The Population/Host Model

 

Table: The list of parameters appearing in the global PDE model for MPB redistribution. Density units are represented with respect to hectares (hec), amounts of pheromone with respect to micrograms ($\mu$g=10^-6g), and numbers of MPB are counted in hundreds (HMPB). The basic time unit is the flight hour (fh), of which there are approximately five per day.

Parameter Definitions and Units
Parameter	Definition	Units
A0	Critical concentration at which pheromones become repulsive	$\mu$g hec-1
A3	Saturation parameter for pheromones	---
a1	Rate of pheromone production by nesting beetles	$\mu$g fh-1 HMPB-1
b1	Rate of pheromone diffusion	hec fh-1
$\beta$	Mortality rate of beetles due to resin outflow	R0-1
$\delta_1$	Loss rate of pheromone	fh-1
$\mu$	Diffusitivity of flying beetles due to random movement	hec fh-1
$\nu$	Strength of directed MPB motion due to pheromone gradients	hec2 $\mu$g-1 fh-1
R0	Rest resin capacity of a healthy tree (resin volume/stem area)	R0
r1	Rate of landing and conversion from flying to nesting beetles	fh-1
r2	Rate of resin replenishment	fh-1
r3	Rate of resin outflow through holes bored by beetles	fh-1
r4	Rate of resin crystallization (tree recovery)	R0-1
$\gamma(x,y,t)$	Emergence rate in time and space	HMPB hec-1 fh-1

 

In previous papers [24,45,46,5,4,16,25,23] we have developed and validated a spatial model for MPB dispersal and mass attack in pine forests. The state variables are:

P(x,y,t) -

population of flying MPB.

Q(x,y,t) -

population of nesting/eating MPB.

A(x,y,t) -

concentration of pheromone suite.

R(x,y,t) -

resin capacity (related to xylem thickness and surface area of tree).

H(x,y,t) -

number of entrance holes bored by attacking MPB.

The equations relating these state variables are presented here only briefly. An equation for density of nesting MPB,

$$\dot{Q} = r_1 \frac{R}{R_0}P -\beta r_3 R Q,$$ (3)

accounts for landings in proportion to unoccupied surface area $$\left(r_1 \frac{R}{R_0}P\right)$$ and `pitch-out' by trees $(-\beta r_3 R Q)$. The factor $$\frac{R}{R_0}$$ measures unoccupied surface area since the resin reservoir is distributed just under the bark and is depleted locally in the vicinity of MPB attack. Constituitive resin responses are described by

$$\dot{R} = \left[ r_2 (R_0-R) -r_3 H \right] R ,$$ (4)

with terms modeling induced resin response (r₂ R(R₀-R)) and depletion due to attack (-r₃ RH). The number of attack holes satisfies

$$\dot{H} = r_1 \frac{R}{R_0}P - r_4 r_3 H R ,$$ (5)

with terms ( - r₄ r₃ H R) describing host recovery from attack through resin recrystallization. While (1) and (3) appear very similar, the progress and success of attack depends sensitively on their competing rates; if H can be driven down more rapidly than Q overcomes tree defenses, then the tree will survive the attack. Thus H measures a tree's current stress, while Q measures success of MPB attacks. The dispersing population itself satisfies a chemotactic reaction-diffusion PDE,

$$\frac{\partial}{\partial t} P = - \nabla \cdot \left\{ \lef... ...right\} -\omega_1 P - r_1 \frac{R}{R_0}P + {\gamma} (x,y,t),$$ (6)

where A is the concentration of the pheromone suite and

$$f(A) = A_3 A_0 \left\{(A_3 + 1) \ln \left[ 1 + \frac{A}{A_3 A_0} \right] -\frac{A}{A_0 } \right\} .$$

This function accounts for MPB attraction to the pheromone plume in low concentrations, when most of the plume is composed of attractants. Later in the attack the suite is composed of a higher proportion of anti-aggregants, and this is modelled as a bias away from larger pheromone levels. The source term for dispersing MPB, $\gamma(x,y,t)$, describes the emergence of young adults from the previous season's successsful attacks. Thus equations (1-4) are purely an in-season dispersal model; reproductive dynamics for MPB are not included or relevant to this particular discussion. Parameter descriptions are given in Table 1; sizes for these parameters and units are given in Table 2.

 

Table: Parametric values for numerical simulation and units. The parameter $\hat{\gamma}$ is the spatially-averaged emergence rate of dispersing adults. Units involving resin are measured relative to R₀. Other units are: $\mu$g (10^-6 grams), hec (10⁴ square meters = hectares), fh (flight hours $\sim$ 5 fh/day). Parameter values are based on observational and anecdotal data discussed in Biesinger, 1998.

Parameter Values
Parameter	Value	Parameter	Value
A3	1	A0	7.8 $\mu$g hec-1
a1	$2\mu g\ \mbox{fh}^{-1}\ \mbox{HMPB}^{-1}$	b1	$0.324 \mbox{ hec}\ \mbox{fh}^{-1}$
$\beta$	$0.43\ R_0^{-1}$	$\delta_1$	$360 \mbox{ fh}^{-1}$
$\mu$	$1 \mbox{ hec}\ \mbox{fh}^{-1}$	$\nu$	5.7 hec2 $\mu$g-1 fh-1
R0	$1\ R_0$	r1	$0.16 \mbox{ fh}^{-1}$
r2	$0.0045 \mbox{ fh}^{-1} R_0^{-1}$	r3	$0.0023 \mbox{ fh}^{-1}$
r4	$0.0045\ R_0^{-1}$	$\hat{\gamma}$	.1 HMPB hec $^{-1}\ \mbox{fh}^{-1}$