An Individual-based model simulator for clonal plant growth developed in 2011 and 2012 in the context of the ANR Syscom project Modecol, in Matlab.
It was faff developed within the ANR Syscomm (SYStèmes Complexes et Modélisation Mathématique) project MODECOL ( MODélisation ECOLogique de prairies virtuelles) [ANR-08-SYSC-012]. The model presented here is detailed and analyzed in:
We propose a stochastic individual-based model of graph-structured population, viewed as a simple model of clonal plants. The dynamics is modeled in continuous time and space, and focuses on the effects of the network structure of the plant on the growth strategy of ramets. This model is coupled with an explicit advection-diffusion dynamics for resources.
At time clonal plant is represented as a set of nodes (ramets) that may be connected by links (rhizomes or stolons). In this simplified representation of a clonal plant, ramets are represented by points in the plane, and connection by lines. The state of the nodes is described by the following finite measure:
where is position of the th node and total number of nodes; denotes the Dirac measure centered on the point . The measure describes the distribution of nodes over the space of spatial positions. For any node at position we define the set of indices of the nodes connected to :
The plant grows in a resource landscape. At each time , this resource landscape is represented by the available resources at position . The nodes accessing high levels of resources are more likely to give birth to new nodes.
Birth and death rates
Each node of in position may disappear at a death rate and give birth to a new node at a birth rate . These rates are per capita rates. Global death and birth rates at population level are respectively: and ; the global event rate is . Basically, the per capita rates depend on the local availability of resources: we suppose that the birth rate is an increasing function of and the death rate is a decreasing function of . When a node is added to the population, it is always linked with the mother node; when a node is removed from population, all connections to are suppressed.
A node at position at time gives birth to a new node at position according to the p.d.f. . is the angle between the preferred direction of reference and the direction of the new shoot . is the p.d.f. of the angle of the new shoot and is the p.d.f. of the length of the associated link. will be a rough approximation of the gradient of given by connected nodes.
Interactions between nodes and resources
For the coupling of the (discrete) individual dynamics with the resource density dynamics is modeled as:
we model with the kernel the fact that resource consumption is not local.
Exact Monte Carlo simulation of the IBM
The only approximation is the numerical integration of the resource dynamics which is performed with a finite difference scheme.
, , given for do compute the rate and , with if then sample according to sample according to [birth] else sample according to end if compute [numerical approximation of ] end for
, , , , :
, , , , :