The Modified Incidence Function (MIF) Model

Bruce Maxwell and Jay Rotella, Montana State University

This web site describes an incidence function model (IF) that was slightly modified from that developed by Hanski (1994). MIF was used to simulate spatial metapopulation dynamics of an invading plant species in a virtual management area. Our primary interest was to determine the value of information about an invasive plant species’ habitat quality for identifying the management approach that would most effectively decrease population spread.

Model Description

The model assumes a landscape with a set of habitat patches and a set of populations that occur in a dynamic colonization-extinction quasi-equilibrium. The habitat is divided into areas that will support (high probability of occurrence habitat) and areas that will not support (low probability of occurrence habitat) a new population. Therefore, all unoccupied map cells i can take on probabilities of occupation (incidence) J_i at time t (year) based on the probabilities of annual colonization C_i and extinction E_i.

One may assume that C_i and E_i depend on environmental variables and on life-history traits of the species. A fundamental assumption of the Hanski (1994) model is that a primary influence on C_i is the size of the habitat patch (A_i) and similarly that E_i is dependent on the size of a population and thus is a function of A_i. Therefore,

where γ is a constant and q is a parameter that determines the strength of the influence of A_i on E_i and can be thought of as the influence of environmental stochasticity on the extinction probability (q decreases with increasing environmental stochasticity). We included the influence of targeted management on population extinction and assumed that management efficacy was also a function of population size (A_i). So that with management

where f is the proportional efficacy of management or the proportion of populations killed by the particular management practice.

In order to make the model more compatible with our previous field studies (Rew et al 2005), we held the area of each habitat patch i constant and assumed that A_i was the habitat suitability or probability of occurrence (POO) so we substituted POO_i for A_i. This type of substitution was suggested by Hanski (1994).

The colonization probability C_i is assumed to be a function of the number of potential immigrants arriving at each habitat patch i per year (M_i) from source populations. Each source population was assumed to respond to habitat quality POO_j that influences the number of immigrants each provides.

In addition, p_j is 1 if a patch is occupied or 0 if not occupied by a possible source population j, d_ij is the distance between habitat patch i and population j. α determines the shape of the dispersal curve (decreasing α increases the area under the tail of the dispersal curve increasing the dispersal distance). β was held constant but is thought to be the product of density in the source population and the rate of emigration from occupied patches. We assumed no further interaction among the immigrants at colonization (Hanski 1997), but deviate from the Hanski (1994) model by adding the influence of habitat suitability or POO_i on the colonization probability. Therefore,

where y is a function of β and the species specific colonization ability y’ (which could be thought of as the viability of the dispersed seed or propagule).

A very nice feature of this model is the potential to estimate the parameter values from resampling presence absence data using

And estimating the parameters γy’, z and q using maximum likelihood estimation (Hanski 1997).

Simulations of invasion can be conducted across a virtual map (management area) by determining the probability of pixel occupation (J_i) at all locations for each time step (e.g. year) starting from an initial source population. Since many plant invasions are thought to begin by introduction to an area along roads (e.g. Pauchard et al., 2003; Pauchard & Alaback, 2004; Hill et al. 2005; Theoharides and Dukes 2007; von der Lippe and Kowarik 2007), we added a road that curves through the management area and has a range of relatively high POO values to demonstrate some preliminary simulations (Figure 1).

Simulations are typically run for 10 years (Figure 2), and with ten replications where the POO patches are redistributed across the map and mean results with standard deviation reported (Figure 3).

Figure 1. Map of virtual management area (50 X 50 pixel) with 10 patches (5 X 5 pixel) of relatively high randomly distributed POO and a road with relatively high POO and an initial population (red pixel) established along the road.