Murray Cod Modelling to Address Key Management Actions Final Report for Project md745
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- Figure 10: Observed variation length at age. Red line is the mean estimate of length given age and the blue lines represent upper and lower plausible bounds.
- Table 4. Probability distributions of length given age of Murray cod, length in cm.
- Size classes Age 30-40 40-45
- Figure 11: Some examples of the probability of being in a size class, given age, where the unit of length is centimetres.
- Figure 12: Minimum population size risk curves (risk increases as the risk curve shifts to the left and risk decreases as risk curves shift to the right).
- 5. Management scenarios 5.1 Workshop 2: Specialist workshop on Murray cod modelling
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4.3 Variable growth Murray cod exhibit variable growth (Fig. 10). Additionally, mark-recapture data indicate inconsistent growth between years. Variable growth can be accounted for directly in a size based model through the estimate of transition rates from the mark-recapture data, however this would restrict the model’s applicability to the mid-Murray. It is also possible to account for variable growth in an age based model by including a plausible range of length given age or a size distribution for each age. Wan et al. (1999) developed a more flexible growth model than the standard von Bertalanffy growth function. 
where Wf, W0 and k are equivalent to L∞, t0 and k in the von Bertalanffy growth curve and the parameter c provides a point of inflexion and therefore slightly more elastic growth. Fitting the growth model (Wan et al., 1999) to the age-length data to obtain parameter estimates for the model: Wf = 150; W0 = 6; c = -103; and k = 0.0011 (see Fig. 8). 
Figure 10: Observed variation length at age. Red line is the mean estimate of length given age and the blue lines represent upper and lower plausible bounds. Selecting k from a bounded distribution between the upper and lower plausible bounds allows lengths to be derived given age: k = 0.0006 + 0.0015*Beta(4.2, 8), where Beta(a, b) is a random deviate drawn from a beta distribution with shape parameters a and b (see Evans et al. 1993). By producing 1000 lengths for each age allows probabilities to be generated for assigning size to an age and therefore the expected size distribution for each age class. This approach is useful in assessing the impact of the removal of certain size class of fish. For example, a five year old fish has probability of being in size class 40-45cm of 0.02 (Pr(40-45|5)=0.019); 45-50cm is Pr(45-50|5) = 0.112; 50-55cm is Pr(50-55|5) = 0.249; 55-60cm is Pr(55-60|5) = 0.304; 60-65cm is Pr(60-65|5) = 0.220; 65-70cm is Pr(65-70|5) = 0.082; 70-75cm is Pr(70-75|5) = 0.013; and the probability for a five year old fish being in any of the other size classes is 0 (see Table 4 and Fig. 11). Table 4. Probability distributions of length given age of Murray cod, length in cm.
Figure 11: Some examples of the probability of being in a size class, given age, where the unit of length is centimetres. 
4.4 Expressions of risk and scenario ranking The model uses a Monte Carlo simulation technique where the user determines the number of iterations produced. Typically, in order to examine the consequences of a potential management actions, each scenario is run (iterated) a minimum of 1 000 times. The purpose of the large number of iterations is to sample sufficiently from the parameter distributions so that a full exploration of the variation of the distribution is undertaken and the likelihood of extreme events can be examined (Ferson et al. 1989; Burgman et al. 1993). The data generated from the simulation can be represented as probability distributions (or histograms) or converted to error bars, reliability predictions, tolerance zones, and confidence intervals (Wittwer 2004). Recording the minimum population size from each iteration or trajectory and then graphing the associated normalised cumulative frequency distribution produces a graph of probabilities versus population size, the minimum population size risk curve. These represent both the chances of extinction (probability of falling to zero) and the chances of falling below some non-zero population threshold (Burgman et al. 1993). Additionally, risk curves can be readily compared and assessed in terms of increasing or decreasing risk by a shift to the left or right respectively of the minimum population size risk curve (Fig. 12). A method for quantifying changes in risks is to calculate the average minimum population size for each curve and compare these values (McCarthy 1995; McCarthy and Thompson 2001; Todd et al. 2002; 2004).
Given that one of the objectives of the project is to examine a number of management scenarios, it is useful to report on the statistics of: risk curves associated with the distribution of the minimum population size (may be specific elements of the population such as fish aged 5–9, 10–14, 15–19, or fish aged 20 or older); the average minimum population size; the absolute difference in the average minimum population size and the percentage change in average minimum population size. 5. Management scenarios 5.1 Workshop 2: Specialist workshop on Murray cod modelling A second workshop on Murray cod modelling was held on April 18th 2007. The participants were managers/policy makers from around the Murray-Darling basin with the focus of the workshop on management outcomes. The invited participants were: Matt Barwick, Mark Lintermans (ACT), Glenn Wilson (NSW/QLD), Ross Winstanley (VIC), Andrew Sanger (NSW), Cameron Westaway (NSW), Travis Dowling (VIC), Julia Smith (VIC), Karen Weaver (VIC), Gary Backhouse (VIC), Peter Kind (QLD) and Anita Ramage (QLD). South Australian representatives were unable to attend on April 18th, on May 2nd, Charles Todd met with Qifeng Ye, Jason Higham and Brenton Zampatti and held a mini workshop at SARDI, in South Australia. A presentation was made to the workshop of the development of the management model and the key areas of uncertainty that remain. It was concluded from the workshop that the Murray cod management model adequately captures: • Density-dependence (depending on parameter setting the model can generate consistent recruitment as observed in the mid-Murray reaches (and tributaries) and pulse type recruitment observed in the lower-Murray); • The variety of current recreational fishing regulations and has the capacity to alter these regulations (options include slot size and closures – temporary or permanent); • A variety of fishing rates that can be applied to two size classes, fish below 1m and fish above 1m, by selecting a choice of fishing rates or applying a set of specific fishing rates from the analysis of mark-recapture data collected from the Murray below Yarrawonga; • The choice of stocking fingerlings or 1 year old fish and the number of years of stocking; • Thermal pollution scenarios; and • Modelling habitat change as either an increase or decrease to the available habitat, the year to implement such changes as well as the length of time that the change will remain. The general conclusion from the workshop was that the software developed met the expectations of the project brief. Modifications/improvements were suggested at the workshop which included: accounting for mortalities of fish released after being angled; some fish kill scenarios to examine both severity and likely time to recovery from when fish kills occur; and changing the thermal pollution section to ‘impacts on early life stages’ that can not only capture thermal impacts but mortalities due to weirs and water off takes such as channels and pumps.
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