**Methods**

Three main components are required for the cost-weighted distance output:

- The rate of spread
- Host density
- Climate suitability

**1. Rate of spread**

The rate of spread represents the ability for the MPB to disperse and succesfully establish in a new area. Conceptully, the expansion of the MPB is like a dispersing semi-circular wave (Fig. 3). Simply, the rate of spread can be estimated as the change in the radius of the semi-circle (Equation 1). Thus, a historical rate of spread can be averaged from the extent of their distribution. This average spread rate can be used to estimate the maximum distance that the MPB can travel given a certain amount of time.

**Fig. 3: Semi-circular radial expansion, corresponding to equation 1**

**Equation 1: Rate of spread, ***r* = radius, *A* = area

**2. Host density**

Not knowing specific host-parasite relationships, it can be assumed that the ability for a parasite to establish is non-linearly related to the density of the host. A rough estimation on the density of hosts (Pine trees) can be done, knowing the percent compostion of volume of vegetation in an area. Thus, the resistance of a cell to the spread of the MPB will be inversly related to the volume of pine trees (Equation 2, Fig. 4).

**Equation 2: Resistance, ***V*_{p} = pine tree volume, *b* = value at which V_{p} is approximately 0

**Fig. 4: Resistance vs. Pine Volume (m**^{3}/ha), with *b* unknown

**3. Climate suitability**

Climate exerts a primary control over the potential range of a species. In particular, the MPB is limited in its northern range by -40^{o}C temperatures. However, under global climate change, MPB populations may have higher over-winter survival rates, allowing for rapid expansion. For the final cost surface, knowing what regions will experience -40^{o}C temperatures will help predict what cells and regions the MPB will most not likely be able to establish. Thus, future predicated minimum winter temperatures will function as a constraint (Equation 3).

**Equation 3: Minimum temperature if function**

**Cost-weighted distance analysis:**

Overall, these three components will make up the inputs for the final cost-distance output. The average rate of spread will be used to calculate the the maximum distance that the MPB can travel given a certain amount of time. The resistance and climate suitability will create a cost surface. In which, each cell's cost is the resistance (volume of pines) multplied by the climate constraint (Equation 4).

**Equation 4: Cost function, ***C* = climate constraint