# Visualizing Models for Spatial Spread of Invasive Species

This application creates dynamic visualizations of mathematical models for the spatial spread of invasive species in North America.

This material is based upon work supported by the U.S. Geological Survey under Grant/Cooperative Agreement No. G15AC00311.

# More Coming Soon

At some point, look here for tools to 'dig into' the biological and mathematical principles of this model.

# Mathematical Models of Spatial Spread among Invasive Species

Note: More details about the modeling will be coming soon...

## Reaction Diffusion Model with Harvesting

A reaction diffusion model considers a continuous spatial population to change in way that has both intrinsic growth and decay at each location, combined with dispersal between locations.

This application uses a basic type of reaction-diffusion model. Presuming there is a differentiable function $$P(x,y,t)$$ tracking the population at longitude $$x$$, latitude $$y$$ and time $$t$$, then the change in population at any given time can be modeled by: $$\frac{\partial P(x,y,t) }{\partial t}=R(x,y,t)+D(x,y,t)\left( \frac{\partial ^2 P(x,y,t)}{\partial x^2}+\frac{\partial ^2 P(x,y,t)}{\partial y^2}\right)$$

where $$R(x,y,t)$$ and $$D(x,y,t)$$ represent the intrinsic rate of change and the dispersal rate, respectively, at each location.

Specifically, this model assumes a logistic model for the intrinsic rate of change: $$R(x,y,t) = r(x,y,t) - d(x,y) P(x,y,t)\left(1-\frac{P(x,y,t)}{K}\right) - \gamma * P(x,y,t)$$

where $$\gamma$$ is the harvest rate and the intrinsic death rate is proportional to the complement of habitat suitability ($$r(x,y) \propto (1-h(x,y))$$).

Growth is modeled by an annual birth cycle proportional to habitat suitability: $$r(x,y,t) = \rho \cdot h(x,y) P(x,y,t) \mbox{ for } t=0,12,\dots$$

Finally, the diffusion model assumes dispersal is proportional to habitat suitability ($$D(x,y) \propto h(x,y)$$), which is assumed to be constant with respect to time during the model period.

## What are Invasive Species?

Ellen Schofield posted a nice video explanation of what ecologists mean by 'invasive'.

## Credits

This web application has been developed as part of a sponsored Great Basin Cooperative Ecosystem Studies Unit project with the aim of making mathematical models of spatial population spread accessible to a broader audience. The purpose is to develop a 'proof-of-concept' application that illustrates new approaches to real-time model computations with ecologically significant implications.

This material is based upon work supported by the U.S. Geological Survey under Grant/Cooperative Agreement No. G15AC00311.

## Data Sources

• EDDMapS - reports of black tegu sightings in Florida.
• Custom habitat suitability estimates extracted from geo-tiffs made by Mark Hayes and colleagues using GLM techniques.
• World map data extracted from the 'mapdata' package in R .

## Development Tools

We developed this application entirely with free and open source tools, hosting the code on github (email Joe Champion for access to the code repository).

We made extensive use of new tools in R for web development, including:

• Shiny - web application framework connecting R with HTML and JavaScript.
• Shiny Server - open source platform for hosting Shiny applications.
• GGVis - for specifying interactive HTML-based canvas layers using the grammar of R data.
• ReacTran - for specifying and computing reaction-diffusion models.
• SDMTools - for post-processing outcomes of species distribution models.
• deSolve - for numerical solutions of initial value problems of systems of ordinary and partial differential equations.
• Other Packages , especially dplyr, rgeos, raster, sp, png, and maptools for data wrangling.

## Technical Details

We plan to update this page with more technical details in the future. Please feel free to contact the development team.