# Pradel models: recruitment, survival and population growth rate

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1 Pradel models: recruitment, survival and population growth rate CHAPTER 13 So far, we have concentrated on estimation related to the general question what is the probability of leaving the population?. Clearly, death marks permanent departure from the population. Absence from the population can be permanent (like death), or temporary (a subject we ll discuss more fully in a later chapter on something known as the robust design ). However, if we re interested in modeling the dynamics of a population, then we re likely to be as interested in the probability of entry into the population as we are the probability of exit from the population. So, where to begin. We ll start with the fundamental model of population dynamics. Usually, the assumption (based on even a casual glance at a typical textbook on the subject) is that population dynamics models are based entirely on high-level mathematics. However, while it isn t difficult to find examples of such models, the fundamental model is quite simple: population dynamics has to do with the change in abundance over space and time ( N) N additions - subtractions That s it,really. The rest is just details (of course,the details can get messy,and that is what often leads to the higher math referred to above). But the basic idea is simple: when the net number of additions is greater than the net number of subtractions, then clearly the population will grow ( N> 0). When the reverse is true, the population will decline. So, population dynamics involves the study and estimation of the relative contributions to the additions and subtractions from the population Population growth: realized vs. projected Usually, the net growth of a population is expressed as the ratio of successive population abundances: N t+1 /N t. This ratio is usually referred to asλ, leading (frequently) to a whole bunch of confusion -λas the ratio of successive population sizes, or λ as the projected growth rate of a population under specified model conditions? As you may recall, the projected growth of a structured population is given as the dominant eigenvalue from a non-negative projection matrix (the Leslie matrix for models structured on age, and the Lefkovitch matrix for situations where some other classification factor often size or developmental state is a better demographic category than age). The word projected is key here it is Cooch & White (2015)

2 13.2. Estimating realizedλ 13-2 the growth rate of the population that would eventually be expected if (and only if) the conditions under which the model are valid are time invariant (i.e., not stochastic). We differentiate between projectedλ and realized λ. We let realized λ be (simply) the ratio of successive population sizes. Projected λ and realized λ will be equivalent if and only if the growth of the population has achieved stationary, ergodic conditions (the familiar stable-age or stable-age structure at equilibrium). Under such conditions, the population (and each age-class in the population) will be growing at rateλ, such thatλ N t+1 /N t. So, we suggest qualifying the used of λ with a prefix either projected, or realized, and noting (if appropriate) when they are equivalent, and when they are different: projected λ: the growth rate of the population expected under time invariance (where λ is commonly derived as the dominant eigenvalue from a projection matrix model) realizedλ: the observed growth rate of the population between successive samples (time steps):λ i N i+1 /N i While projected λ is often of considerable interest (especially for prospective management studies), in a retrospective study, where we re interested in assessing the pattern of variation in growth that has occurred, it is realizedλ that is of interest (although there are a variety of analytical methods out there for retrospective analysis that somewhat blur this convenient distinction: the life table response experiment (LTRE) developed by Caswell is a hybrid of a retrospective technique using prospective perturbation analysis of deviation in the projectedλunder a variety of experimental conditions). For our purposes though, we ll keep to the simple distinction we ve just described we want to explore changes in realized growth, λ Estimating realized λ Sinceλ N t+1 /N t, then it seems reasonable that as a first step, we want to derive estimates of abundance for our population at successive time steps, and simply derive the ratio to yield our estimate ofλ. Simple in concept, but annoyingly difficult in practice. Why? In large part, because the estimation of abundance in an open population is often very imprecise. Such estimates often suffer rather profoundly from violation of any of a number of assumptions, and are often not worth the effort (abundance estimation in open populations was covered in Chapter 12 estimation in closed populations is covered in Chapter 14). So we re stuck, right? Not exactly. The solution comes in several steps. We start with the recognition that our basic purpose in characterizing the change in abundance between years rests on assessing the relative number of additions and subtractions. This basic idea was introduced in Chapter 12. With a bit of thought, you should realize that an individual which dies, or permanently emigrates, is clearly a subtraction from the population. As such, we can reasonably state that the number of individuals in the population next year is going to be a function, at least in part, of the number of individuals in the population this year that survive and return to the population next year. However, we also know that there may be additions to the population, either in the form of permanent immigration, or births (in situ recruits). Let the number of individuals this year be N t. Letϕ i be the probability of surviving and returning to the population (ϕ i S i F i, where S is the true survival probability, and F is the fidelity probability see chapter 10). Let B i be the number of new individuals that enter the population between (i) and (i+1) - in other words, B i is the number of individuals in the population at (i+1) that were not there at (i). Thus, we can write: N i+1 N i ϕ i + B i

4 Putting it together: derivingλ 13-4 Let s pause to highlight the distinctions betweenϕ i (going forward in time) andγ i (estimated from the reverse encounter history, going backward in time): forward in time ϕ i probability that if alive and in the population at time i (e.g., this year), the you will be alive and in the population at time i+1 (e.g., next year) backward in time γ i probability that if alive and in the population at time i (e.g., this year), that you were also alive and in the population at time i-1 (e.g., last year) Understanding backwards encounter histories begin sidebar Consider the following encounter history: What this history is telling us, going forward in time, is that the individual was initially encountered and marked at occasion 1, not seen at occasion 2, but then seen on the next 3 occasions (3, 4, and 5), then not seen on occasion 6. The probability expression corresponding to this history would be: ϕ 1 ( 1 p2 ) ϕ2 p 3 ϕ 3 p 4 ϕ 4 p 5 ( 1 ϕ5 p 6 ) Now, if we reverse the encounter history ( ), then we see that, conditional on being alive and in the population at occasion 5, that the individual was also in the population at occasion 4, and at occasion 3. Given that it was also there at occasion 1, but not encountered at occasion 2 (with probability 1 p 2 ), then the probability expression (in terms ofγ i and p i ) corresponding to is γ 5 p 4 γ 4 p 3 γ 3 ( 1 p2 ) γ2 p 1 end sidebar Putting it together: derivingλ Still with us? We hope so,because the parameterγ i features prominently in what comes next. Remember, our interest in this parameterγ i comes from it having something to do with recruitment. Now, for the next big step pay close attention here. Remember from our previous discussion that B i is the number of individuals entering the population between time (i) and (i+1). If N i+1 is the number of individuals in the population at time (i+1), then B i /N i+1 is the proportion (or probability) that an individual in the population at i + 1 is one that entered the population between (i) and (i + 1). So, if (B i /N i+1 ) is the probability that an individual entered the population,then 1 (B i /N i+1 ) is the probability that it was already in the population. Does this sound familiar? It should. Think back what isγ i? It is the probability that if you re in the population at time (i) that you were also there at time (i-1), which is precisely the same thing! In other words, Thus, since γ i+1 1 B i N i+1 N i+1 N i ϕ i + B i

5 Putting it together: derivingλ 13-5 then we can write Or, re-arranging slightly, γ i+1 1 B i 1 N i ϕ i N i+1 ϕ i λ i N i+1 ( Ni+1 N i ϕ i ) N i+1 λ i ϕ i γ i+1 In other words, all we need are estimates ofϕ i andγ i+1, and we can derive estimates ofλ i, all without ever measuring population abundance (N)! Here we have a technique where, using simple mark-recapture data, we derive an explicit estimate of population growth, with considerable precision (as it turns out), without the need to estimate abundance. An alternative derivation begin sidebar We can also derive this expression forλ using a slightly different approach. Let the size of the population at risk of capture (encounter) on occasions i and i+1 be N i and N i+1, respectively. A subset of these two sets (populations) of animals is the subset of all animals that are alive in the population at both times. At time i+1, the size of this set can be given as N i ϕ i. At time i, the size of that set can be given asγ i+1 N i+1. Since both relationships give the size of the same set (population size alive at both times), then we can write N i ϕ i γ i+1 N i+1 Sinceλ N i+1 /N i, then it follows clearly thatλ i ϕ i /γ i+1, which is exactly the same relationship we derived above. end sidebar We also note that sinceλ i ϕ i + f i, then we can clearly also derive the following 2 expressions: γ i+1 ϕ ( i ϕ ) i ϕ i + f i λ i f i ϕ i ( 1 γi+1 γ i+1 ) This is the essence of the Pradel models in MARK. They rest on the duality between estimating ϕ i going forward in time, andγ i going backward in time. Moreover, becauseϕandγ can both be estimated in a general likelihood expression, not only can we estimateλand f directly, but we can also derive estimates of the variance in both. Perhaps more importantly, we can address hypotheses about variation in one or more of these parameters using the standard linear models approach we ve used throughout constraining various parameters to be linear functions of one or more covariates.

7 13.4. Pradel models in MARK 13-7 At this point, you need to decide which analysis you want to do. You could focus on estimation of theγ i values alone (i.e., estimate the seniority probabilities). Estimation ofγcan be useful in analysis of age-specific variation in recruitment to breeding state (i.e., the transition from pre-breeder to breeder). The next 3 models (and their robust design equivalents) consist of different pairs of parameters (for example, you could estimate survival and λ by selecting the Survival and Lambda data type). You cannot estimate all 3 of the primary parameters simultaneously (i.e., you can t estimateϕ,γ, f, andλ in one data type), since they are effectively a linear function of each other (such that estimating any two of them can provide estimates of the remaining parameters). Let s run through all 4, starting with the seniority only data type, providing estimates of seniority γ and encounter probability p. Once you click the OK button, you ll be presented with the PIM for the seniority parameter γ: Why only 1 row in the PIM? Why not the triangular PIM we ve seen for standard recapture models? The answer is because the Pradel models don t allow for age effects. As noted by Franklin, A.B. (2001), the reason for this is that the likelihood for estimating γ is conditioned on the entire encounter history, not just the portion following first capture as is the case when estimatingϕunder the CJS model. For example, MARK conditions on the full encounter history to estimate f andλ, whereas it conditions on only the 1101 portion to estimateϕand p. Therefore, age cannot be included because age cannot be estimated back to the initial zeros of the encounter history. Thus, you have now more than one row in the PIM (since each row corresponds to a different release cohort, and cohort and age are collinear). If age-specific estimates are desired, groups of animals can be created based on age. The results browser (below) shows the results of fitting the 4 standard models to the capsid data. Franklin. (2001) Exploring ecological relationships in survival and estimating rates of population change using program MARK. pp in R. Field, R. J. Warren, H. K. Okarma & P. R. Sievert (editors). Wildlife, Land, and People: Priorities for the 21st Century. The Wildlife Society, Bethesda, MD.

8 13.4. Pradel models in MARK 13-8 Based on these 4 models, and with a default ĉ of 1.0, it appears as though a model with time-variation in γ and a constant encounter probability p is overwhelming supported by the data. However, since we ve mentioned ĉ, what about it? If you try to run a bootstrap, or median ĉ, MARK will quickly tell you that a neither of these GOF tests are available for the Pradel recruitment only model (and indeed, they aren t available for any of the Pradel models). At first glance, it might seem that you could simply flip the encounter histories back around the normal forward direction (as if you were going to do a CJS analysis), and simply use the CJS bootstrap GOF test. However, this is inappropriate in general, since the likelihood is based on the full encounter history. Still, it may be reasonable if your only interest is in estimating the γ parameters. For the moment, we can cautiously suggest that if all you re interested in is γ, and are using the Pradel seniority only model, then since this model is identical to taking your encounter histories, flipping them, and running them through the CJS model, then the ĉ from the CJS model may be appropriate. But don t quote us (this is still a work in progress). For the other open population Pradel models ( survival and seniority, survival and lambda, and survival and recruitment ), this is likely to be incorrect. These difficulties notwithstanding, it is likely these other models that will hold the greatest interest to you, since they provide information on parameters related to the growth of a population. We will focus in particular on the Survival and Lambda and Survival and Recruitment models. Let s restart our analysis of the capsid data. This time, we will select the Survival and Lambda model. After we click the OK button, we are immediately presented with the open PIM for the apparent survival parameter f. If you open up the PIM chart (shown below),you ll see that there are 3 structural parameters involved in this model type: ϕ, p and λ. If you look at each PIM separately, you ll see that each of them consists of 1 row (the reason for this has been discussed earlier). So, again, the modeling is relatively straightforward you can apply constraints

10 13.4. Pradel models in MARK The Clogit function has 2 arguments: the first is the particular set (or pair) of parameters you want to apply the link to, and the second identifies the member of the pair. For example, CLogit(1,1) and CLogit(1,2) refer to the first pair of parameters,corresponding toϕ 1 andλ 1,respectively.CLogit(2,1) andclogit(2,2) refer to the second pair of parameters, correspondingtoϕ 2 andλ 2, respectively. And so on. Here is the completed link function specification window: Note: if you are applying the CLogit link to only a subset of theϕandλparameter pairs, you must remember to specify the log link function for the λ parameters you are not constraining using the CLogit. In this case, we specify the log link for parameters 14-20, and (shown above). If we run the model with the CLogit link function applied toϕ 8 andλ 8, we ll see that ˆϕ ˆλ However, applying the CLogit only to those parameters which are identified (based on a first analysis) as violating the logical constraint that ˆϕ i ˆλ i might appear rather post hoc. What about a priori applying a CLogit link to all of theϕandλparameters? We ll see in a moment why this is not a good idea. For now, we ll plunge ahead as if it were.

11 13.4. Pradel models in MARK Here is the completed link specification window if we apply the CLogit link function to all successive pairs of ϕ and λ parameters: Pay attention to how successive pairs of parameters are indexed the first argument in CLogit(n,p) is n=which pair, while p=which parameter in the pair. Click the OK button, and add the results to the browser. Hmmm. Something has definitely gone wrong. The first two models in the browser are the model with and without the CLogit link constraint applied to [ϕ 8,λ 8 ], respectively. However, when we apply the CLogit link to all of theϕandλparameters, we see that not only is the model deviance quite different ( versus 750), but the number of estimated parameters is much lower (10 versus 23 & 25). If we look at the estimates,

12 13.4. Pradel models in MARK we see that the problem is that many of the estimates ofλ are estimated at the boundary, even though λ was previously estimated as>1 for many of the estimates (see listing of estimates on p. 10). What has happened? Well, the answer is somewhat explicit in the name of the link function: Clogit. The cumulative logit function is still a logit link, meaning, it constrains estimates to be bounded [0, 1]. While this isn t a problem for some parameters, it is clearly a problem for parameters such asλ, which are not upper-bounded at 1. So, you could/should apply the CLogit link function to enforce the constraint thatϕ i λ i only for those pairs of parameters whereλ i 1. end sidebar So, at this point, this looks like perhaps the simplest thing we ve done so far with MARK. However, there are several issues to consider. First, several parameters are confounded under the fully timedependent model. For example, in model {ϕ t p t λ t } the first and lastλare inestimable becauseϕ 1 is confounded with p 1 andϕ k 1 is confounded with p k 1 (for k encounter occasions). Here are reconstituted estimates from the fully time-dependent model {ϕ t p t λ t }: We see clearly that the first and last estimates ofλ are problematic (very large or impossibly small SE for both estimates). We also have the usual issues of inestimability of terminal survival and encounter parameters. However, when constraints are placed on either ϕ or p, some of the variation in these parameters is taken up by λ, γ, or f, which often solves the estimability problem. For example, if you look at the estimates from a model where the encounter probability p is constrained to be constant over time (e.g.,

13 13.4. Pradel models in MARK model {ϕ t p. λ t }), you will see that all of the estimates forλappear reasonable. Unfortunately, the issue of applying constraints to models where λ is a structural parameter is somewhat more complex than the preceding example suggests. For example, Franklin (2001) suggests that sinceλ,γ and f are (often) the parameters of biological interest in the Pradel models, that it is often best to model ϕ and p as completely time-dependent, and apply the constraints to λ, γ, or f. While this might seem reasonable, there s a catch. If you do specify constraints onλ,γ, or f, you need to be a bit careful when interpreting the meaning behind constraining these parameters. Since λ i ϕ i + f i, if a model is fit with time invariant (i.e., constant)λ, or whereλis constrained to follow a linear trend, but with time varying f, then this implies a direct inverse relationship between survival and recruitment (e.g., ifλis held constant, then ifϕ i goes up, then f i must go down). While this may be true in a general sense, it is doubtful that the link between the two operates on small time scales typically used in mark-recapture studies. As noted by Franklin (2001), models where ϕ is time invariant while λ is allowed to vary over time are (probably) reasonable, as variations in recruitment are the extra source of variation in λ. More complex models involving covariates have the same difficulty. Population-level covariates (e.g., weather) are interpretable, but it is potentially difficult to interpret individual-based covariates as operating on population growth. The root of the problem is that while individual covariates could apply to survival probabilities, the recruitment parameter is not tied to any individual it is a population-based, average recruitment per individual in the population. What is needed is a generalization of the general JS model where new entrants to a population are tied to existing members of the population, for example, if newborns were identified with their parents. So, as long as you take some care, then you ll be OK. There are always challenges in modeling parameters that are clearly linear functions of each other be advised think carefully. The Pradel models have great potential, for a number of applications - if you re careful, then you can apply them to a fairly broad set of problems. We introduce once such application in the following section. An alternative approach based on random effects is discussed in Appendix D (section D.4.4).

14 13.5. an extension using the S and f parametrization an extension using the S and f parametrization... Suppose you are interested on the relative degree to which recruitment or survival influence the dynamics of a population. For those of you with any background in matrix population models, this is an old concept: you have a metric describing the growth of the population (λ), and you want to determine the degree to which λ is sensitive to variation in one or more elements of the demography of the population. For the matrix modelers among you, such a sensitivity analysis is usually expressed in terms of the rate of change ofλgiven a certain change in one of the matrix elements (a ij ). Done on a log scale, this sensitivity analysis becomes an elasticity analysis (the log scale expresses relative proportional contributions to λ). Now, in the typical sensitivity or elasticity analysis, the point is to determine what would happen to growth if a specified change is made in one or more vital rates. So, a prospective analysis. In the prospective context, sensitivity and elasticity are together referred to as perturbation techniques, since the goal is to see how much projected growth would change as the system is perturbed by changing one of the vital rates which contributes to population growth. There is a very large literature on the application of prospective perturbation techniques in conservation and population management. However, in the retrospective context, the story is a little bit different. In this situation, we have an observed time series ofλ i, which we might estimate from our mark-encounter data. We want to know what the relative contributions of ϕ and f have been to the observed variation in λ. In other words, what has driven the basic pattern of variation in λ over time. Several years ago, Jim Nichols and colleagues addressed this very question. The approach they developed is (typically) very intuitive, and the result is rather elegant. The basic idea is that sinceγ i+1 is the probability that an individual in the population at time i+1 (and thus contributing to N i+1 ) was also there at time i (and thus included in N i ), and since N i+1 N i ϕ i + B i, then N i ϕ i N i+1 γ i+1 and B i ( 1 γ i+1 ) Ni+1 So, sinceλ i N i+1 /N i, then E(λ i ) [ γi+1 N i+1 + ( 1 γ i+1 ) Ni+1 ] E(N i ) Since the abundance term, N i+1, is the same for both product terms in the numerator, then theγ i+1 in the first term in the numerator is interpretable as the contribution of survivors from time i to time i+1, while the (1 γ i+1 ) in the second term of the numerator is the contribution of new recruits into the population. So, the two terms give the relative contributions of survivors and new recruits to λ, which is conceptually analogous to the elasticities of both terms. The details (and several very clever extensions to multi-state and robust design models) are discussed at length in the Nichols paper. However, this approach involves doing algebra with estimates of γ only. A more direct, and ultimately more flexible approach is to re-parameterize the likelihood in terms of survival and recruitment directly. This is the basis of the Survival and Recruitment model. Re-start the analysis of the capsid data set, this time selecting the Survival and Recruitment model. If you look at the PIM chart, you ll see that there are only 3 structural parameters:ϕ, p and f. For this model type, the realized growth λ is estimated as a derived parameter (meaning, it is derived by algebra, outside the likelihood). This has one immediate implication, which refers back to our earlier discussion of applying constraints to Pradel models. Because λ is a derived parameter (not in the likelihood), you cannot put a constraint on Nichols, J. D., J. E. Hines, J-D. Lebreton, R. Pradel. (2000) Estimation of contributions to population growth: a reverse-time capture-recapture approach. Ecology, 81,

15 13.5. an extension using the S and f parametrization λ. Rather, you might apply constraints to the processes which contribute to λ (i.e., S and f ). Let s proceed by fitting a single model to the capsid data: model {ϕ t p. f t }. We ll use constant encounter probabilities over time to eliminate some of the confounding problems inherent in fully time-dependent models. Here are the reconstituted parameter estimates from this model and the reconstituted estimates of λ (on the real scale): To reinforce the relationship between survival, recruitment, and λ, compare (i) the derived estimate of λ provided by MARK (above), and (ii) the value of the sum of the estimates of survival and recruitment (shown at the bottom of the preceding page). Since λ i ϕ i + B i N i ϕ i + f i

16 13.5. an extension using the S and f parametrization then (i) and (ii) should be identical, as they are shown to be (to within rounding error) in the following table for a sub-sample of intervals: interval derived ˆλ ˆϕ f ˆ ˆϕ + ˆ f We could express the proportional contribution of (say) survivalϕ to realized growthλsimply as ϕ/(ϕ + f ). The variance of this proportion can be estimated using the Delta method (Appendix B). An important point to make here is that interpreted as elasticities in this fashion, the approach Nichols and colleagues describe partitions variation in realized λ due to contributions from survivors and recruits, without the (somewhat) limiting assumptions of some matrix models approach (there is a growing literature debating whether or not some of the ergodic assumptions underlying matrix methods really are a limit, but that isn t something we want to get into here). One final point: note that all we ve done up until now is talk about net additions and subtractions. We haven t partitioned these any further. For example, we haven t partitioned additions into in situ recruits and immigrants. We may, in fact, not be satisfied with simply using a summary accounting like total recruits or total subtractions - we may want to know how many of each are due to underlying, lower-level processes (like births, immigration, deaths, or emigration). However, to do that, we d need to consider different approaches: the Jolly-Seber (and related) models (introduced in Chapter 12), and the robust design (which we introduce in Chapter 15). But, if partitioning λ into summary contributions of total recruits and total losses is what you re after, then the Pradel models may be of some use. λ = survival + recruitment : be careful! begin sidebar Another example of a potential pitfall. In the preceding, we made use of the fact that realizedλcan be estimated as the sum of survival and per capita recruitment, both estimated over a given interval. However, you need to be careful in how you are handling recruitment. Consider, for example, a population with 2 age classes: babies, and adults. Assume that adults start breeding at age 2 years (i.e., they don t breed as yearlings), and on average produce C babies. Babies survive with probability S o, and thus become adults, which survive with probability S a. Assuming the population is censused after breeding, the population can be described by the lifecycle diagram shown on the next page, where node 1 refers to babies, and node 2 refers to adults (age 1 yr and older). The self-loop on node 2 indicates that survival does not vary with age among adults. The fertility arc (connecting node 2 back to node 1) represents the expected contribution of each individual in node 2 at time (i) to the baby age class at time (i+1). If we letρ ϕ/(ϕ + f ), then var( ˆρ) fˆ ( ˆϕ + f ˆ ) 2 fˆ ˆϕ ( ˆϕ + f ˆ var( ˆϕ) ĉov( ˆϕ, ˆ f ) ) 2 ĉov( f, ˆ ˆϕ) var( f ˆ ( ˆϕ + f ) ˆ ) 2 ˆϕ ( ˆϕ + f ˆ ) 2 where the variance and covariance of ˆϕ and ˆ f can be output from MARK (see Appendix B).

18 13.6. average realized growth rate the temporal symmetry approach to estimating λ is relatively simple, you do need to pay close attention to what is going on (which we suppose is a general truism for most things, but we ll make the point again here). end sidebar average realized growth rate Following estimation of the time-specific realized growth rates,λ i,there is natural interest in the average growth rate over the course of the study. You might recall from section 6.14 in Chapter 6 that estimating the average for a parameter can be somewhat more complicated than you might expect. This is especially true in the present situation, where we are interested in estimating ˆ λ. Here, the complication is that we re calculating the mean of the ratio of successive population sizes, where the population sizes at each time step are outcomes of an underlying, geometric stochastic generating process. As such, the most appropriate average to report is the geometric mean of the individual ˆλ i, not the more familiar arithmetic mean. You might recall that the geometric mean of a set of n numbers {x 1, x 2,...,x n } is given as y n n An important result is that unless the set {x 1, x 2,...,x n } are all the same number (i.e., {x 1 x 2 x n }), then the geometric average is always less than the arithmetic average. [Note: if the geometric mean, in general, is new to you, it is worth consulting the following- sidebar - before proceeding much further.] i 1 x i begin sidebar arithmetic mean, geometric mean, and population growth... The first statistical calculation you usually learn how to do is the computation of an average. This is how the teacher comes up with grades (in many cases) so it s a basic bit of math most students have an inherent interest in. What you learned to calculate is what is known as the arithmetic average. For example, given three random values x 1, x 2, and x 3 then arithmetic mean y is y x 1 + x 2 + x 3 3 However, one of the most important averages in ecology is something known as the geometric mean, or geometric average, which given the same three random values x 1, x 2, and x 3 is y 3 x 1 x 2 x 3 In other words, the geometric mean of three numbers is the cube-root of their product. If you have n numbers, the geometric mean is y n n i 1 ( 1n ( )) log(x) x i e

19 13.6. average realized growth rate For example,let x 1 2, x 2 2.5,and x 3 4. Given these values,the arithmetic mean is ( )/ , whereas the geometric mean is y Consider the simplest possible case of two numbers: x 1 and x 2. Unless x 1 and x 2 are the same number, then the geometric average is always less than the arithmetic average. Here is a proof attributed to Cauchy. Assume we have 2 values: x 1 and x 2. Assume that x 1 x 2. Thus x 1 x 2 0, and ( x1 x 2 ) 2> 0 x 2 1 2x 2 x 2 + x2 2 > 0 [add (4x 1 x 2 ) to each side] x x 1 x 2 + x2 2 > 4x 1 x 2 (x 1 + x 2 ) 2 > 4x 1 x 2 ( ) x1 + x 2 2 > x 1 x 2 2 x 1 + x 2 2 > x 1 x 2 where the LHS is the arithmetic mean of x 1 and x 2, and the RHS is the geometric mean of the same two values. The LHS (arithmetic mean) is greater than the RHS (geometric mean). Q.E.D end sidebar Now, why do we care about the distinction between an arithmetic and geometric mean? We care because the geometric mean is the appropriate average for stochastic population growth. We ll illustrate the reason why we make this statement, by means of a simple numerical example. Consider two successive years of population growth, withλ 1 being the growth factor for the first year, andλ 2 being the growth factor for the second year. Then, the population size after the first year, starting at size N 0, is N 1 λ 1 N 0 after 1 year while after 2 years, the population size is given as N 2 λ 2 N 1 λ 2 λ 1 N 0 after 2 years Thus,theλ i values for each year i are simply multiplied together when projecting of geometric growth through time. What can we use this for? Well, let s step back a moment, and recall from some population biology class you might have taken that under time-invariance, such that λ is constant over time, we can write N t N 0 λ t

20 13.6. average realized growth rate Let t 3. Thus N 3 N 0 λ 3 N 0 (λ λ λ ) Clearly, in this case the growth rate over each year is the same, and is given by λ. So, the expected change in size over t 3 time steps is simply the average annual growth rate, raised to the 3 rd power. And, thus in reverse, the average growth rate in a given year would simply be the 3 rd root of the product of the individualλ values. For example, if N 0 10, and N , then since N t N 0 λ t under time invariance, thenλ 3 ( /10) , and thus the average growth rate (again, assuming time invariance) is simply Now, consider the more typical case whereλvaries from year to year. Ifλ 1 is the geometric growth factor for the first year,λ 2 is the geometric growth factor for the second year, andλ 3 is the geometric growth factor for the third year then N 3 λ 3 λ 2 λ 1 N 0 So, what is the average growth rate over the 3 years? Consider the following simple numerical example: letλ ,λ , andλ The arithmetic mean of these three growth rates is Is this the appropriate growth rate to project population size in 3 years? Let s see what happens. Let N 0 =10. Thus, after three years, we project the population size in 3 years will be N 3 ( ) Now, if instead we had used the arithmetic average in the projection equation, we would project the population size after three time steps to be ( ) , which is higher than it should be (10.405>10.393). But, what if instead we use the geometric mean? The geometric mean of our annual growth rates is 3 3 ( ) So, the projected size of the population after three time steps, using the geometric mean growth rate, would be ( ) , which is exactly what it should be, based on our earlier projection using the individualλ i values. So, the constant-environment equivalent of the fluctuating environment is an environment with a constantλthat is the geometric average of theλ s in the fluctuating environment. We refer to this mean stochastic growth rate asλ S. And, as noted earlier, the geometric mean is always smaller than the arithmetic mean, and so, we expect that the stochastic growth rate will be lower than the deterministic growth rate (this is a very important result in population biology). OK, back to Pradel models, and calculating the most appropriate average growth rate over the time series of estimated growth rates, ˆλ i. From the preceding, we see that using a simple arithmetic mean calculated over the set of ˆλ i would be incorrect it would overestimate the stochastic growth rate, which is more appropriately estimated using the geometric mean. How can we estimate the geometric mean growth rate using MARK? In fact, you have two options. You can either (i) derive estimates of the geometric mean over the set of ML estimates of ˆλ i (a fairly straightforward exercise using a spreadsheet of the estimates), or, (ii) you can use derived estimates of λ on the log scale. Why is this second approach useful? Recall again from some earlier population biology class that λ e r, where r is the instantaneous (intrinsic rate) of growth, whereasλis the ratio of population sizes at 2 discrete points in time.

21 13.6. average realized growth rate We can write the stochastic growth rateλ S over T time steps as λ S ( λ 1 λ 2 λ 3 λ T ) 1/T ( e r 1 e r 2 e r3 e ) r 1/T T which, taking logs, can be rewritten as lnλ S r 1 + r 2 + r 3 + r T T So, we can calculate lnλ S simply as the arithmetic mean r. Let s explore application of these algebraic relationships using the capsid data set we introduced earlier in this chapter. Using the Pradel survival and Lambda data type, we ll fit model {ϕ t p. λ t } to the data, using the CLogit link applied to parametersϕ 8 andλ 8 to enforce the logical constraint that ϕ i λ i (this was discussed in the- sidebar - starting on p. 9 of this chapter). Here are the estimates of realized growth rate for the 12 intervals for the capsid data set: For our subsequent calculations, we ll exclude the estimate over the first interval, ˆλ , as being suspicious (see earlier discussion on the potential for bias and confounding for initial estimates from Pradel models). Now, how do we calculate the best estimate of average growth rate? From the preceding discussion, we know that the simple arithmetic mean of the set of estimates ˆλ 2 ˆλ 12 (1.0074) would be incorrect as a measure of this average growth, since it would overestimate the average expected stochastic growth of the population over any given interval. Instead, we should focus on the geometric mean of the estimates. As also noted on the preceding page, we can do this in one of several. First, we could simply take our set of estimates ˆλ i, export them to a spreadsheet, and calculate the geometric mean by hand. Alternatively, and perhaps more conveniently, we could export the derived estimates of ln( ˆλ i ) from MARK, and take the arithmetic mean of these derived estimates, recalling that ln(λ i ) r i, and that ln(λ S ) is simply the arithmetic mean r. Recall that MARK generates estimates of ln( ˆλ i ) as derived parameters for all of the Pradel data types, which makes this approach very straightforward. Simply select Output Specific Model Output Parameter Estimates Derived Estimates Copy to Excel. The exported estimates are shown at the top of the next page. The first 12 rows are the estimates of ˆλ i, while the next 12 rows are the same estimates, but reported on the log scale, ln( ˆλ i ).

22 13.6. average realized growth rate For example, ln( ˆλ 2 ) ln( ), where ˆλ We focus here on the set of estimates ln( ˆλ 2 ) ln( ˆλ 12 ), shown above with light green shading. The arithmetic average of this set is r ln(λ S ) Thus,our estimate of the stochastic,geometric growth rate on the real scale is ˆλ S e ( ) , which as expected is less than the arithmetic mean of Not only that, the geometric mean suggests a slow decline in population size over the long-term, whereas the arithmetic mean suggests a slow increase in population size. At this point, you may (and should) also be wondering about the variance of our estimated stochastic growth rate. We could simply take the variance calculated on the log scale, and using the Delta method (see Appendix B), back-transform our estimated variance to the usual real scale. But, recall from section 6.14 in Chapter 6 that such an approach is generally biased, since it fails to take into account the conditional sampling variances of our time-specific estimates of growth rate. The preferred approach is to use a random effects, variance components approach, which could be implemented using either a moments-based approach (Appendix D), or using MCMC (appendix E). For (relative) simplicity, we will briefly demonstrate the steps for the using the moments-based method. We will fit a random effects intercept only (i.e., mean) model to the capsid data. The intercept only model assumes that the parameters in the model (say,λ i ) are drawn from some underlying distribution specified by meanµand varianceσ 2 (see Appendix D for complete details our intent here is to demonstrate the mechanics only). First, retrieve the model {ϕ t p. λ t } in the browser. Then, select Output Specific Model Output Variance Components Derived Parameter Estimates. This will spawn another window (shown at the top of the next page) asking you whether you want to use ˆλ i ( Lambda Population Change, or ln( ˆλ i ) ( log(lambda) Population Change ) in the calculations.

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