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# Looking Backwards to Know Where You’re Going

#### Using Dynamical Statistical Models to Understand and Forecast Biological Invasions

Biological invasions are constantly shaping and reshaping the world around us. Invasions unfold in many forms, including the introduction of exotic species on a new continent, reintroduction of endangered or extirpated species, or spread of disease. The scientific community upholds a strong desire to manage biological invasions depending on their anticipated consequences. But before scientists can generate effective action plans, they need to develop a basic understanding of the invasion, including the location of introduction, rate of spread, and birth and death rates of the invading organism or pathogen. Scientists obtain this information by collecting data during an invasion and extract it by using these data as input into dynamic statistical models.

Dynamic statistical models are mathematical representations of the real world, with connected layers that enable scientists to estimate features of interest. They also account for scientists’ inability to completely observe invasions in nature, thus creating data that are “messy” representations of the underlying invasion processes. The “process layer,” which describes how the invasion progresses through time, is of most interest to scientists. For example, one can use partial differential equations (PDEs)—mathematical models commonly developed by applied mathematicians to explain how spatial processes (like the spread of disease) change over time—as the “process layer” within a dynamic statistical model. A benefit of a PDE approach is the ability to explain how the current state of invasion evolved from the past, which allows forecasting of how the invasion will progress in the future. It also permits “backcasting” to identify where the invasion began. The key is selecting or creating an appropriate PDE that mimics the invasion.

Ecological diffusion is a PDE applicable to biological invasions. To understand ecological diffusion, picture a grid of equal-sized cells with an individual animal placed in one cell. During some time interval, the animal will choose to remain at its current location or move into an adjacent cell. The probability of the animal remaining or moving into the cells is based on “habitat” within each cell. This sequence of choices is repeated over many time steps (see Figure 1). Overtime, the individual will spend more time in “good” habitats. The ecological diffusion PDE emerges when one expands this idea to include an infinitely-large number of animals, and shrinks the grid cells and time step between choices until they are infinitely small. The result is an equation that describes a heterogeneous spatial distribution of a population that evolves through time. One can modify the ecological diffusion model to allow an invasion that not only moves over a landscape but incorporates combined birth and death rates; this in turn lets the invasion grow in magnitude. This modification of the PDE is called a reaction-diffusion PDE. Reaction-diffusion PDEs allow scientists to estimate the location of introduction, rate of spread of an invasion, growth of invasion at invaded locations, and influence of landscape and individual animal characteristics on invasion dynamics.

Figure 1. An example of an individual deer’s movement into different cells based on the habitat types (i.e., forest, soy bean or corn fields, or wetland) within each cell. Arrow color represents the probability of moving into each cell (scale for probabilities depicted at left). This is the basis of the ecological diffusion model.

Once the process layer is chosen within the dynamic statistical model, the data layer is constructed. This layer connects the data collected from the invasion to the process layer using probability distributions; such probability-based linkage is critical for connecting the clean mathematical world of PDEs to the messy reality of biological data.

To demonstrate the application of a dynamic statistical model in characterizing biological invasions, we investigated the dynamics of chronic wasting disease (CWD) in white-tailed deer in southwestern Wisconsin. CWD is a fatal, transmissible, neurodegenerative disease that negatively affects deer populations. It was discovered in Wisconsin in 2001, and since that time the Wisconsin Department of Natural Resources (WIDNR) has routinely tested deer for CWD throughout the state (see Figure 2a). The WIDNR is interested in understanding factors that permitted the disease to invade and grow within the deer population. To meet this need, we developed a dynamic statistical model with a process layer comprised of a reaction-ecological diffusion model with parameters that were functions of landscape features and parameters for the effects of individual deer characteristics. Our data layer took the form of a binomial model because the data consisted of binary CWD test results for individually-sampled deer. The results from this model demonstrated that CWD prevalence, particularly among male deer, is increasing, and landscape features such as human development, forest cover, and large water bodies can impact the speed at which the disease can spread and grow (see Figure 2b and 2c). The model has also allowed us to forecast how CWD prevalence rates will change on the landscape and determine via backcasting the region in which the disease was most likely introduced. This information will help the WIDNR optimize their CWD surveillance and disease management activities.

Figure 2. 2a. Location of samples collected by the Wisconsin Department of Natural Resources for chronic wasting disease surveillance in white-tailed deer, and the study area (yellow square) used in our analysis. 2b. Estimated chronic wasting disease prevalence in male white-tailed at least four-years old in southwestern Wisconsin using our dynamic statistical model. 2c. Map depicting the growth and spread of chronic wasting disease in our study area, estimated and forecasted by our dynamic statistical model.

Dynamic statistical models are powerful tools for understanding and managing biological invasions. They permit scientists to link mathematical models with messy biological data while properly accounting for uncertainty in our understanding of the natural world. Given the ever-increasing number of biological invasions (often driven by human activities), the importance of these analytical tools will continue to grow.

Daniel P. Walsh is a quantitative ecologist at the United States Geological Survey’s National Wildlife Health Center. His research focuses on the development and application of statistical and machine-learning approaches to understand and manage diseases in wildlife populations and model population dynamics. Trevor J. Hefley is an assistant professor in the Department of Statistics at Kansas State University. His research focuses on developing and applying spatio-temporal statistical methods to inform fish and wildlife management decisions.