Results 1 to 2 of 2

Math Help - Chaos and Dynamical Systems

  1. #1
    Senior Member
    Joined
    Oct 2008
    Posts
    393

    Chaos and Dynamical Systems

    The equation  P_n_+_1 = rP_n(1-P_n/C) -k may be considered a density - dependent population model with constant harvesting. If the growth rate is r = 3 and the carrying capacity is C = 6000, what is the largest number k that could be harvested each generation so that the population has a stable positive equilibrium, and so may not become extinct. I thought you needed to differntaite it but this would get rid of k?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by adam_leeds View Post
    The equation  P_n_+_1 = rP_n(1-P_n/C) -k may be considered a density - dependent population model with constant harvesting. If the growth rate is r = 3 and the carrying capacity is C = 6000, what is the largest number k that could be harvested each generation so that the population has a stable positive equilibrium, and so may not become extinct. I thought you needed to differntaite it but this would get rid of k?
    If r=3 and c=6000 the difference equation can be written in the form...

    \Delta_{n} = p_{n+1}-p_{n}= -\frac{p^{2}_{n}}{3000}+2\ p_{n}-k = f(p_{n}) (1)

    Necessary condition for convergence of p_{n} is the existence of an 'attractive fixed point', i.e. a solution x_{0} of the equation f(x)=0 with the condition f'(x_{0})<0 and that is true for k<3000. In that case the solutions are...

    x= 3000\ \pm \sqrt{3000}\ \sqrt{3000-k} (2)

    ... and x_{0} corresponds to the sign '+'...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Dynamical systems
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: January 23rd 2011, 08:17 PM
  2. Dynamical Systems
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: January 21st 2010, 07:12 PM
  3. Dynamical Systems
    Posted in the Advanced Applied Math Forum
    Replies: 6
    Last Post: September 17th 2009, 12:02 AM
  4. Dynamical Systems
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 5th 2009, 12:29 AM
  5. Dynamical Systems
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 4th 2005, 11:46 AM

Search Tags


/mathhelpforum @mathhelpforum