{"id":16,"date":"2012-08-06T16:08:04","date_gmt":"2012-08-06T16:08:04","guid":{"rendered":"http:\/\/www-sop.inria.fr\/members\/Fabien.Campillo\/?page_id=16"},"modified":"2023-10-06T13:52:06","modified_gmt":"2023-10-06T13:52:06","slug":"ibm-clonal","status":"publish","type":"page","link":"http:\/\/localhost:8888\/wordpress\/software\/ibm-clonal\/","title":{"rendered":"IBM of clonal plant dynamics"},"content":{"rendered":"

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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.<\/em><\/p>\n\n\n\n

Context<\/h2>\n\n\n\n

It was developed within the ANR<\/a> Syscomm (SYSt\u00e8mes Complexes et Mod\u00e9lisation Math\u00e9matique) project MODECOL ( MOD\u00e9lisation ECOLogique de prairies virtuelles) [ANR-08-SYSC-012]. The model presented here is detailed and analyzed in: F. Campillo and N. Champagnat, Simulation and analysis of an individual-based model for graph-structured plant dynamics, Ecological Modeling 2012. [PDF<\/a>].<\/p>\n\n\n\n

Description<\/h2>\n\n\n\n

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.<\/p>\n\n\n

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Network structure of the plant.<\/figcaption><\/figure><\/div>\n\n\n

At time \"t\" 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:

  <\/span>   <\/span>\"\[\nu_t=\sum_{i=1}^{N_t}\delta_{x^i_t}\]\"<\/p>
where \"x^i_t\in\mathcal{D}\" is position of the \"i\"th node and \"N_t\" total number of nodes; \"\delta\" denotes the Dirac measure centered on the point \"x\". The measure \"\nu_t\" describes the distribution of nodes over the space \"\mathcal{D}\subset\mathbb{R}^2\" of spatial positions. For any node at position \"x\" we define the set of indices of the nodes connected to \"x\":

  <\/span>   <\/span>\"\[J(t,x) = \{i = 1,\dots,N_t; x \textrm{ and }x^i_t \textrm{ are connected} \}\,.\]\"<\/p><\/p>\n\n\n\n

The plant grows in a resource landscape. At each time \"t\", this resource landscape is represented by \"\mathbf{r}(t, x)\geq 0\" the available resources at position \"x\in\mathcal{D}\". The nodes accessing high levels of resources \"\mathbf{r}(t, x)\" are more likely to give birth to new nodes.<\/p>\n\n\n\n

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Birth and death rates<\/h2>\n\n\n\n

Each node of \"\nu_t\" in position \"x\" may disappear at a death rate \"\mu(t, x)\" and give birth to a new node at a birth rate \"\lambda(t, x)\". These rates are per capita rates. Global death and birth rates at population level are respectively: \"\bar\lambda_t=\sum_{i=1}^{N_t}\lambda(t,x^i_t)\" and \"\bar\mu_t=\sum_{i=1}^{N_t}\mu(t,x^i_t)\"; the global event rate is \"\bar\lambda_t+\bar\mu_t\". Basically, the per capita rates depend on the local availability of resources: we suppose that the birth rate \"\lambda(t, x)\" is an increasing function of \"\mathbf{r}(t, x)\" and the death rate \"\mu(t, x)\" is a decreasing function of \"\mathbf{r}(t, x)\". When a node is added to the population, it is always linked with the mother node; when a node \"x\" is removed from population, all connections to \"x\" are suppressed.<\/p>\n\n\n\n

Dispersion kernel<\/h2>\n\n\n\n

A node at position \"x\" at time \"t\" gives birth to a new node at position \"y = x + v\" according to the p.d.f. \"D_{t,x} (v)= f (\kappa, (d_{t,x} , v)),g (|c|)\". \"(d_{t,x},v)\" is the angle between the preferred direction of reference \"d_{t,x}\" and the direction of the new shoot \"v\". \"f\" is the p.d.f. of the angle of the new shoot and \"g\" is the p.d.f. of the length of the associated link. \"d_{t,x}\" will be a rough approximation of the gradient of \"\mathbf{r}(t,x)\" given by connected nodes.<\/p>\n\n\n

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Dispertion kernel.<\/figcaption><\/figure><\/div>\n\n\n

Interactions between nodes and resources<\/h2>\n\n\n\n

For the coupling of the (discrete) individual dynamics with the resource density dynamics \"\mathbf{r}(t,x)\" is modeled as:

(\"\triangle\") <\/span>   <\/span>\"\[\partial_t \mathbf{r}(t,x)=\textrm{div}(\mathbf{a}(x),\nabla \mathbf{r}(t,x))+\mathbf{b}(x)\cdot\nabla \mathbf{r}(t,x)-\alpha,\mathbf{r}(t,x),\sum_{i=1}^{N_t} \Gamma_{x^i_t}(x)<\/p><\/p>\n\n\n\n

we model with the kernel \"\Gamma_{x^i_t}(x)\" the fact that resource consumption is not local.<\/p>\n<\/div>\n<\/div>\n\n\n\n

Exact Monte Carlo simulation of the IBM<\/h2>\n\n\n\n

The only approximation is the numerical integration of the resource dynamics \"(\triangle)\" which is performed with a finite difference scheme. <\/p>\n\n\n\n

\"T_0\leftarrow 0\", \"\nu_0\", \"\mathbf{r}(t,x)\" given\nfor \"k=0,1,2,\dots\" do\n   compute the rate \"\lambda(T_k,x^i_{T_k})\" and \"\mu(T_k,x^i_{T_k})\"\n   \"\bar\lambda \leftarrow\sum_{\xin\nu_{T_k}}\lambda(T_k,x)\", \"\bar\mu \leftarrow\sum_{\xin\nu_{T_k}}\mu(T_k,x)\"\n   \"T_{k+1}\leftarrow T_k+S\" with \"S\sim \exp(\bar\lambda+\bar \mu)\"\n   if \"\textrm{\rand}()<\bar\lambda/(\bar\lambda+\bar\mu)\" then\n      sample \"x\" according to \"{\lambda(T_k,x)/\bar\lambda;x\in\nu_{T_k}}\"\n      sample \"v\" according to \"D_{T_k,x}(v)\"\n      \"\nu_{T_{k+1}}\leftarrow \nu_{T_k}+\delta_w\" [birth]\n   else\n      sample \"x\" according to \"{\mu(T_k,x)/\bar\mu;x\in\nu_{T_k}}\"\n      \"\nu_{T_{k+1}}\leftarrow \nu_{T_k}-\delta_x\"\n   end if\n      compute \"\mathbf{r}(T_{k+1},x)\" [numerical approximation of \"(\triangle)\"]\nend for<\/pre>\n\n\n\n
Simulation<\/h2>\n\n\n\n
Here the angle p.d.f. \"f\" is a\u00a0Von Mises distribution<\/a> of parameters \"\mu_{\textrm{\tiny\rm f}}\" and \"\kappa_{\textrm{\tiny\rm f}}\";\u00a0the length p.d.f. \"g\" is a log-normal distribution<\/a> of parameters \"\mu_{\textrm{\tiny\rm g}}\" and \"\sigma^2_{\textrm{\tiny\rm g}}.\" The maximum link per node is\u00a0\"N_{\text\rm{\tiny\rm links}}\".<\/p>\n\n\n\n
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Guerilla<\/h2>\n\n\n\n
\"N_{\textrm{\tiny\rm links}}=3\",\u00a0\"\mu_{\textrm{\tiny\rm g}}=-1.8\", \"\sigma^2_{\textrm{\tiny\rm g}}=0.1\",\u00a0\"\mu_{\textrm{\tiny\rm f}}=0\", \"\kappa_{\text\rm{\tiny\rm f}}=400\":<\/p>\n\n\n\n