Test effects of including summer harvest

[ChloeRN]

So far, our model is set up to include only harvest happening between the start of October and the end of May (= "main hunting season"). In reality, there is some hunting going on over the summer months (start of July to end of September) as well, and this is presently ignored.

Ideally, we would model summer harvest as well but there are some complications.

First, we expect that the effort for summer harvesting is substantially more variable than for winter harvests, and we do know that there is likely to be an aging bias in the carcass data as juveniles (young-of-the-year) can be identified without tooth analysis, resulting in a larger proportion of young individuals aged as opposed to older individuals. We confirmed the presence of this bias by looking at the age distributions from the data, as well as the proportion of foxes below and above 3.5 kg aged. However, we noted that the bias only became apparent after 2012 (before that, most individuals were aged anyways), so there is a chance for analysing age structure data from 2005 to 2012.

[ChloeRN]

I have tested out two additional models that account for summer harvest in different ways:

Model 1

• Lumps summer and winter harvest together
• Uses summer harvest counts but assumes age structure equal to winter harvest
• Unchanged model structure but adjusted "pData" (proportion aged in harvest)

Model 2

• Models summer harvest explicitly and separate from winter harvest
• Summer harvest is assumed to happen after reproduction and before immigration, and no natural mortality is assumed to occur during that time period
• Also assumes that all summer harvest is reported, that aging is representative, and that summer harvest has not changed systematically post 2012 (with regards to variation among years and relative differences among age classes)
• Uses summer age-at-harvest matrix for years 2005-2012
• Extended model structure with an additional census at the start of October

[ChloeRN]

The results for Model 1 show that simply adding the count of summer harvest to the winter AaH matrix has limited effects. It results in slightly higher population size estimates overall but mostly increases uncertainty for estimates of population-level quantities and harvest mortality. The results from that model are shown in the comparison below, too, but not discussed further as there is little of interest here.

[ChloeRN]

Model comparisons (Note that population sizes plotted correspond to June census prior to immigration, i.e. survivors plus locally born young-of-the-year)

It's very evident that Model 2 produces substantially lower population size estimates (typically between 100 and 450 as opposed to between 300 and 1800 in the model ignoring summer harvest).

[ChloeRN]

This seems to be due to a combination of lower denning survival (~40% versus 65% in the model ignoring summer harvest) and higher winter harvest mortality for all age classes but the oldest.

The average estimates of summer harvest mortality hazard rates (Mu.mHs) are low compared to winter harvest mortality.

[ChloeRN]

Interestingly, the changes above do not seem to propagate into the estimates of average natural mortality (Mu.mO) at first glance. However, we have to remember that these parameters are given very strong informative priors (especially in this example that uses the rather precise values for red foxes in Northern Sweden) and therefore are not swayed easily. Taking a closer look, we do see that yearly natural mortality (mO) responds, and quite drastically so:

(Similar for the other age classes)

Looking at whole posteriors, we see that we are dealing with posterior distributions with a median close to 0, but quite long (and in some cases heavy) tails, e.g.:

So, while the estimates of average mO do not change due to strongly informative priors, the annual estimates are pushed towards 0 and this - unsurprisingly - also shows in the estimates of parameters linked to temporal variation in mO:

We have a random effect SD that now pushes against the (unrealistically high) upper prior bound of 5, and have very high uncertainty in the covariate effects for mO. In practice, this means that including summer harvest (at least as done here), results in a model with zero predictive ability when it comes to natural mortality.

[ChloeRN]

Based on the above, we should probably test what happens to the summer harvest model when using a less "strong" informative prior on natural mortality such as the prior from the meta-analysis or the (otherwise poorly performing) Hoening prior.

[ChloeRN]

Turns out my suspicion about an issue with "too informative" survival/natural mortality priors was correct. Looking additionally at results from a model including summer AaH but using the more diffuse survival priors from the meta-analytical approach instead restores our ability to model temporal variation in natural mortality by allowing more "room" for the estimates of mean natural mortality:

This model produces similar estimates of denning survival as the original model, but higher estimates of winter harvest mortality. Average estimates of natural mortality may be slightly lower but also hold more uncertainty.

As a consequence, the population-level estimates of this model fall between those of the original model and the much lower ones from the first attempt of the summer harvest model (typical range between 130 and 650):

[ChloeRN]

(On a side note: using the Hoening model prior in the summer harvest model still does not work very well)

[ChloeRN]

Things get really interesting when comparing the summer harvest model using the meta analysis prior to the "baseline model" (no summer harvest) using the same prior.

We do still see the expected shifts towards higher winter harvest mortality and lower natural mortality when accounting for summer harvest:

Including summer harvest still also produces lower estimated population sizes, but the difference is less extreme than when using the North Sweden survival prior and the posteriors mostly overlap.

The most striking difference between the models though is that by accounting for summer harvest in the model using the meta analysis prior, we get a massive increase in precision in the estimation of population level quantities across the board!

[ChloeRN]

At this point, we need to discuss if we want to further pursue models including summer harvest (in which case we should be working with the model using the survival prior from the meta analysis).

To make that decision, we should carefully consider the assumptions of the model with summer harvest (vs. ignoring summer harvest), as well as the potential gains in terms of inference vs. the costs of added complexity (and more work to implement it for the entire workflow.

[ChloeRN]

Following up on a comment from @stijnhofhuis, I also re-tested the effect of adding the summer harvest counts to the winter AaH matrix while using the meta-analysis survival/natural mortality priors. With that setup, we still get similar population size estimates than when not accounting for summer harvest, but precision is better - not worse as it was with the North Sweden prior:

We also get slightly lower denning survival and slightly higher harvest mortality of younger age classes:

[ChloeRN]

Another consideration brought up by @stijnhofhuis is regarding the use of summer harvest count data in addition to summer AaH data. The latter is only reliable for years 2005-2012 as we expect substantial aging bias afterwards. Irrespective of aging, the summer harvest counts from 2013-2021 are usable data though, and could be included in the model.

A quick comparison of predictions (from model using only 2005-2012 summer AaH data) and observed harvest count data shows that data do fall within the predicted range, but there is clearly some time-variation that may warrant modelling:

However, all in all that fit is looking quite decent...

[ChloeRN]

I've tried including a (rather rudimentary) likelihood for the summer harvest counts from 2013 onwards. I tried out two slightly different approaches to parameterising the lambda parameter for a Poisson likelihood, with quite similar results.

As expected, including the summer harvest count data improves the predictions of number harvested in summer for the later years to better track the pattern in the data:

However, beyond that it did not really improve the model substantially, and also led to much longer runtimes and poor mixing for several parameters.

Population-level estimates were quite similar to those from the model not using the harvest counts:

It did also not improve precision of any of the average vital rates:

And encountered some issues with estimating time variation in summer harvest mortality or natural mortality, depending on the likelihood chosen:

[ChloeRN]

The conclusion is that we will continue with a model that uses the summer Age-at-Harvest data for years 2005 to 2012. The testing out of further extensions modelling harvest counts post 2012 will be archived on the branch "summerHcount_test" for the time being as we'll not pursue that any further.