Lab. 7. Stability analysis of the Ricker's model

   Ricker's model simulate the dynamics of a population with intra-specific
   competition. It is a discrete-time analog of the logistic model. Ricker's model
   is:

                                     [eqrick.gif]

   where K is carrying capacity (=equilibrium density) and r is the intrinsic rate
   of population increase. Use Microsoft Excel to simulate population dynamics with
   the Ricker's model. In all simulations, the carrying capacity K= 1000.

   1.Run the model with r=0.2 for 80 generations starting from No = 1. Plot the
   graph, explain why it has the S-shape.

   2.Estimate relaxation time (time until the population density approaches the
   equilibrium with specific accuracy, e.g., 0.1) as a quantitative measure of
   stability. Run the model with parameter r=0.05; 0.1; 0.2, 0.3;... 1.9, 1.95 and
   find relaxation time using No = 900 as initial density. Plot relaxation time
   versus r and explain the shape of the graph. Note: the number of generations
   should be greater than the relaxation time.

   3. Run the model with parameter r changing from 1.8 to 3.0 with step 0.025 and
   plot the bifurcation diagram as follows. Run the model for 200 generations with
   each value of r, discard first 100 generations and plot population numbers in the
   final 100 generations versus the r-value. Describe the plot: at what value of r
   the population dynamics changes from a stable equilibrium to a 2-point limit
   cycle, to a 4-point limit cycle, to chaos? Plot population dynamics versus time
   for r=l; r=1.8; r=2.2; r=2.6; r=2.8. Indicate the type of population dynamics.

   4. Add random fluctuations to the model:

                                     [eqrand.gif]

   where [eqxi.gif] is a random variable with normal distribution, mean = 0 and
   standard deviation = s. The random variable with normal distribution (m =0, and
   SD=1) can be obtained by summing 12 random number (function RAND()) with the
   uniform distribution ranging from 0 to 1 and subtracting 6 from the sum. Run the
   model with parameter r changing from 0.1 to 1.9 with 0.1 steps, K=1000, s=10, No
   = 1000, for 500 generations. Estimate the coefficient of v-stability:

                                     [eqvstab.gif]

   where sigma is the standard deviation of population density and s is the standard
   deviation of the noise. Plot VS versus r and compare it with the theoretical
   graph: [eqvstab.gif] . Compare the graph with the graph of relaxation time.

   [1][back.gif] [2][up.gif] [3][forward.gif]
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   [4]Alexei Sharov 12/4/98

References

   1. http://alexei.nfshost.com/PopEcol/lab6/lab6.html
   2. http://alexei.nfshost.com/PopEcol/popecol.html
   3. http://alexei.nfshost.com/PopEcol/lab8/lab8.html
   4. http://alexei.nfshost.com/~sharov/alexei.html