updates docs

README.md

 # ABMEv.jl
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docs/src/examples/delta_competition_example.md

@@ -3,4 +3,86 @@
 In this script, we model the evolution of a population where agents are simply defined by their position on some landscape. We implement the simplest possible birth and death function.
 
 ## The landscape
-Let's start by a linear landscape. We define a discrete segment of length `9`, with reflecting boundary conditions.
+Let's start by a linear landscape. We define a discrete segment of length `9`, with reflecting boundary conditions. In fact, reflecting boundary conditions are implemented for any finite space.
+
+!!! warning "1D Reflections"
+  As of v1, only reflections on one dimensional space are implemented. We have a two dimensional reflection method that will be released in the future.
+
+There are two ways of implementing a linear landscape. The first one uses a `DiscreteSegment` while the second relies on LightGraphs.jl library. Both are *almost* equivalent.
+
+### DiscreteSegment
+```julia
+using ABMEv
+nodes = 10
+mysegment = DiscreteSegment(1,nodes)
+```
+
+### grid
+```julia
+using ABMEv, LightGraphs
+nodes = 10
+g = grid([nodes,1])
+mysegmentgraph = GraphSpace(g)
+```
+Here is how you can visualise the landscape.
+
+```julia
+using GraphPlots
+gplot(g,locs_x = 1:nodes,locs_y=1:nodes)
+```
+
+## Defining competition processes
+We propose that any individual have a constant birth rate, and competes with all the individuals present in the same patch. Let ``i \in \N``,``x_i \in \{1,2,\dots,9\}``.
+The competition pressure experience by individual ``i`` is such that
+
+```math
+d(x_i) = \sum_j \delta(x_i-x_j)
+```
+where ``\delta`` is the dirac function.
+
+In this way, we recover a logistic growth function for subpopulation within a patch.
+
+```julia
+K0 = 1000 # We will have in total 1000 individuals
+b(X,t) = 1 / nodes
+d(X,Y,t) = (X[1] ≈ Y[1]) / K0
+```
+At equilibrium, population size in each deme will reach `K0 / nodes`.
+
+## Dispersal
+We assume that anytime an offspring is born, it is given a chance to move (`\mu = 1`).
+```julia
+mu = [1.]
+D = (1.5,)
+```
+## Running the world
+We initialise the world with initial population of size ``K_0 / 9`` located on patch `5`. We keep track of individuals' ancestors by setting `ancestors=true`. Because we wish to use `Gillepsie` algorithm, we need `rates=true` as agents' internal birth and death rates are updated at every time step.
+!!! note "Warning"
+  `rates` treatment is something we might implement in the library internals.
+
+```julia
+NMax = 2000
+tend = 2
+p = Dict{String,Any}();@pack! p = d,b,D,mu,NMax
+myagents = [Agent(myspace,(5,),ancestors=true,rates=true) for i in 1:K0/nodes]
+w0 = World(myagents,myspace,p,0.)
+@time sim = run!(w0,Gillepsie(),tend)
+```
+## Analysis
+Let's verify that the population's growth is logistic. We will plot the population size over time.
+To do so, one need to define `dt_saving`.
+
+```julia
+using Plots
+@time sim = run!(w0,Gillepsie(),tend,dt_saving=1)
+Plots.plot()
+@animate for (i,t) in tspan(sim)
+  Plots.
+end
+```
+
+One might be tempted to plot the world for some time steps. This requires some knowledge of the library Plots.jl.
+Agents values are accessed by `get_x(w::World)`.
+
+!!! notes "Plotting"
+  This will be proposed as a blackbox with PlotRecipes.jl in the near future