. Berryman et al. (1989) developed a model of the beetle–host interaction that explicitly describes the processes of beetle attack, host resin defense, and immigration from flying beetles. The model is given by
$$ \frac{dA}{dt}=\overbrace{b_{\!1}(b_{\!2}-b_{\!3}R)AR}^{\rm{beetle\mbox{ }recruitment}}-\overbrace{b_{\!4}AR}^{\rm{mortality}} $$
(1)
$$ \frac{dR}{dt}= \overbrace{b_{\!5}(1-b_{\!6}A)}^{\rm{resin\mbox{ }production}}-\overbrace{b_{\!7}R}^{\rm{resin\mbox{ }loss}} $$
(2)
where A(t) is the density of attacking beetles per tree and R(t) is the volume of resin per beetle gallery. The first term of Eq. 1 is recruitment from flying beetles due to host attractiveness, where b1 is the size of the local flying beetle population and parameters b2 and b3 reflect the attractiveness/repellency of the as a function of resin volume. The second term is beetle mortality, where b4 is the rate at which per capita mortality increases with resin volume. Equation 2 describes resin dynamics in the host, where b5 is the maximum rate of resin production of the host, b6 is the decrease in resin production caused by attacking beetles, and b7 is the resin loss rate through the attack holes.
Because we are interested in studying beetle–host dynamics in the absence of immigration from flying beetles, we remove the recruitment term from Eq. 1. Introducing the following dimensionless variables
$$ \tilde{A}=A b_{\!6}, \: \tilde{R}=R \frac{b_{\!7}}{b_{\!5}}, \: \tilde{t}=t b_{\!7} $$
and the dimensionless parameter
$$ \alpha=\frac{b_{\!4} b_{\!5}}{{b_{\!7}}^2} $$
we can write a dimensionless version of the modified model as (after dropping the tildes)
$$ \frac{dA}{dt}=-\alpha A R $$
(3)
$$ \frac{dR}{dt}=1-A-R $$
(4)
The model given by Eqs. 3 and 4 has two steady states. The first is (A*,R*) = (0,1), which is a steady state where the tree is alive and all attacking beetles are dead. The second is (A*,R*) = (1,0), which is a steady state where the beetles have successfully killed the host tree. The Jacobian of Eqs. 3 and 4 is given by
$$ \mbox{\bf{J}}_{(A^*,R^*)}= \left( \begin{array}{cc} -\alpha R^* & -\alpha A^* \\ -1 & -1 \end{array} \right) $$
(5)
At the steady state (A*,R*) = (0,1), both eigenvalues are negative, which means that the tree-alive steady state is stable. At the steady state (A*,R*) = (1,0), one eigenvalue is negative and one is positive. Thus, the tree-dead steady state is an unstable saddle. An unstable tree-dead steady state means that beetles do not kill host trees in the absence of continual recruitment from flying beetles, regardless of the initial density of attacking beetles. However, because continual beetle immigration is not a requirement for host death in nature, one or more of the key biological interactions is missing from the model of Berryman et al. (1989). Figure 2 shows the isoclines, steady states, and example phase plane trajectories.
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig2_HTML.gif
Fig. 2

Example trajectories for the model of Berryman et al. (1989) in the absence of recruitment from flying beetles. Dashed lines show the attacking beetle and resin isoclines. Gray circles depict the steady states; the solid circle is a stable steady state and the open circle is unstable. The black lines are example trajectories for α = 0.5, and solid black circles denote initial conditions

Powell et al. (1996)

Powell et al. (1996) developed a mechanistic model of beetle dispersal and attack that describes the dynamics of flying beetles, attacking beetles, host resin, and pheromones. The model is spatially explicit, and it is described by a set of six partial differential equations. To extract the beetle–host interaction in the absence of flying beetle recruitment, we assume that the density of attacking beetles changes only as a result of mortality from host defenses. With this assumption, the beetle–host interaction is described by the following set of equations
$$ \frac{dA}{dt}=-p_1AR $$
(6)
$$ \frac{dH}{dt}=-p_5HR $$
(7)
$$ \frac{dR}{dt}=p_2(p_3-R)R-p_4HR $$
(8)
where A(t) is the density of beetles attacking a host, H(t) is the number of holes in a host produced by the attacking beetles, and R(t) is the abundance of resin in a host. From Eq. 6, the attacking beetle density declines from resin-caused mortality, where p1 is the rate at which the per capita beetle mortality rate increases with resin. Resin is lost through the holes bored by attacking beetles, where p5 is the rate at which the hole-filling rate increases with resin abundance. If we assume that each attacking beetle bores a single hole, then H(0) = Ho = Ao. Resin production is described by the first term of Eq. 8, where p2 is the maximum per capita rate of resin renewal and p3 is the maximum amount of resin in the absence of attacking beetles. Powell et al. (1996) consider that host vigor is described by p3; the maximum capacity of the host to hold resin. However, as we show below, p3 drops out in the process of nondimensionalization. As an alternative, we consider p2 a measure of host vigor because it reflects the ability of the host to produce resin and does not drop out in the process of nondimensionalization.
Because the resin dynamics given by Eq. 8 are independent of attacking beetle density, the model dynamics can be understood by studying the coupled Eqs. 7 and 8. We introduce the following dimensionless variables
$$ \tilde{H}=\frac{H}{p_3}\frac{p_4}{p_5}, \: \tilde{R}=R \frac{1}{p_3}, \: \tilde{t}=t p_3 p_5 $$
and dimensionless parameter
$$ \beta=\frac{p_2}{p_5} $$
to write the modified dimensionless model as (after dropping the tildes)
$$ \frac{dH}{dt}=-HR $$
(9)
$$ \frac{dR}{dt}=\beta(1-R)R-HR $$
(10)
The model given by Eqs. 910 has two steady states. The first is (H*,R*) = (0,1), which is a steady state where the tree is alive and the attacking beetles are dead. The second steady state is \((H^*,R^*)=(\bar{H},0)\), where \(\bar{H}\) can be any value between zero and the initial number of holes (i.e., \(0 \leq \bar{H} \leq H_o\)). More precisely, \(\bar{H}\) is an infinite set of steady states that all satisfy Eqs. 9 and 10 at equilibrium (Fig. 3). Biologically, the infinite set of steady states reflects a situation where the host has been killed and a variable number of holes and attacking beetles remain alive. The number of holes remaining depends on the outcome of the dynamical process and, thus, on the initial number attacking beetles and the parameter β. For brevity, we refer to the infinite set of steady states simply as a steady state in the text below. The stability analysis is presented in Appendix B. Similar to the modified Berryman et al. (1989) model, the tree-alive steady state given by (H*,R*) = (0,1) is stable. The tree-dead steady state is also stable if \(\bar{H}>\beta\), but it is unstable if \(\bar{H}<\beta\). As a result, the host–pathogen interaction at the core of the Powell et al. (1996) model does not have the characteristics observed in nature. Figure 3 shows the isoclines, steady states, and example phase plane trajectories for the model given by Eqs. 10 and 9.
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig3_HTML.gif
Fig. 3

Phase plane diagram for the beetle–host model of Powell et al. (1996) in the absence of recruitment from flying beetles. Dashed lines show the attacking hole and resin isoclines. The gray symbols are steady states; the circle represents a point steady state and the thick line represents an infinite set of steady states. Solid symbols are stable steady states and open symbols are unstable. The transition between unstable and stable regions of the dead host steady state (thick gray line) occurs when \(\bar{H}=\beta\). The black lines show example trajectories for β = 0.5, and the separatrix is shown in bold

A new model for the host–pathogen interaction

Empirical data on the threshold for bark beetle survival (e.g., Mulock and Christiansen 1986; Fig. 1) suggest that the process of aggregation, whereby flying beetles are attracted to new hosts by the pheromones emitted from attacking beetles, does not have a direct influence on the beetle–host interaction beyond determining the initial attack density. Our reanalysis of the published models by Berryman et al. (1989), Stenseth (1989), and Powell et al. (1996) reveals that, when flying beetle recruitment is removed from the beetle–host interaction, the dead-host steady state becomes unstable for some parameter values. Biologically, this implies that trees cannot be killed without continuous beetle immigration, which is not what occurs in nature. In the next section, we develop a new model of the beetle–host interaction that resolves this problem by explicitly including host damage caused by attacking beetles. As above, we assume that the process of beetle aggregation can be subsumed into the initial attack density (see Appendix E for validation of this assumption). We begin by developing a general model that describes the beetle–host interaction with undefined functions and study the steady states. In the next section, we consider a linear form of the general model and study the characteristics and stability of the emergent threshold for bark beetle survival.

General model

Resin production by a host tree is considered the primary mechanism of defense against bark beetles (Christiansen et al. 1987; Franceschi et al. 2005). Prior to an attack, the phloem tissue contains some amounts of preformed resin, which is referred to as the constitutive defense. At the start of an attack, resin is produced in the phloem tissue around the attack site (referred to as the induced defense) (Christiansen et al. 1987; Lewinsohn et al. 1991; Raffa and Smalley 1995). Induced resin is derived from carbohydrates in the phloem tissue surrounding the attack site (Trapp and Croteau 2001). Because carbohydrates are produced by photosynthesis in the leaves, both the production and the transport of carbohydrates to the attack site become determinants of resin production (Christiansen et al. 1987). This process of induced defense is well supported by empirical evidence, which demonstrates that interfering with the host’s ability to supply or transport carbohydrates has a large influence on resin production and the ability of the host to defend against bark beetle attack (Lombardero et al. 2000; Wallin and Raffa 2001; Miller and Berryman 1986). Phloem tissue is composed of densely packed sieve tubes and is the main transport path for carbohydrates from the leaves to the attack sites. As a result, attacking beetles can interfere with host defense by either diluting resin production among numerous attack sites or by damaging phloem tissue and thereby reducing carbohydrate transport.

These empirical observations suggest that a dynamic model of both constitutive and induced defenses requires three components: resin volume, attacking beetles, and the undamaged phloem tissue necessary for carbohydrate transport. Similar to the previously published models, the model proposed here includes the ability of the host to produce resin. The key modification is an explicit representation of damage to the carbohydrate transport pathway. In general, such a model can be described by
$$ \frac{dA}{dt}=-A h(R) $$
(11)
$$ \frac{dS}{dt}=-S k(A) $$
(12)
$$ \frac{dR}{dt}=S f(R)-R g(A) $$
(13)
where A(t) is the number of attacking beetles still alive in the tree, S(t) is the number of undamaged sieve tubes in the tree, and R(t) is the resin volume of the host tree. The function f(R) describes the rate of resin production per undamaged sieve tube. As such, it is a phenomenalistic representation of carbohydrate production, carbohydrate transport, and conversion of carbohydrates to resin.
Carbohydrate transport through sieve tubes is governed by osmoregulatory flow (Thompson and Holbrook 2003), which suggest that resin production should be at a maximum fm when no resin is present (i.e., f(0) = fm) and decrease with increasing resin volume (i.e., f(R) < 0) until it reaches zero at the maximum resin capacity Rm (i.e., f(Rm) = 0). The function g(A) describes resin metabolism by beetles, which we assume is a monotonically increasing function of beetle density (i.e., g(A) > 0) that goes to zero when there are no attacking beetles still alive (i.e., g(0) = 0). Attacking beetle dynamics are given by Eq. 11, where the function h(R) describes beetle mortality due to resin. Empirical data suggest that greater amounts of resin cause greater mortality rates in beetles (e.g., Raffa and Smalley 1995), so we constrain the function h(R) to be a monotonically increasing function of resin abundance (i.e., h(R) > 0) with the constraint that h(0) = 0. We model the sieve tube damage caused by attacking beetles (including any associated fungus) as a per capita sieve tube loss rate k(A) that increases monotonically with the number of attacking beetles still alive (i.e., k(A) > 0), and with the constraint that k(0) = 0. The host is dead when all the sieve tubes are damaged (i.e., S = 0). The definitions and constraints for the general model are summarized in Table 1.
Table 1

Symbol definitions and constraints for the general model

Symbol

Definition

Constraints

A

Number of attacking beetles per tree

A ≥ 0

S

Number of sieve tubes per tree

S ≥ 0

R

Resin volume per tree

R ≥ 0

t

Time index

Rm

Maximum resin volume per tree

Rm ≥ 0

fm

Maximum resin production rate

fm ≥ 0

f(R)

Per capita resin production function

f(0) = fm, f(Rm) = 0, f(R) < 0

g(A)

Per capita resin loss rate

g(0) = 0, g(A) > 0

h(R)

Per capita beetle mortality function

h(0) = 0, h(R) > 0

k(A)

Per capita sieve tube damage rate

k(0) = 0, k(A) > 0

Steady state solutions

The constraints described above are sufficient to determine steady states of the general model, without the need to specify functions. The model given by Eqs. 1113 has three steady states: (A*,S*,R*) = \((\bar{A},0,0)\), \((0,0,\bar{R})\), and \((0,\bar{S},R_m)\). \(\bar{A}\) is the final number of attacking beetles and can be any value between zero and the initial number of attacks (i.e., \(0 \leq \bar{A} \leq A_o\)). Similarly, \(\bar{R}\) is the final resin volume between zero and the initial resin volume (i.e., \(0 \leq \bar{R} \leq R_o\)), and \(\bar{S}\) is the final number of sieve tubes between zero and the initial number (i.e., \(0 \leq \bar{S} \leq S_o\)).

Biologically, the steady states are qualitatively similar to those predicted by the previously published models. At the steady state \((A^*,S^*,R^*)=(0,\bar{S},R_m)\), all attacking beetles are dead and the host tree is at maximum resin capacity with some amount of undamaged sieve tubes remaining. The proportion of damaged sieve tubes indicates the amount of permanent damage caused by the unsuccessful beetle attacks (i.e., \(1-\bar{S}/S_o\)). At the steady state \((A^*,S^*,R^*)=(\bar{A},0,0)\), the attacking beetles have damaged all sieve tubes and killed the host tree. The last steady state is given by \((A^*,S^*,R^*)=(0,0,\bar{R})\), which reflects a situation where the attacking beetles damaged all of the host sieve tubes and the remaining resin was sufficient to kill the beetles, with the end result that both the beetles and host are dead.

Linear model

We consider a model with linear functions. Let \(f(R)=f_o\left(1-\frac{R}{R_m}\right)\), g(A) = goA, h(R) = hoR, and k(A) = koA. Parameter fo is the maximum resin production rate per sieve tube, Rm is the maximum resin volume, go is the rate at which the resin loss rate changes with the number of attacking beetles, ho is the rate at which beetle mortality changes with resin volume, and ko is the rate at which the sieve tube loss rate changes with the number of attacking beetles. Introducing the following dimensionless variables
$$ \tilde{A}=\frac{A g_o}{h_oR_m}, \: \tilde{S}=\frac{S}{S_o}, \: \tilde{R}=\frac{R}{R_m}, \: \tilde{t}=t h_oR_m $$
(14)
and dimensionless parameters
$$ \gamma=\frac{S_o f_o}{R_m^2 h_o}, \: \zeta=\frac{k_o}{g_o} $$
(15)
allows us to write the following dimensionless model (after dropping the tildes)
$$ \frac{dA}{dt}=-AR $$
(16)
$$\frac{dS}{dt}=-\zeta AS $$
(17)
$$ \frac{dR}{dt}=\gamma S (1-R)-AR $$
(18)

The nondimensional steady states are \((\bar{A},0,0)\), \((0,0,\bar{R})\), and \((0,\bar{S},1)\), where \(0 \leq \bar{A} <\infty\), \(0 \leq \bar{R} \leq 1\), and \(0 \leq \bar{S} \leq 1\). The initial conditions are (Ao,1,Ro), where 0 ≤ Ro ≤ 1 and 0 ≤ Ao ≤ ∞. The two steady states of biological interest are the live beetle – dead host state \((A^*,S^*,R^*)=(\bar{A},0,0)\) – and the dead beetle—live host state \((A^*,S^*,R^*)=(0,\bar{S},1)\). Both of these steady states are stable for γ and \(\bar{A}\) slightly greater than zero (Appendix C). Thus, and in contrast to our reanalysis of published models, the host–pathogen model developed here has steady states that are consistent with empirical observations.

Dynamics of the linear model given by Eqs. 1618 are governed by two parameters γ and ζ. Biologically, γ is the ratio of maximum resin production to maximum beetle mortality and ζ is the ratio of the phloem damage rate to the resin removal rate. To understand the magnitude of initial attack density required to kill a host tree, and how this is influenced by the parameters, we study the threshold behavior using numerical simulations of the model trajectories. Figure 4 shows example trajectories of the linear model. At high initial beetle abundance, beetles quickly kill the host tree. As the number of initial beetles decreases, the host is still killed but the number of beetles remaining also decreases. Below the critical initial beetle abundance Aoc, the host repels the attack and all beetles are killed. Further decreases in the initial beetle abundance have the effect of reducing the level of damage sustained by the host before the beetles are killed. The critical initial beetle abundance Aoc is defined as the value of Ao that results in (A*,S*,R*) = (0,0,0) and represents the threshold for beetle survival. Because there is no analytical solution to Eqs. 1618, we determined Aoc numerically. When host trees survive attacks, the resulting damage and resin densities depend on the initial density of attacking beetles (Fig. 5). Even if attack densities are lower than the critical threshold, the host tree can still suffer substantial damage.
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig4_HTML.gif
Fig. 4

Example trajectories for the linear model given by Eqs. 1618. Each line shows a trajectory starting from different initial beetle densities Ao. The dynamics are shown for γ = 8 and ζ = 1, and arrows show the direction of time. a Trajectories in the full 3D phase space. Circles are initial conditions, and the bold lines are steady states. Solid lines are stable steady states, and the dashed line is the unstable steady state. b Projection of the 3D trajectories onto the R-A plane. The gray line shows the trajectory emerging from the critical initial beetle abundance Aoc, above which beetle attacks kill the host tree and below which the host kills the attacking beetles

https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig5_HTML.gif
Fig. 5

Asymptotic host responses. Proportion of host damaged after the attack process ((1 − S*); black line) and the minimum resin abundance (gray line) for parameter values shown in Fig. 4. The critical attack density (\(A_o^c\)) is shown with the dashed line

The critical threshold Aoc depends on parameters γ and ζ. The empirical observations of Fig. 1 indicate that greater numbers of beetles are required to kill hosts with higher levels of vigor. For these data, vigor is defined as the volume of wood produced per year relative to the leaf area, which reflects growth efficiency of the host (Waring et al. 1980). The rationale for studying the influence of host vigor on the threshold for successful beetle attack is that trees able to produce greater amounts of wood have higher carbohydrate production and, thus, should also be able to produce greater amounts of resin if attacked. We develop this link quantitatively in the next section. Qualitatively, however, greater host vigor corresponds to greater resin production. In the linear model, this is the γ parameter. Figure 6 shows how the critical threshold changes as a function of γ.
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig6_HTML.gif
Fig. 6

Critical threshold of initial attacking beetle abundance Aoc as a function of host resin production for the linear model given by Eqs. 1618. Thresholds are shown for ζ = 0.5 (black line), ζ = 1 (dark gray line), and ζ = 2 (light gray line). The gray point shows the critical attack density (Aoc) for the gray trajectory of Fig. 4

The linear model predicts that the threshold Aoc should increase with increasing ability of the host to produce resin. Qualitatively, this agrees well with the empirical data (Fig. 1). In the following section, we link resin production to host vigor quantitatively and fully parameterize the model for mountain pine beetles using independent literature data. To assess the relevance of the mechanisms described by the model, we then compare the quantitative predictions from the linear model with the empirical threshold of Fig. 1.

Parameterization of the linear model

Linking host vigor to resin production

Carbohydrate production for a tree can be described by loL, where L is the leaf area of the tree and lo is the per-unit leaf area rate of carbohydrate production that includes the cumulative influence of environmental factors such as light levels and temperature (e.g., Zhang et al. 1994). Carbohydrates produced by the tree will eventually get used for respiration, growth, and resin defense. Tree respiration is partitioned into two components: maintenance respiration and growth respiration (e.g., Vanninen and Mäkelä 2005). Maintenance respiration is proportional to tree mass W, and growth respiration is proportional to carbohydrate production. Thus, the net amount of carbon available Ψ for growth and defence can be described by
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Figa_HTML.gif
(19)
where μ is the per-unit leaf area rate of carbohydrate production prior to discounting growth respiration, and σ is the per-unit mass maintenance costs.
Much research suggests that the net photosynthate is preferentially allocated to growth prior to damage (e.g., Lombardero et al. 2000; Turtola et al. 2003). If we assume that growth is limited by the availability of carbohydrates (rather than nutrients), that μ and σ are average yearly rates, and that L and W can be considered constant during the year (growth incremented between years), then the amount of wood mass added to a tree Gw in a given year is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Figb_HTML.gif
(20)
where mw is the per capita cost of producing wood and cw is the conversion from carbon mass into wood mass. Host vigor ν is often measured as the amount of mass added to a tree each year, divided by the leaf area (Waring et al. 1980). From Eq. 20, this can be expressed as
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Figc_HTML.gif
(21)
Once beetles begin attacking a host, we assume that all available carbohydrates are directed to resin production as required, which is known as the growth-differentiation hypothesis (Loomis 1932). This means that the maximum rate of resin production is governed by the maximum carbohydrate production. The total amount of resin produced per year Gr is then given by
https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Figd_HTML.gif
(22)
where mr is the per capita cost of producing resin and cr is the conversion from carbon mass into resin mass. In the linear model, the maximum rate of resin production is given by foSo, where fo is the per-sieve rate of resin production when resin abundance is zero and So is the initial number of sieve tubes. Thus, the maximum rate of resin production in the linear model foSo can be related to net carbohydrate production by
$$ f_oS_o=\frac{G_r t_r}{ \delta_r} $$
(23)
where tr is the conversion from years to growing days and δr is resin density. Combining Eqs. 2023, we can write the relationship between host vigor and the maximum rate of resin production as
$$ f_oS_o=\nu L \frac{(1-m_r)c_r t_r}{(1-m_w)c_w \delta_r} $$
(24)
The dimensionless attack density \(\tilde{A}\) and dimensionless parameter γ can be transformed to the empirical scales of Fig. 1 as follows. Let \(\hat{A}\) be the number of beetle attacks per unit bark area and ν be host vigor as presented in the data. Using the relationships given by Eqs. 14 and 24, we can relate the model predictions to the empirical data by
$$ \hat{A}= \tilde{A} \frac{h_or_ox}{g_o} $$
(25)
$$ \nu=\gamma \frac{{R_m}^2h_o}{L}\frac{(1-m_w)c_w \delta_r}{(1-m_r)c_rt_r} $$
(26)
The parameter ro is the maximum resin volume per volume of phloem tissue, which emerges from expressing the maximum resin volume as Rm = roxB, where x is phloem thickness and B is the tree bark area.

Parameterization

Many of the parameters in Eqs. 25 and 26 are well established in the literature, while others can be estimated within a range. To compare our model predictions with the empirical data shown in Fig. 1, we parameterize our model from studies that are independent of these data (Table 2). The only exception to this are the parameters L and B, which collectively describe the size of an average tree in the study. Because we obtain estimates or ranges for all model parameters, no model fitting is required and the contrast between model prediction and data is a quantitative and independent test of the model. Using the values from Table 2, we can write as \(\hat{A}=427 \tilde{A}\), ν = 24186 γ, and 0.47 ≤ ζ ≤ 1.89, where the range in ζ reflects the estimated range in the sieve tube loss rate ko. The predicted threshold from the parameterized linear model is shown in Fig. 7.
Table 2

Literature parameters for the linear model

Parameter

Description

Value

Source

Ro

Initial resin volume

0.01Rm (l)

1

ro

Maximum resin concentration

327 (l m − 3)

2

Rm

Maximum resin volume

46.1 (l)

3

ho

Beetle mortality rate per resin

0.0003869 (l − 1 day − 1)

4

go

Resin loss rate per beetle

4.44e-6 (A − 1 day − 1)

5

ko

Sieve tube loss rate per beetle

2.1e-6 – 8.4e-6 (A − 1 day − 1)

6

L

Leaf area

20 (m2)

7

mw

Wood production cost

0.305 (g g − 1)

8

cw

Wood mass per carbon mass

1.96 (g g − 1)

9

mr

Resin production cost

0.69 (g g − 1)

10

cr

Resin mass per carbon mass

1.14 (g g − 1)

11

tr

Time conversion to growing days

0.0056 (y day − 1)

12

δr

Resin density

854.7 (g l − 1)

13

x

Phloem thickness

0.015 (m)

14

Source details are given in Appendix D

https://static-content.springer.com/image/art%3A10.1007%2Fs12080-008-0017-1/MediaObjects/12080_2008_17_Fig7_HTML.gif
Fig. 7

Predicted threshold for successful attack of Mountain Pine Beetles as a function of host vigor (dashed lines). All rate parameters in the model are estimated independently using literature data. The range reflects the uncertainty in the sieve tube loss rate ko. The upper line is ζ = 0.47 and the lower line is ζ = 1.89. Empirical data are shown with circles as in Fig. 1

Discussion

The work of Berryman et al. (1989), Stenseth (1989), and Powell et al. (1996) provide the first dynamic models of the interaction between bark beetles and host trees that explicitly describes resin defenses. These models also include continuous recruitment of flying beetles to the population of beetles that are attacking each host. However, data on the threshold for successful attack in natural systems suggest that the dynamics of flying beetles are not involved in the beetle–host interaction, except to establish initial conditions (Christiansen et al. 1987; Mulock and Christiansen 1986, Fig. 1). To understand the core beetle–host interaction in these models, we reanalyzed the models of Berryman et al. (1989), Stenseth (1989), and Powell et al. (1996) in the absence of recruitment from flying beetles. Our results show that the modified models still predict a steady state with live beetles and a dead host, but the steady state is unstable. We therefore conclude that the stability observed in the original models is the result of continuous recruitment from the flying beetle population. While such recruitment is an important process, the beetle–host interaction in natural systems is stable at the scale of an individual tree in the absence of such recruitment, suggesting that the representation of the core beetle–host interaction needs further development.

One process absent from the models of Berryman et al. (1989), Stenseth (1989), and Powell et al. (1996) is an explicit representation of host damage from attacking beetles. The model developed here extends previous models to include host damage. We demonstrated that including explicit host damage produces the same qualitative steady states as previous models (i.e., living beetles with a dead host and dead beetles with a living host) but that both are now stable. Our new model also predicts that the steady states are infinite sets, which implies that the final density of attacking beetles in a dead host, and the final level of host damage, can take on a range of values that depends on the dynamics of the system and the initial abundance of resin and beetles. Biologically, there is good evidence to suggest that this is more realistic than a point steady state (e.g., Christiansen et al. 1987).

The steady states of our new model are in good qualitative agreement with empirical data. However, because all parameters are estimated independently from the literature, the model can be further validated by comparing the predicted threshold for successful beetle attack against data. Our analysis is the first work to quantitatively predict the successful attack threshold for bark beetles based on underlying biological mechanisms. Model predictions are superimposed over the empirical threshold in Fig. 7. The model underestimates the attack threshold over the range of uncertainty in the point estimate of parameter ko, but the upper predicted threshold is reasonably close to the observed threshold. Given the assumptions that are required to link host vigor to resin production, and the assumption that all trees have the same average characteristics (e.g., bark area, phloem thickness, and leaf area), the agreement between model prediction and data is encouraging and gives us some faith that the model reflects the biological processes in natural systems.

One key assumption of our model is that aggregation dynamics occur over sufficiently fast time scales that they can be subsumed into the initial conditions of the host–pathogen interaction. To assess the appropriateness of this assumption, we compared our simplified model with one where aggregation dynamics were modeled explicitly (Appendix E). Using aggregation dynamics obtained from an empirical study, the simplified model produces dynamics that are in excellent agreement with the full model (Fig. 9), which provides strong support that our assumption is appropriate. Moreover, this comparison indicates how studies of the bark-beetle system can benefit from separating ecological dynamics into two stages: the first stage considers the aggregation dynamics that map densities of emerging beetles to densities of attacking beetles on host trees, and the second stage considers the fate of the host–pathogen interaction given a set of attack densities.

Host resistance to bark beetles is described by the attack threshold. In our model, the threshold is determined by the parameters γ and ζ. The parameter γ is the maximum rate of resin production per resin volume divided by the maximum beetle mortality rate. Thus, hosts with greater ability to produce resin on a per capita scale are more resistant to beetle attacks. The model also suggests that host vigor may be refined further if it is calculated relative to the susceptibility of different beetle species to resin defenses. The second parameter ζ is the ratio of the sieve tube damage rate divided by the resin loss rate. Beetles that damage host trees faster reduce the threshold required to kill a host. Thus, the model suggests that beetle success is improved by increasing how quickly they can damage the host tissue and tolerate resin rather than by the ability to remove resin.

As a result of the economic impact of bark beetles, there is a wealth of empirical data that can be used to validate models of the host–pathogen interaction. The agreement between model prediction and the empirical threshold found here suggests that our new model captures aspects of the natural host–pathogen interaction that occur between aggregation events. However, a more rigorous test of any model is the ability to predict not only the final state abundances but also the time required to reach steady state. Empirical observations of successfully attacked host trees suggest that attacking beetles can colonize the entire bark area of a host over a time span of four to six weeks (e.g., Kirisits and Offenthaler 2002). Using the estimated parameters, the model predicts that the time for half of the bark area to be damaged is over 200 days, which is much longer than the colonization time observed in natural systems. Thus, while the model developed here captures the threshold for attack, there still remains scope for further development. Our present analysis focuses on model predictions using only point estimates of the parameters. As new empirical data become available for this system, more robust point estimates can be developed and the analysis can be expanded to incorporate the variance and covariance in parameter estimates. Furthermore, by developing resin production functions (Eq. 24) that include more detailed physiological processes, the model presented here can be expanded to consider situations where tree growth is limited by both carbohydrates and nutrients.

The general model developed here describes the fate of the beetle–host interaction through the processes of resinous defenses and host damage. For a wide class of functions, the model predicts that the steady states are infinite sets, which is more consistent with empirical observations than previous models. The linear version of the model does reasonably well at predicting the empirical threshold when parameterized to independent data. However, there is clearly room for improvement. In particular, we feel that a better understanding of the functions and parameters that describe the biological processes will be invaluable towards assessing the utility of the model proposed here. The advantage of establishing a single framework to investigate beetle–host systems is to compare across species that have markedly different success in attacking host trees. For example, are some species less successful because they have less tendency to aggregate, or because they are less able to damage a host? Because beetle survival is determined to a large extent by the ability to kill a host tree, the threshold for successful attack generates a strong Allee effect in bark beetles at the population scale. Thus, understanding how the characteristics of each beetle species determine the successful attack threshold will help us understand and predict the population dynamics of different bark beetle species.

Acknowledgements

We would like to thank Alex Potapov and Frank Hilker for independently solving the phase-plane trajectories used in Appendix B, and two anonymous reviewers who helped improve the manuscript. This study was funded by Natural Resources Canada–Canadian Forest Service under the Mountain Pine Beetle Initiative. Publication does not necessarily signify that the contents of this report reflect the views or policies of Natural Resources Canada–Canadian Forest Service. Additional support was provided by Natural Sciences and Engineering Research Council (NSERC) and Alberta Ingenuity Postdoctoral fellowships to WAN and NSERC Discovery grants and Canada Research Chairs to MAL.

Copyright information

© Springer Science+Business Media B.V. 2008