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BASE

Random walks: the drunk and the canyon

drunkard_walking.jpg

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.

Parâmetros

Simulation parameters:

Option Parameter
Number of Species S Number of drunks
Step Size step Number of steps the drunk moves sideway at each time interval
Maximum Initial Distance xlmax maximum distance between the drunks and the canyon when the simulation starts
Initial Distance Equal 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
Maximum timetmax total simulation time

Example

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.

Step effect

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

Time effect

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

Question

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?

Virtual population

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 in the Maximum time parameter.

Parametros estocasticidade

The options here control the simulation of populations under a continuous time random walk:

Option Parameter Definition
Enter name for last simulation data setR object R object in which to save the simulation results
Maximum timetmax Maximum simulation time, in the same time unit as the rates used
Number of simulationsnsim Number of different populations to simulate
Initial sizeN0 Initial population size
birth rateb instantaneous birth rate ($b$)
death rated instantaneous death rate ($d$)

Example

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.

Questions

  1. Which are the corresponding parameters between the drunk random walk and the birth and death dynamics?
  2. The effects you have noted on the drunk simulation (step size and end time) affect the population dynamics in the same way?
  3. What is the consequence of this result for the conservation of natural populations?

To learn more

1)
we are only interested in the lateral movement