Population cycles generated with a very simple logistic equation, X(t+1)=rX(t)[1-X(t)], r set to 3 (upper figure) and 3.6 (lower figure). The lower graph exhibits deterministic chaos. Blogged about at Climate Change Blog.


The simulation was programmed in Octave and plotted on Grace, both running on Linux.

  • Practicing Troublemaker 7y

    I'm rubbing my eyes right now.
  • Peter Roopnarine 7y

    Yeah, chaos can do that to you :-)
  • DavidMXG 7y

    I wrote a program in C to do this a few months back & a few years back did another to map how the population equation gave chaos at different values of r. So I am reminded to upload the images sometime.
  • Peter Roopnarine 7y

    A figure of the logistic map would be great David. I'm surprised that no one has posted that yet to the Emergence group (apologies to anyone if I'm wrong on that). I wrote a similar program a few years ago in very old Visual Basic, and converted the population size values to musical notes as time elapsed. The contrast between stationary, cyclic, quasi-cyclic and fully chaotic patterns was very cool. Got students thinking about the fun side of quantitative approaches!
  • DavidMXG 7y

    Are there any instances of actual biological populations where the constant r is big enough to obtain chaos?

    According to my dated reading of this subject no biological examples had been found in nature, whether in the wild or in a lab. Insects/invertebrates might be a likely candidate since some species can breed quickly creating vast numbers of a organisms. (Whether there is chaos might be relevant for ecological surveys of invertebrates - as to whether or not the populations of some species are chaotic year by year etc.)
  • Peter Roopnarine 7y

    Good question David! Until very recently, no confirmed instances, though there are now suggestions for populations of plankton in some new studies. But, we actually know r with any accuracy for very few species, relative to the number out there in Nature. There seems to be a bias against raising chaos as a potential explanation for observed population fluctuations, invoking instead stochastic or random noise. There is also a common assumption that r is fixed, while in reality it is likely variable, perhaps quite variable for some species. I suspect that with all the work being conducted, we will have a better answer within the decade!
  • DavidMXG 7y

    Thank you. I am thinking what kind of function could be substituted for r.
    r=f(X(t)) & maybe with some other parameters too.
  • Peter Roopnarine 7y

    That would work, particuarly if r was considered to be influenced by an external, say climatic, factor. You could also use a mean r plus some stochastic variation (random number here), or, maybe interestingly, make r a function of X, i.e. a function of population size itself. This last one would definitely be a possibility if one thinks that, e.g. birth rate increases or decreases on the basis of population size. I'm sure that there is literature out there exploring some of this, but it could also be programmed fairly easily. Any Emergence takers? :-)
0 faves
Uploaded on March 14, 2008
This photo is in 6 groups
This photo is in 2 albums

Additional info

  • Viewing this photo Public
  • Safety level of this photo Safe
  • S Search
    Photo navigation
    < > Thumbnail navigation
    Z Zoom
    B Back to context