eWalk, an R function to simulate two-dimensional evolutionary walks

   
   
eWalk <- function(sigma,lon,lat, generations,alpha_x,theta_x,alpha_y,theta_y, color,levy=FALSE, plot=TRUE) {
  route=data.frame(1:generations,1:generations)                             # a data.frame to store the route
  names(route)<-c("lon","lat")
  generation= generations                                               # generation counter
  
  while (generation > 0){                                                   # stops the simulation when generation counter is 0
    if (levy == TRUE){                                                      # only if levy=TRUE: Levy Flight!
      rndm=abs(rcauchy(1))                                                  # process to select if I should jump or not
      if (rndm > 15){                                                       # Levy jump probability threshold
        l=15                                                                # length of the jump
        x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))    # Brownian or OU process for longitude
        y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))    # Brownian or OU process for latitude
        loc=cbind(x,y)
        route[generation,]<- loc                                            # store new location
        generation= generation - 1                                        # advance to next generation
        lon<- x                                                        # go to the new position!
        lat<- y
        
      }
      else{                                                                 # if Levy jump is not selected, l=1 i.e. proceed with  
        l=1                                                                 #normal Brownian motion or OU process
        x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))
        y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))
        loc=cbind(x,y)
        route[generation,]<- loc
        generation= generation - 1  
        lon<- x      
        lat<- y
      }
    }
    else{                                                                   # if levy=FALSE, l=1, only Brownian motion or OU
      l=1
      x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))
      y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))
      loc=cbind(x,y)
      route[generation,]<- loc
      generation= generation - 1  
      lon<- x      
      lat<- y
    }
  }
  
  if (plot == TRUE) {                                                       # if plot=TRUE, plto the route!
    lines(route, col= color, lwd=3)  
  }
  return(route)                                                             # print the data.frame with each location during the walk!
  
}
>

In addition, I have added a complement to simulate Lèvy flights, a non-gaussian process similar to Brownian motion but allowing for jumps during the motion. Lévy flights pattern has been found in searchingmovements of predators (Bartumeus et al., 2005), or have been used to simulatepunctuated equilibrium in traitsevolution (Gould and Eldredge, 1977; Landis et al., 2013). To perform Lèvy flight, eWalk selects a random number from Cauchy distribution and if it is higher than a threshold (15 by default), Brownian motion term is multiplied by a constant representing a jump (also 15 by default). Although this is not the formal expression of a Lèvy process, this is a simple way to simulate a negative exponential rate in for the length of each step.

   
 par(mfrow=c(2,2))

a=eWalk(0.15,0,0,1000, 0, 0, 0, 0, color="blue", levy=FALSE, plot=FALSE)
plot(a,type="n", main="Brownian motion")
lines(a,lwd=3, col="blue")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0, 0, 0, 0, color="red", levy=TRUE, plot=FALSE)
plot(a,type="n", main="Lêvy flight")
lines(a,lwd=3, col="red")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0.01, 20, 0, 0, color="orange", levy=FALSE, plot=FALSE)
plot(a,type="n", main="OU one optimum")
lines(a,lwd=3, col="orange")
abline(v=20, col="red", lwd="2")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0.01, 20, 0.01, 20, color="green", levy=FALSE, plot=FALSE)
plot(a,type="n", main="OU two optimum")
lines(a,lwd=3, col="green")
points(a, cex=.3, col="black", pch=21)
abline(v=20, col="red", lwd="2")
abline(h=20, col="red", lwd="2")