discretelogistic {MQMF}

discretelogistic example and figure 3.2 Discrete logistic model

Description

discretelogistic is an implementation of equation 3.1 in the Simple Population Models chapter 3. It enables the exploration of the dynamics of the Discrete logistic model, based around the classical Schaefer model.

The time-series nature of population growth is clear from the fact that Nt+1 is a function of Nt. One can thus expect serial correlation. Setting the r parameter to values <= 1.0, would generate monotonically damped equilibria. r values between 1 < r < 2.03 would generate damped oscillatory equilibria, r values from 2.03 < r < 2.43 should generate stable limit cycles on a cycle of 2, 2.43 < r < 2.54 gives stable limit cycles of cycle 4, then 2.54 < r < 2.57 gives cycles > 4, and ~2.575 < r gives chaos (though r = 2.63 appears to generate a repeat period of six!). discretelogistic should be used in conjunction with plot, for which an S3 method has been defined plot.dynpop. As the dynamics are obviously sequential (i.e. n at t+1 is dependent upon n at t) the last line of the dynamics is removed to avoid an empty nt1 row.

Usage

discretelogistic(r = 0.5, K = 1000, N0 = 50, Ct = 0, Yrs = 50, p = 1)

Arguments

r

intrinsic rate of population increase; default = 0.5

K

carrying capacity; default = 1000.0

N0

Initial population size; default=50.0 = 5 percent depletion. Note that the term 'depletion' can be confusing. Surely 50 remaining from 1000 should be a depletion of 95 percent? But no, it is deemed to be the complement of 5 percent. Fisheries jargon can be confusing.

Ct

annual catch default = 0.0

Yrs

years of population growth, default=50

p

the production curve asymmetry parameter. the default value of 1.0 gives the classical Schaefer model. A value of 1e-08 approximates the Fox model where the mode is skewed to the left of centre.

Value

invisibly returns a matrix of year, nt, and nt1

Examples

  discretelogistic(0.5,1000.0,25,0.0,50) # asymptotic
  discretelogistic(2.5,1000.0,25,0.0,50) # 4-phase stable limit
  ans <- discretelogistic(r=2.55,K=1000.0,N0=100,Ct=95.0,Yrs=100)
  plot(ans)  # uses an S3 plot method for dynpop objects