ODD protocol for model description
The model description presented here follows the ODD (Overview, Design concepts, Details) protocol (Grimm et al. 2006; Grimm et al. 2010).
- Purpose
To gain detailed insight on how landscape structure influences coexistence and gain understanding of the underlying mechanisms and their relative importance. We hypothesize that when habitat availability and habitat clumping are high, coexistence will be most stable - although species in a weak competitive position might benefit from fragmentation.
- Entities, state variables, and scales
The first type of entities we use are individuals which could be insects or plants. State variables are location (x,y coordinates), species (1 or 2), reproductive rate in either habitat type (λpreferred= 2.5, λunpreferred= 1.5) and dispersiveness (i.e. the standard deviation of a Gaussian kernel which is kept constant within simulation runs). Evidently the core and edge specialists prefer core and edge habitat respectively.
The second type of entities are the grid cells which represent the local spatial environment. They have a location (x, y coordinates), a habitat type (0 = unsuitable habitat (matrix), 1 = edge habitat, 2 = core habitat) and a local density(the total number of individuals from both species sharing the same coordinates as the patch) .
Since this is a purely theoretical study the spatial and temporal scales are depending of the specific system we have in mind. We suggest that our model applies to both insects and plants but the scales would differ between them. In order to avoid any further projection on real specific systems of this generic model, we deliberately do not assign very specific units.
- Process overview and scheduling
Within one time step individuals disperse and, if they survive the dispersal phase (i.e. if they land in a suitable patch), they reproduce (density dependently), after this they die. The offspring then go on the do the same in the next time step.
A higher level module first loops over all the individuals to handle the dispersal phase. In this phase individual’s locations are updated and unsuccessful dispersers are removed. Hereafter the local density in each grid cell is assessed and assigned to that grid cell. In a subsequent loop the individuals then reproduce according to this local density and the local habitat type.
- Design concepts
Basic principles.
The population dynamics used are similar to work by several authors (Travis and Dytham 1999; Poethke and Hovestadt 2002; Kubisch et al. 2010; Fronhofer et al. 2011; Boeye et al. 2013). However, the difference to these models is that our model each individual has very few traits and no evolution or plasticity etc. takes place. The complexity in our model lies in the landscapes structure. On this aspect our approach is most similar to Wiegand et al. (1999) and Wiegand et al. (2005). We try to expand the field of knowledge by combining competition between two species based on simple population dynamics and complex landscape structures.
Emergence.
As the spatial structure of the landscape changes over different simulation results we expect the species abundances and levels of local co-occurrence (see later) to vary.
Adaptation, objectives, learning, prediction, sensing, collectives.
These play no role in our model.
Interaction.
Each individual indirectly and equally interacts with all other individuals in its patch through increasing the local density. The local density is the only driver of local competition and is negatively correlated to the reproductive output of those in the patch.
Stochasticity.
Both the dispersal and the reproductive process have stochasticity embedded in them. During the dispersal process distances in both x and y direction are randomly selected from a Gaussian kernel. The reproductive output is selected from a Poisson distribution.
Observation.
The first main result observation taken from each simulation run after the 1000th time step are total individual count of both species from which the global coexistence is calculated. The global coexistence is at its maximum (100%) when on a global scale both species are equally abundant whereas a value of 0% indicates the total exclusion of one species.The second value is the percentage of inhabited grid cells occupied by individuals of both species (i.e. the local co-occurrence).
- Initialization
During initialization the landscape is created according to the parameters given for: P (proportion of suitable habitat, i.e. of the core and edge type), H (Hurst exponent, level of clumpedness), Pcore (proportion of the suitable habitat that is of the core type, Pedge = 1 – Pcore), the size of the landscape. The landscapes are created with a separate module using the Diamond-square algorithm (Miller 1986) which returns a 2D matrix with the habitat type values (0 = unsuitable habitat (matrix), 1 = edge habitat, 2 = core habitat). Once the landscape is created the main model is initialized with one thousand individuals of each species randomly distributed over the suitable habitat.
- Input data
Except for the landscape created by a separate module there is no input data.
- Submodels
Dispersal:
During the dispersal phase a loop goes over all individuals and for each one of them samples a distance in both x and y direction from a Gaussian kernel with a fixed standard deviation. These distances are then rounded to the closest integer. Subsequently the distances are added to the x and y coordinates of the individual. If the new coordinates lie within the boundaries of the landscape and the habitat on that location is of a suitable type (core or edge) then the individual is appended to a list of survivors.
Reproduction:
Before reproduction takes place the local density in each cell is assessed. Next the mean number of offspring µ which each individual will produce in its local patch is calculated as follows:
µ = λ(1+ aNt)-1
with
a = (λ-1)/N*
Here, λ specifies the net reproductive rate which depends on whether the individual prefers the local habitat type (λpreferred= 2.5, λunpreferred= 1.5), N* is the population equilibrium density for a single patch and is a constant set to 2, Nt is the summed local density of both species at time t; if Nt is higher than N* the mean number of offspring (µ) will decrease below 1 due to competition and the local population will shrink. The actual number of offspring is drawn from a Poisson distribution with mean µ(Travis and Dytham 2002; Kubisch et al. 2011).