This R script generates annual time series of proportional recruitment while accounting for the cyclical nature of Antarctic krill recruitment. Its outputs may be used as inputs in population dynamics models. Stochasticity is included in recruitment periodicity and the amplitude of extremes. The aim of this simulator is to produce recruitment series with defined statistical properties that also capture the important characteristics of empirical time series.
The two longest published Antarctic krill proportional recruitment time series originate from the LTER and AMLR programmes (Fig. 1).
Figure 1. Antarctic krill proportional recruitment time series as reported by the AMLR (F36 is the proportion of individuals smaller than 36mm, Kinzey et al, 2013) and LTER (F30 is the proportion of individuals smaller than 30mm, Ryabov et al, 2017; supplement).
To generate proportional recruitment time series, RecMaker follows these steps:
-
Create a vector of years in which maximum recruitment occur where the period between years is randomly generated between chosen bounds.
-
Generate a vector of years at the midpoints between consecutive years of maximum recruitment, in which minimum recruitment occur.
-
Generate maximum and minimum recruitment values between chosen bounds.
-
Interpolate values between the maximum and minimum using a local logistic function (a logistic function between each consecutive extreme; i.e., min to max or max to min).
The local logistic functions used in RecMaker to interpolate recruitment in year y (Ry) between extremes (min and max), are computed as:
Where Slope controls the slope (is positive for increasing sections and negative for decreasing sections), and, Mid controls the midpoint of the logistic functions, and is computed as:
Where wmax is the weight put on the year of maximum recruitment (Ymax) and may be modulated to adjust the midpoint of the logistic function, which controls the relative duration of periods of high recruitment and low recruitment.
Using the empirical series (Fig. 1), time series of minima (a value that was lower than the preceding and succeeding values) and maxima (a value that was greater than the preceding and succeeding values) were used to calibrate Slope and wmax.
The calibration of Slope and wmax was done by finding the values that minimized the Root Mean Square Error between the empirical time series and the simulated ones. As a result (Fig. 2), these values were determined to be:
- Slope=2.5
- wmax=2.4
Figure 2. Published recruitment time series (blue: AMLR; red: LTER) and simulated ones (black).
The figure below (Fig. 3) shows 9 time series generated by RecMaker using the following parameters:
- Slope=2.5
- wmax=2.4
- Mins=c(0,0.3)
- Maxs=c(0.7,1)
- Periods=c(4,7)
Figure 3. Examples of proportional recruitment time series as simulated by RecMaker.