ANTONIUS SILO

6.3.1

Sensitivity analysis

An essential part of model development is sensitivity analysis to understand the effect of error in parameter estimates on the model output. Therefore an initial series of model runs were conducted to test the sensitivity of the model to variations in the input parameters. The default values used are listed in table 6.1. The range of values over which the input parameters were varied depending on the estimated error in the input parameters, each parameter was varied over the full range of plausible values for that parameter. As the model contains non-linear relationships parameter values were co-varied, since basing the sensitivity analysis on individually varied parameters may hide effects of the non-linear relationships within the model (Kremer, 1983). A quantitative analysis of the sensitivity of the model output to individually varied parameters was conducted (Table 6.1). However, in order to avoid over complication, the results of co-varying parameters were examined graphically (Figure 6.1), and the pertinent features of the graphical analysis are discussed. For the quantitative analysis the sensitivity parameter, S, was calculated following (Fasham et al., 1990) with the slight modification that the absolute value was taken to simplify interpretation. Thus;

S = ¯ ¯ ¯ ¯ ¯ ¯ ³ OP−OD OD ´ ³ P−PD PD ´ ¯ ¯ ¯ ¯ ¯ ¯ (6.3)

whereOD is the model output with parameters set to default values,OP is the model

output with the given parameter changed to the new value P, and PD is the default

parameter value. The sensitivity of the model prediction of fertilisation success was examined. Where there is uncertainty in the choice of upper and lower parameter values, following (Fasham et al., 1990), the limit values were taken to be half and twice the standard value respectively, apart from the surfzone residence time where a wider range of values was considered to reflect the lack of information available regarding surfzone hydrodynamics (see section 3.5.2).

In relation to the physical parameters that define the volume of the surf zone, the model displayed far greater sensitivity to the height of breaking waves than beach an-

A)

B)

C)

Figure 6.1: Effect of covarying parameters on the proportion of eggs fertilised. The mixed volume was directly manipulated rather than individually manipulating parameters that control the mixed volume. A) varying mixed volume of surf zone for given gamete release time values; B) varying mixed volume of surf-zone for given values of surf-zone residence time; C) varying residence time

Table 6.1: The default, and range of parameter values used in the model. The default values were used for each model run unless indicated otherwise. The basis for the parameter values refers to the section in the thesis where the selection of that parameter value is discussed. The sensitivity values S are the sensitivity of fertilisation success compared to the maximum/minimum parameter values. Theβ values used in the model is 106_{larger than the value given in chapter 4 as the model}

works in m3 _{whereas the original calculation measured sperm per ml.}

Module Parameter Symbol Default Value Basis Min/ Max S Physical Beach angle σ 14◦ _{section 3.5.2 7/28 0.04/0.10} dimensions Height of breaking waves h 4m section 3.5.2 2/8 1.80/0.47

Volume V – – ±50% 1.80/0.64

Surface roughness ² 1.6 section 3.5.2 1.3/1.9 1.74/2.03 Surfzone residence time τ 600s section 3.5.2 60/3600 1.09/1.07

Spawning Length of gamete release R 900s section 6.2 450/1800 0.44/0.28 behavior Proportion of eggs released γ 0.83 section 6.2 0.60/0.98 1.00/1.00

Minimum landing size MLS 70mm chapter 5 – – Population egg production E On the basis of chapter 5 – –

exposure and MLS

Population sperm S On the basis of chapter 5 – – production exposure and MLS

Fertilisation Bimolecular rate constant β 2.0×10−15 _{section 4.4 1/4×10}15 _0.97/0.89

model Proportion eggs viable α 0.88 section 4.4 0.83/0.96 0.93/0.96

Model Run time T 18000s section 6.2 – –

conditions Time step t 10s section 6.2 – –

gle. To ease interpretation of the sensitivity analysis the volume of the surf zone was also itself directly manipulated. For the graphical analysis of covaried parameters the surf-zone volume was directly manipulated rather than varying the parameters that define the surf zone volume (Figure 6.1).

The model is sensitive to variation in all parameters. The degree of sensitivity varied between parameters (Table 6.1). The model shows a non-linear response to variations in some parameter values: this is particularly evident for the mixed volume of the surf-zone (Figure 6.1.B) and highlights the importance of conducting sensitivity testing across the full range of possible parameter values.

The model shows greater sensitivity to variation in the physical parameters defining the size and mixing in the model than the biological parameters defining rate and proportion of gamete release and the fertilisation model parameters. The single parameter analysis and covaried parameter analysis both indicate that the model is

insensitive to the duration of gamete release. Figure 6.1.B shows that the model is, however, particularly sensitive to combinations of surf-zone residence time and mixed volume. For the analysis these parameters were covaried independently, however in reality there is likely to be a positive relationship between mixed volume and residence time (Denny et al., 1992).

The sensitivity analysis indicates that the model is particularly sensitive to the physical parameters defining the model, and that considerable uncertainty exists in the selection of appropriate parameter values for the physical parameters. This indicates that the main priority for further research is to improve knowledge of the hydrodynamics of the surf-zone rather than the biology of limpet spawning. However this does not deny that further research into the biology of limpet spawning is necessary to improve the accuracy of the model.

6.3.2

Simulation runs

To address the aims of this study a variety of model runs were conducted. The model was run for the populations at each level of wave exposure to see the total proportion of eggs fertilised. The simulated fishery was applied to the populations and in each case the model was re-run and the proportion of eggs fertilised and total zygote production calculated. As in chapter 5 the reduction in population SSB and the available fishery yield was calculated for each simulation run, this enabled a comparison of the relative variation in population zygote production, population egg production, SSB and yield as simulated fishery pressure was increasingly applied. The model was also run to simulate the effects of managing on the basis of minimum landing size and\or MPAs.