Thought experiment: density dependence with simultaneous mortality and migration

[ebuhle]

OK @marksorel8 @mdscheuerell @conversesj, humor me if you will. After our last Zoom meeting, I spent some time fiddling around with a toy model to help me better understand what our "null expectation" is when looking at these patterns of density dependence in stage-specific migration. Specifically, this is a simple ODE system designed to get at the two-bathtub-drains problem.

I finally got around to writing it up as a vignette. The results are interesting and somewhat counterintuitive (to me). I was especially struck by how closely the patterns of apparent stage-specific density dependence mirror @marksorel8's results, despite the fact that there is no plasticity or density dependence or life-history strategy in migration whatsoever -- all the action is on the resident side.

Anyway, have a look and let me know what you think.

[mdscheuerell]

This is really interesting, @ebuhle, and certainly worthy of more discussion.

With respect to this comment:

The plot of ΔMt vs. R0 in the first pulse is not depensatory, but if compensation were weaker than in this example and in the presence of correlated environmental stochasticity and observation noise, it could easily be mistaken for linear.

I'm wondering what combination of compensation/depensation, correlated environmental stochasticity, and observation error might lead to the apparent pattern that @marksorel8 found where the number of migrants increased exponentially?

[ebuhle]

I'm wondering what combination of compensation/depensation, correlated environmental stochasticity, and observation error might lead to the apparent pattern that @marksorel8 found where the number of migrants increased exponentially?

I wondered the same thing. I haven't played around extensively with alternative parameter values or functional forms, and it's not immediately obvious to me how you could get this apparent depensation without some actual depensation in survival and/or migration, but I wouldn't want to hang my hat on that without further exploration (which any of y'all are welcome to try, or I'm happy to take requests). The timing and relative magnitude of the migration pulses is another factor to consider.

[marksorel8]

@ebuhle ! This is beautiful, and could be a really useful model for evaluating different hypotheses. For example, I think we expect that the strength of density dependence in daily survival increases through summer as fish get larger. This would accentuate the pattern of increasing density dependence between R0 and deltaM with time. Modeling this whole system as continuous rather than discrete is really cool and could have a number of uses. Maybe funding could be found to support work on this continuous model at some point.

Regarding @mdscheuerell 's questions . I'd say there can be, as I believe the saying goes, no depensation without depensation.

There is one feature of the Wenatchee spring Chinook system that could affect the patterns that we see though - hatchery affects. Hatchery-origin spawners in the Chiwawa tend to spawn lower in the stream closer to the acclimation facility and the trap and could be genetically different in ways that affect juvenile life history expression. I've included the proportion of hatchery origin spawners for a given brood year as a covariate on process errors in the past, and it does considerably reduce support for the exponential model of summer subyearling production over a density-independent (linear) model. I had planned to just talk about potential hatchery effects in the discussion, but perhaps folks think it would be worth including in the analysis?

[ebuhle]

I've included the proportion of hatchery origin spawners for a given brood year as a covariate on process errors in the past, and it does considerably reduce support for the exponential model of summer subyearling production over a density-independent (linear) model.

That's interesting. Do you have a sense of why this happens? Is it just a matter of p_HOS being confounded with S and also associated with (I'm guessing) lower per capita parr production, or does the distribution of migration pulses change as a function of hatchery influence, or...?

[marksorel8]

There is a positive relationship between pHOS and S, so you are probably right that that confounding of S a pHOS explains some of it. I suspect it could also be due in part to your second suggestion, that the distribution of migration pulses change as a function of hatchery influence.

Now that you have jogged my memory, I think the reason I decided not to include pHOS in the analysis was because of the confounding of S and pHOS. I guess I made a judgement call, but the only alternative I can think of would be to ask someone with local knowledge like Andrew, "how much do you think pHOS really affects the distribution of migration pulses?" Perhaps the necessarily data to test this would be some sort of genetic parentage on a sample of the juveniles emigrants.

[ebuhle]

Now that you have jogged my memory, I think the reason I decided not to include pHOS in the analysis was because of the confounding of S and pHOS.

Maybe it's beyond the scope of the present manuscript, but this harkens back to an analysis of hatchery vs. wild RRS in Wenatchee spring Chinook spawner-recruit data that I did for a 2015 AFS talk with Andrew, Todd Pearsons, Mike Ford and others. I used my old S-R regression approach where R is modeled as a "response surface" over S_W and S_H, rather than p_HOS. Results on RRS were inconclusive, with wide uncertainty, but I haven't revisited using the state-space approach (even though, ironically, this mixed H/W model was the original motivation for what became salmonIPM). Of course, S_W and S_H can also be confounded (and were in this case, IIRC).

Does Andrew have PBT data on juveniles? I know his group has done some of this for adults.

Going back to @mdscheuerell's comment, I think the really interesting question is what combination of (time-varying?) compensation, depensation, and/or process and observation noise would give you a pattern of apparent depensation in one migration pulse but compensation in the others.

[mdscheuerell]

I think the really interesting question is what combination of (time-varying?) compensation, depensation, and/or process and observation noise would give you a pattern of apparent depensation in one migration pulse but compensation in the others.

This would definitely be worth exploring in some detail, although I suspect it's beyond the scope of this immediate project.

[marksorel8]

I'm not sure if Andrew has PBT data on Juveniles. I really haven't delved into their RRS project.

I agree that, while important, a detailed exploration of mechanism that could produce the observed patterns is outside the scope of this immediate project.

[ebuhle]

I agree, except insofar as it relates to our priors on alternative explanations for the observed patterns. (E.g., are they spurious, perhaps driven by hatchery influence? Or indicative of underlying density dependence in survival and/or migration, and if so, is one more plausible than the other?)

[ebuhle]

You might be interested in this update with a simple model of density-dependent emigration.

[mdscheuerell]

That's really interesting, especially where

initial exploration failed to find parameter values producing “realistic” dynamics, for which the relationships are convex in the early stages but strongly concave in the later ones.

[ebuhle]

Yeah. Emphasis on "limited", but my hunch is that you can't get these sharply divergent patterns without a similarly dramatic seasonal shift in the density dependence of emigration and/or resident survival. (Barring other confounding factors, that is.)

[marksorel8]

Very very cool! This is really informative and elegant. Do you have ideas for further application/development of this model @ebuhle or @mdscheuerell?

For what they are worth (perhaps not much), here are some thoughts regarding

It is also notable that the apparent strength of compensation, at least with respect to initial cohort size R0, increases across successive migration pulses or stages regardless of density dependence in migration. It seems somewhat unlikely, based on limited parameter exploration, that this shift would be strong enough to convert an accelerating density-dependent relationship in stage 1 to a saturating one by stage 4. This suggests that such a pattern would require another explanation – for example, that the instantaneous migration rate becomes less strongly density-dependent over the course of the season, either gradually or abruptly.

I know Andrew thinks of the fall migration as a kind of pre-winter redistribution that isn't necessarily density dependent, which seems reasonable. It's a bit of a head scratcher to me that I saw evidence of density-dependent migration of sub-yearlings at a time in spring when they are quite small and the river is quite large, but maybe there is some response to density that plays out at small scales starting when fish are born and diminishes at some point in the season. I really enjoyed the 2006 paper by Sigurd Einum et al. in Oikos, which posits about how the physiological constraints of juvenile salmonids change throughout ontogeny and affect how density-dependence is partitioned among mortality, movement, and growth.

Depending on whether there is interest in pursuing this further, perhaps it could be useful to do a little literature review and summarization of the state of knowledge on density-dependent juvenile movements and survival in anadromous salmonids. I could potentially work on this if you thought it would be worthwhile.

I'm off to another upper Columbia River basin with spring Chinook adult and juvenile monitoring data (the Methow) to get married. I'll be back online next Wednesday. Thanks for this really interesting work and discourse.

[ebuhle]

It's a bit of a head scratcher to me that I saw evidence of density-dependent migration of sub-yearlings at a time in spring when they are quite small and the river is quite large, but maybe there is some response to density that plays out at small scales starting when fish are born and diminishes at some point in the season.

This is what's puzzling me too. I typically think of fry migration / displacement as being largely passive, and as you pointed out earlier, one would expect per capita density dependence to increase, not decrease, as parr grow and use larger territories in a seasonally shrinking habitat.

I really enjoyed the 2006 paper by Sigurd Einum et al. in Oikos, which posits about how the physiological constraints of juvenile salmonids change throughout ontogeny and affect how density-dependence is partitioned among mortality, movement, and growth.

That is a very good paper that I should revisit; it's been a while. I do think a lit review on density-dependent juvenile movement in anadromous salmonids would be an important part of the Introduction and/or Discussion in the present manuscript, and interesting in general. I don't really know the current state of the science on this. Slightly tangential but related, there are some old papers by Grant and colleagues on territory size, and an associated literature on "self-thinning", that predict potentially abrupt changes in mortality and/or migration as a function of effective density.

As for this little simulation exercise, I didn't have any grand plans beyond formalizing the intuitions and hypotheses that we kicked around on our Zoom call to see what patterns they would actually imply. My conclusion thus far is that simple process assumptions probably can't account for the observed patterns. I may keep tinkering, especially if you guys have ideas that you think would be particularly illuminating. Also, I would be interested in seeing the log-ratio plots for the real Wenatchee data.

I'm off to another upper Columbia River basin with spring Chinook adult and juvenile monitoring data (the Methow) to get married. I'll be back online next Wednesday. Thanks for this really interesting work and discourse.

Mazel tov! :tada: Definitely don't do anything fish biology-related, unless it's hook-and-line sampling.