The SPEED model

SPEED is an individual-based, spatially-explicit model where individuals are born, disperse, reproduce and die, and population dynamics are related to the local environment. Carrying capacity is determined by the amount of each type of suitable habitat available locally and

reproductive rate by local climate suitability (Fig 4.1). Each individual experiences the surrounding population density, local habitat and climatic environments (at spatial scales of m, ha or km, as appropriate), with the climatic conditions adjusted at each time step (annual in our example, although any relevant time step can be used). These data are combined within the model to determine the likelihood of survival to reproduction, and number of offspring produced. Thus SPEED integrates four major drivers of species’ range shifts: population dynamics (carrying capacity and reproductive rate), habitat availability, climate suitability, and dispersal. We describe methods for computing these parameters, and sources of data. Access to the executable for running SPEED, the parameter list and user manual for running the model are provided in the Supplementary Information, and other data sets are available from the sources we cite.

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Figure 4.1. The structure of the SPEED model. The data required to run the model are detailed

under ‘Data Inputs’ and arrows indicate how the data feed into the model processes. Ki denotes the cell-specific carrying capacity. Nit denotes the number of individuals in a cell (i) at time (t).

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4.3.1.1 Mapping climate suitability

Any climate suitability surfaces can be used as inputs to SPEED, scaled from zero (unsuitable) to one (optimum) and inputted as gridded data at a resolution appropriate to the simulation. In our example, climate suitability was incorporated from climate envelope models at a 10 km grid square resolution and was varied annually.

4.3.1.2 Rate of reproduction

SPEED requires an estimate of the maximum population growth rate (Rmax). Reproduction occurs

in the SPEED model as a single event per time step (in the case of P. aegeria we have used an annual time step). The actual reproductive rate achieved is assumed in SPEED to be dependent on climate suitability, derived from the climate suitability layer. This allows the reproductive rate to vary both spatially and temporally, as individuals in different cells (spatial) or different time steps (temporal) experience different climatic conditions. The relationship between climatic suitability and population growth is defined as follows. We assumed that the realised population growth rate (R) increases linearly between two climatic thresholds: the minimum climate suitability for reproductive replacement (the ‘break-even’ point, where R = 1) and the optimum climate suitability (where Rmax is achieved and beyond which there is no further increase in reproductive

rate; See Fig A4.1). The optimum climate suitability (where Rmax is achieved) can be specified by

the user, or (as in our example using P. aegeria) can be taken as the maximum climate suitability observed in any grid cell within the study landscape at the start of the simulation. In our example, we assumed that the ‘break-even’ point (where R = 1) occurred when climate suitability values were equal to the AUC (Area Under the receiver operating characteristic Curve) threshold value generated from the species’ observed starting distribution (1970-1982 distribution in Britain) and its projected probability of occurrence, based on a downscaled projection from a continental European climate-distribution model (see below). When climate suitability is lower than the ‘break-even’ point, R declines linearly to zero when climate suitability is zero (Fig. A4.1). In order to include a stochastic element representing natural variation in success and failure, the number of offspring produced by each individual is taken from a random draw of a Poisson distribution with a mean (mu) equal to the estimated reproductive rate (R) per grid cell (Travis & Dytham, 1999).

4.3.1.3 Determining habitat availability and carrying capacity

Carrying capacity (K) is determined by the amount of habitat available locally and the relative suitability of that habitat. SPEED requires an input surface of habitat availability, which could be

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based on field surveys or remotely-sensed data, at a spatial resolution appropriate to the study organism and which can be at a different resolution than climate suitability. Habitat availability data give the proportion of each grid cell that contains habitat suitable for reproduction. A maximum grid cell carrying capacity is set, which is the maximum number of individuals a grid cell can support if it contains 100% suitable habitat. The specific carrying capacity of each individual grid cell is calculated by multiplying the proportion of suitable habitat within the cell by the maximum carrying capacity.

Multiple habitat types can be incorporated into SPEED, and each habitat type can be weighted, assuming the modelled species may reach different densities in different habitat types. In this case, each habitat type is assigned a proportional value, which reflects the densities reached in that habitat type relative to the maximum density achieved in the most suitable habitat. When multiple habitat types are used, the cell-specific carrying capacity is the sum of the carrying capacities for each habitat type present in that cell (based on the area of each habitat type and the relative density of the species in each).

4.3.1.4 Dispersal ability

SPEED includes three dispersal parameters, although the modular structure of the programme would allow other dispersal functions to be incorporated. Two negative exponential dispersal kernels describe short-distance and long-distance dispersal, which together capture short- distance routine movements (e.g. foraging) and also longer-distance movements resulting in displacement (e.g. Van Dyck & Baguette, 2005). The third dispersal parameter quantifies the proportion of individuals allocated to long-distance versus short-distance dispersal. Thus dispersal is incorporated in SPEED as short distance dispersal, long distance dispersal, and the proportion of dispersing individuals allocated to long-distance dispersal (ranging between 0 = all short distance and 1 = all long distance). At each time step of the model, an individual disperses a random distance in a random direction in continuous space according to the long-distance or short- distance dispersal kernel it is allocated to.