Imagine that a drunkard is walking in a huge plains, that has a canyon on one of the sides. Every time the drunkard steps forward, he also staggers to the left or to the right, with equal probability.
This is a very simple Markovian process, called random walk, on one dimension1). When the drunkard falls in the canyon, the random walk is over (and so is the drunkard), so we call this condition as absorbing boundary.
| ||S||Number of drunks|
| ||step||Number of steps the drunk moves sideway at each time interval|
| ||xlmax||maximum distance between the drunks and the canyon when the simulation starts|
| ||alleq=TRUE||when selected (TRUE), all drunks start at the maximum distance. When unselected, the drunks start at random position from 1 to the maximum distance above|
| ||tmax||total simulation time|
Let's simulate ten drunkards, staggering for 10 steps at each time interval, for ten thousand time steps:
S =10 step = 10 xlmax = 200 alleq = TRUE tmax = 10000
As with every stochastic process, the results may vary at each simulation. Thus, repeat the simulation some times to make sure you understand the results.
What happens if the drunks are more or less staggering? Try reducing the step size to 2:
S = 10 step = 2 xlmax = 200 alleq = TRUE tmax = 10000
Drunks that stagger less are less prone to end in the bottom of the canyon, or is it only a matter of time? Try increasing the final time:
S = 10 step = 2 xlmax = 200 alleq = TRUE tmax = 50000
The drunk has equal probability to stagger left or right, so on the average he only walks forward. Given enough time, this random walk with absorbing boundary has only one possible outcome. What is it?
The same model of random walk can be used to model the dynamics of a population under demographic stochasticity. Here, we suppose that at every moment the population may lose one individual due to it dying, or gain one individual due to one birth. If both the instantaneous birth rate $b$ and the instantaneous death rate $d$ are equal, the probability of death is equal to the probability of birth.
The instantaneous growth rate if the difference between birth and death
rates ($r=b-d$). The time unit used in $r$ gives the time scale of the dynamics, represented in our simulation
Maximum time parameter.
The options here control the simulation of populations under a continuous time random walk:
|R object||R object in which to save the simulation results|
|tmax||Maximum simulation time, in the same time unit as the rates used|
|nsim||Number of different populations to simulate|
|N0||Initial population size|
|b||instantaneous birth rate ($b$)|
|d||instantaneous death rate ($d$)|
Simulate the trajectory of 20 population with equal birth and death rates, starting with 10 individuals each. Let the time continue up to 50 units:
tmax = 50 nsim = 20 N0 = 10 b = 0.2 d = 0.2
You should see a random walk graph, very similar to the drunks. The number of extinctions
up to the
Maximum time is recorded in the upper left corner of the graph.