PVR_invas.R User’s Guide

Brett van PoortenBritish Columbia Ministry of Environment, Conservation Science

Section

2202 Main Mall, Vancouver, BC, V6T 1Z4

Brett.vanPoorten@gov.bc.ca

December, 2017

Table of Contents

Introduction.............................................................................................................................. 3

Model description................................................................................................................... 3

Parameters and controls.................................................................................................... 11

Running the model............................................................................................................... 12

Evaluating a single removal strategy..........................................................................................12

Evaluating multiple removal strategies....................................................................................14

References............................................................................................................................... 20

Introduction

PVA_invas.R is a population viability analysis simulator designed to quickly and

accurately reflect uncertainty in the fate of invasive populations. Specifically, the

model evaluates how candidate removal strategies targeting different life stages will

affect population persistence. Further, given a list of potential removal strategies

and removal gears or combination of gears, the model can run each and provide a

multicriteria utility decision table where factors such as probability of eradication,

cost and probabilistic range of abundance after nT years is reported. Each model

evaluation is relatively rapid (~ 6 seconds for 1000 simulations of the smallmouth

bass simulation described here), so the model can be used in a workshop-type

discussion to evaluate different options in real-time, promoting effective

communication among stakeholders as to the efficacy of all methods.

The model can be used for any species or population with indeterminate growth

(e.g. most fish, reptiles, amphibians and invertebrates). Populations are separated

into two life stages: pre-recruits and juveniles/adults. Pre-recruits are subject to

density-dependent mortality and can in turn be separated into multiple stanzas,

where the strength of density dependence in each stanza is user defined. Recruited

animals are age-structured; mortality of these animals decreases with length. Age at

recruitment can be adjusted and the number of stanzas can be increased to

represent prolonged density dependent mortality over multiple juvenile ages.

This user’s guide first describes the biological model so user-provided inputs can be

understood within the context of the model. User-provided parameters and controls

are then defined. Finally, a step-by-step guide to using the functions is provided

with examples to show users how to evaluate new populations with the model.

Model descriptionAll model controls, indices, parameters and derived variables are listed in Table 1;

equations are listed in Table 2. The model begins by establishing stock-recruitment

parameters from calculations of equilibrium population structure when

unexploited. Length- (T1.1), weight- (T1.2) and fecundity-at-age (T1.3) are first

calculated. Instantaneous mortality of recruited animals is assumed to be inversely

related to length, following the Lorenzen (2000) model. Asymptotic mortality rate is

calculated from A, the maximum age observed in the population, which is assumed

to have a lifetime survivorship of 1% (i.e. 1% of recruited animals live to A). Back-

calculation of M∞ proceeds similar to the method of Hoenig (1983; T1.4). Survival in

each time-step follows Lorenzen (2000; T1.5), which is used to calculate

survivorship (T1.6) to each age. The product of survivorship and fecundity-at-age

for females provides equilibrium spawners per recruit (T1.7), which is used to

calculate Beverton-Holt recruitment parameters (Walters and Martell, 2004).

Beverton-Holt parameters are modified by relative mortality and habitat capacity

for each pre-recruit stanza (T1.8-T1.9) to give stanza-specific Beverton-Holt

recruitment parameters (which scale mortality by density of competitors within a

cohort; Walters and Korman 1999, Pine et al. 2013). If cannibalism is thought to

occur in pre-recruit stanzas (i.e. pcann>0), the maximum survival of each stanza is

modified as a log-linear relationship with cannibal abundance (where cannibalistic

ages are specified by the user; T1.12-13). This formulation is based on the

reformulation of the Beverton-Holt model of the form (Walters and Korman, 1999)

(1)R=

N 0 e−M 0

1+M1

M 0 (1−e−M 0 )

where e−M 0 is maximum survival at low spawning stock and M 1

M 0(1−e−M 0 ) is the

carrying capacity parameter. In the case of cannibalism, M 0 can be modified as a

function of likely cannibalistic age-classes:

(2) M 0=ρ+τC

Table 1: Model controls, indices, parameters and derived variables used in PVA_invas. Parameter values shown are biological parameters used for the smallmouth bass example.Symbol Value

Spreadsheetname Description

Controlsdt 0.25 dt Length of time-step (years)nT 50 nT Number of time-stepsns 2 nS Number of stanzasAR 1 AR Age-at-recruitmentng 2 n.gear Number of capture gears for recruited animalsnsim 1,000 n.sim Number of population simulationssampt 1 samp.t Time-step when each control gear for recruited animals is usedr 0.05 r Standard economic discount rateG 20 G Human generation time (set large to ignore inter-generational discounting

qR 0.1; 0.1 q.R Maximum catchability of each pre-recruit gear (proportion of stanza captured per unit sampling effort at low abundance)

qA 0.1, 0.1 q.A Maximum catchability of each size-selective gear for recruited animalststart 1 t.start Time-step when sampling begins (allows for delayed start time)EfR,s 0; 0 E.R Sampling effort per time-step for pre-recruit stanzasEfA,g 0; 0 E.A Sampling effort per time-step for recruited animals

va 25; 15 v.a Proportional to rate of increase of ascending limb of dome-shaped selectivity function

vb 0.7; 0.5 v.b Proportional to length at 50% selectivity on ascending limb of selectivity functionvc 0; 0.5 v.c Proportional to rate of decline on descending limb of selectivity function (0 < vc < 1)Cf,R 2,000; 2,000 C.F.R Annual fixed cost of using each pre-recruit removal gearCf,A 2,000; 2,000 C.F.A Annual fixed cost of using each post-recruit removal gearCE,R 100; 100 C.E.R Cost per unit effort for each pre-recruit removal gearCE,A 100; 100 C.E.A Cost per unit effort for each post-recruit removal gear

Indicest {1, 2, …, nT/dt} Time-step (can be sub-annual if dt < 1)a {AR/dt,…,A/dt}dt Age from age at recruitment to oldest age-class in dt stepsi {1, 2, …, Et} Individual numbers {1, 2, …, ns} Recruitment stanzag {1,2,…,ng} Capture gear for recruited animals

Model parametersR0 1000 R0 Unexploited recruitsκ 9.24 reck Compensation ratio in recruitmentpcann 0.2 p.can proportion of pre-recruit mortality at equilibrium due to cannibalismK 0.25 K Metabolic rate parameter of von Bertalanffy functionA 15 A Maximum age (e.g. probability of survival to age A=1%)a f 6,000 afec Fecundity multiplier on weightwm 0.4 Wmat Weight at maturity relative to asymptotic weightt spn 0.0; 0.1 t.spn Start and end of spawning time as proportion of yearM s

¿5; 8 Ms Maximum survival for each pre-recruit stage (relative values)

Bs¿ 10; 9 Bs Available habitat for each pre-recruit stage (relative values)

V 1 200 V1 Initial number of vulnerable animalscanna {5,…,15} cann.a Ages that may be cannibalistic on pre-recruits (if pcann>0)σ R 0.5 sd.S Standard deviation in recruitment

Derived variablesla Lengthwa WeightM∞ Minimum instantaneous mortalityf a Fecundityspnt Time-steps when spawning occurs (0/1 flag)

spna Ages when spawning occurs (0/1 flag)Sa Survivalva ,g Selectivity of gear-glxa Survivorshipφ0 Unexploited eggs-per-recruitα ¿ Maximum survival of Beverton-Holt recruitment functionβ¿ Carrying capacity parameter of Beverton-Holt recruitment function for each stanza-sα s

¿ Maximum survival of Beverton-Holt recruitment functionβ s

¿ Carrying capacity parameter of Beverton-Holt recruitment function for each stanza-sRt Annual recruitment to age-AR

N t ,s Population abundance for pre-recruited animals in year-t and stanza-sN t ,a Population abundance for recruited animals in year-t and age-aEt Annual egg productionC t , g Annual catch by gearqN Density-dependent catchabilityF t , a Instantaneous sampling mortality rate by age-class

Table 2: Functions used to initialize the population viability analysis model

T2.1 la=1−e−Ka

T2.2 wa=la3

T2.3 f a=af (wa−wm )

T2.4M∞=

ln (0.01 ) K

ln (l a=A R )− ln [la=A R+e

K (A−AR )dt −1]

T2.5 Sa=( lala+e

K ∙ dt−1 )M∞

K

T2.6 lxa={ 1 a=AR

∏a=A R

a−1

Sa a>AR

T2.7 φ0=∑a=1

A lxa f a spna2

T2.8 α s=κφ0e( M s

∑ M s)

T2.9βs=B s

¿ κ−1

R0φ0∑s'

( βs'¿ ∏s' '=0

s' '=s'−1

α s' ')T2.10 M s

0=− ln (α s )=ρ s+τ s[R0 ∑a=canna

(lx a spna ) ]T2.11 M s

1=βsM s

0

1−α s

T2.12 ρ s=M s0 (1−pcann )

T2.13 τ s=M s0 pcannR0 ∑

a=canna(lx a spna )

T2.14 Et=1=∑i=1

N t=1

f a

where ρ is mortality independent of cannibals, τ is the cannibalism-dependent

parameter and C is the sum of animals in age-classes where cannibalism may occur

(user defined). This altered formulation results in approximately Ricker-type

recruitment (e.g. overcompensation at high spawner abundance).

Abundance of recruited animals in the first year is determined in one of two ways. If

the population is close to carrying capacity (i.e. V 1≥0.9 R0∑ lxa), abundance is

randomly allocated among age-classes assuming a multinomial distribution

(3) N t=1, a MN (V 1 ,lx aeε a );εa N (0 , σR )

where random year-class strength is given by ε a. This approximation is useful if the

population is close to carrying capacity but will under-represent early year-classes

if the population is still growing because of the steady-state assumption.

Additionally, if initial abundance is low, there is a chance that all mature year-

classes will be empty, causing a lag in egg production in early years; this will result

in population growth being low in the first several years (depending on generation

time). Therefore, if the initial abundance is below 90% of carrying capacity, initial

age-structure is established by deterministically simulating the population from a

low abundance (starting with 10e-15 recruits) until the population reaches the user

defined starting abundance (V1). This initialization proceeds by first calculating the

abundance of each age-class

(4) N t ,a¿ =N t−1 ,a−1

¿ Sa−1

where N* refers to numbers in the initialization. Eggs (Et ,a¿ ) in each initial time-step

are calculated as the product of numbers-at-age and fecundity-at-age (T1.3) and

recruitment is calculated using the stage-independent Beverton-Holt function

(5)N t ,a=1

¿ =E t−1

¿ e−(ρ+τ C t )

1+ e−( ρ+τ C t )

M 1(1−e−(ρ+τ C t) ) ,

where

ρ=−ln( κφ0 )(1−pcann ),

τ=−ln ( κφ0 )pcann

R0 ∑a=canna

(lx a spna)

and

M 1=

(κ−1 ) ln( κφ0 )R0 (κ−φ0 )

.

These parameters are the stage-independent analogues to T1.11-T1.13. Regardless

of how the population is initiated, egg production in the first year will be given by

T1.14.

Recruitment in subsequent years is based on survival through each pre-recruit

stanza. Survival from one stanza to the next is given by the product of focused

removals and the stanza-specific stock-recruitment function

(6) N t ,s+1 BIN ¿,

where

(7) M s , t0 = ρs+τ s ∑

a=canna(N t , a )

and

(8) F s=qs Es;

that is, the product of capture efficiency of gear focused on stanza-s and the units of

effort used each time-step. Survivors from the last pre-recruit stanza become

N t+AR−1 ,a=AR.

Survival of each recruited age-class each time-step is given by the Baranov equation

(Ricker, 1975)

(9) N t+1 ,a+1 BIN (N t ,a , e−Z t ,a )

where

(10) Zt ,a=M a+Ft ,a

and

(11) F t , a=∑g=1

ng

qN , gE t , g va , g.

Et , g is set to zero except for time-steps corresponding with time of year that gear is

used, which is specified by the user.

Each control gear is described by a density-dependent constant of proportionality

(termed catchability in fisheries literature) that relates catch-per-unit effort to

abundance, or equivalently, removal effort to instantaneous removal mortality

(Arreguin-Sanchez, 1996). Catchability may be density dependent for a number of

reasons, but a likely mechanism in aquatic invasive species in due to the change in

individual home range size as the population grows (e.g. the population is more

densely packed as it fills the population spatial range). This results in non-linear

increase in invaded area with abundance. The area occupied by the population is

determined by

(12) AN=γ N β,

where γ is the density-independent rate of increase in the area occupied by the

population as the population grows and β is the rate at which animals reduce their

density with population abundance. Catchability is the proportion of area affected

by the removal gear

(13) qN=δAN

,

where δ is the area swept by one unit of removal gear. These two equations can be

combined into the standard equation for hyperstability:

(14) qN=q N−β.

The q parameter is probability of capture at low abundance. To determine the two

unknown parameters φ and β , it is best to think about how catch per unit effort

would change as the population grows. For example if fishing with a single unit of a

gear at population abundances N1=50 and N2=200 were to yield CPUE1=5 and

CPUE2=15, β could be calculated as

(15) β=1− ln (5 )−ln (15 )ln (50 )−ln (200 )

=0.207.

Noting that qN=CPUEN , Eq. 15 can be substituted into Eq. 14 to give q=0.225.

Equations 6-11 are repeated for nT years. This entire procedure is repeated n¿

times; The proportion of n¿ simulations that resulted in eradication by the end fo the

simulation (nT) gives the probability of extirpation given the sampling procedures

and removal effort specified by the user.

Parameters and controlsThe biological population viability function in the R script (the PVA() function) runs

given inputs of controls and parameters. Parameters for the population being

modelled are set in a .csv file with the naming convention “species parameters.csv”

(e.g. “SMB parameters.csv”); Controls are similarly set in a separate .csv file (e.g.

“SMB controls.csv”). The prefix to these file names (“SMB” in this example) are used

to differentiate populations/species. This prefix is called when running the R script.

Parameters define the biology of the population being modeled and are described in

Table 1. Parameter values listed in Table 1 are those used for the smallmouth bass

(Micropterus dolomieu) case study shown below, but another population or species

could be quickly substituted. In cases where multiple values are indicated for a

single parameter, they must be separated in the spreadsheet using a semi-colon (e.g.

Ms: 5;8). The number of values provided for Ms and Bs must equal the number of

pre-recruit stanzas (nS). Any number of age-classes can be provided for canna as long

as they are between AR and A.

Controls are used to define important indices (e.g. nT, ns, etc.) and provide important

changes to the model associated with removal intensity and cost of removal (Table

1). As in parameters, multiple values in the spreadsheet must be separated using a

semi-colon (e.g. q.R: 0.1; 0.1). The number of values provided for q.R, E.R, C.F.R and

C.E.R must equal the number of stanzas (nS); the number of values provided for q.A,

E.A, C.F.A and C.E.A must equal the number of gears used to sample recruited age-

classes (ng).

Running the model

Evaluating a single removal strategyThere are two functions of importance to most users of PVA_invas.R:

heat.proj() and decision(). Other functions (getLHpars(), set.pars(),

init(), PVA(), show.vuln() and vwReg2()) support the main functions (e.g.

reading values from .csv files), but understanding them is not needed to run the

model.

The first step in running the model is providing parameters in the .csv files. Use

the .csv files provided as a guide. The naming conventions are important, especially

within the files themselves. For example, all names provided in column A of the

controls and parameters .csv files must be maintained, as they correspond to

parameters and indices called within the R code. Controls should be relatively easy

to determine and multiple control options can be evaluated in the Scenarios .csv file.

Parameters may be more difficult to determine, but most can can be obtained from

either the literature or a good biological understanding of the species.

To begin running the PVA_invas.R script, open an R console (or an R user interface

like RStudio) and source the program PVA.R (source('C:/PVA_invas.R')).

This loads the script into memory so R can be run quickly.

A simple model run can be completed and visually evaluated by entering

heat.proj(species=”SMB”). This calls the heat.proj() function using baseline

controls and parameters provided in ‘SMB controls.csv’ and ‘SMB parameters.csv’,

respectively. An optional vector of R0 values of length nsim can also be entered if the

asymptotic recruitment density (carrying capacity) is uncertain (e.g

heat.proj(species=”SMB”,R0s=runif(1000,1000,10000)). The

heat.proj() function (details below) calls the init() and PVA() functions, which

calculate the population initial parameters and size-structure and subsequent

trajectories across each of nsim simulations. The probability of population extirpation

is reported and results are sent to vwReg2(), which plots the distribution of

population projections over T years. Model output and plot can be saved to an object

such as pva by entering pva<-

heat.proj(species=”SMB”,R0s=runif(1000,1000,10000). pva is now a

list with entries: dat and plot. The population projection plot can be drawn by

entering pva$plot. pva$dat is also a list including objects described in Table 3.

To view any of these objects, simply enter pva$dat$NAME where NAME is the

object name.

> source('~/Dropbox/Invasive removal/PVA_invas.R')New species or populations can be evaluated by changing parameters.To update parameters, controls, effort, etc., update your csv filesType 'heat.proj(species,R0.vec) to plot results for a single parameter.Type 'decision(species,R0.vec)' to run all scenarios and evaluate.

> pva<-heat.proj(species="SMB",R0.vec=runif(1000,1000,10000),set.ymax=50000)Initializing populations ...Time difference of 0.6698229 secsCalculating population projections ...Time difference of 5.941278 secsProbability of extirpation after: 12.5 years - 0 % 25 years - 0 % 50 years - 0 %Computing density estimates for each vertical cut ... |===========================================================================| 100%Build ggplot figure ...>

Controls can be varied to evaluate different removal strategies. For example, q.R is

used to set the proportion of each pre-recruit stanza that can be removed with each

unit of sampling effort; E.R sets the number of sampling units used per year and

C.f.R and C.E.R set the fixed annual costs and effort-based costs of using each

sampling gear each year. Setting E.R=0 means no removal strategy is used on the

stanza being referred to. Likewise, q.A sets the proportion of vulnerable recruited

animals that can be removed per unit of each sampling gear focused on recruited

animals. The number of different gears (or the same gears used at different times of

year (samp.t)) is set using n.gear. Similar to pre-recruit gear, E.A, C.f.A and C.E.A set

the amount of effort, as well as fixed and effort-based costs for each gear focused on

recruited animals. The dome-shaped length-based selectivity function is set for each

gear using v.a, v.b and v.c. Finally, a time delay in control measures can be examined

by using t.start > 1, which allows users to evaluate how delayed response may affect

the final abundance.

Evaluating multiple removal strategiesTo evaluate a new removal strategy, values in the control.csv file must be updated,

saved and heat.proj() re-ran. Alternately, multiple removal strategies can be

compared against one another using decision(). The decision() function will

evaluate as many combinations of sampling methods as necessary and provide a

multi-attribute decision table that shows annual cost, net present value of sampling

until the time series is complete or the population is eradicated, time to eradication

and final abundance (if the population persists). Running decision() requires

creating a scenarios.csv (using the same naming convention as above, i.e. ‘SMB

scenarios.csv’). The scenarios.csv contains only controls that differ between

scenarios. For example, if only effort varies across scenarios, that is all that needs to

be included. The controls.csv must indicate the maximum number of gears that will

be used (even if some are not used in some scenarios) and include catchability (q.R;

q.A), effort, fixed and effort-based costs for all gears.

Now you are ready to run your own invasive species eradication evaluation. I have

purposely left my raw code online so users with more advanced R skills can

customize the functions to suit their needs.

Table 3: Objects returned in PVA$datObject DescriptionA.s Stage-specific maximum survival of Beverton-Holt function

B.s Stage-specific carrying capacity parameter of Beverton-Holt function

cost.1 Annual cost of sampling

cost.T Total cost of sampling up to end of time series or time to 100% eradication

Et Matrix of eggs produced per year for each simulation

NPV Net present value of sampling, accounting for intergenerational discounting

Nt Abundance array (dimensions: time-steps, ages, simulations)

NT Abundance in the final time-step (reported as 5th percentile, mean and 95th percentile of distribution)

p.extinct Vector of proportion of simulations eradiated for each time-stepp.extinct.50 Proportion of simulations eradicated by 50th time-step

p.extinct.100 Proportion of simulations eradicated by 100th time-step

p.extinct.200 Proportion of simulations eradicated by 200th time-step

phie Equilibrium eggs-per-recruit

R.A Beverton-Holt maximum survival parameter (stage-independent)

R.B Beverton-Holt carrying capacity parameter (stage-independent)runtime Time to complete simulationt.extinct Years to 100% eradicationVfin Vector of total abundance in the final year across simulations

Figure 1: Smallmouth bass trajectory distribution plot using values described in

Table 1. Hot colours indicate higher density of population abundance probability.

The first column is a list of scenario names, and all subsequent columns are controls

that will change among scenarios. As an example, Figure 2 shows a scenarios.csv file

created for smallmouth bass removal. In this example, there are two pre-recruit

stanzas and two gear to remove recruited fish (the vulnerabilities for each are

defined in the controls.csv file). In this example, 24 units of removal effort will be

applied, but which gear and stanza is targeted is varied. Additionally, I compare

starting removal immediately with delaying for 5 years (20 time-steps). The

decision() function runs each scenario in turn and produces a decision table

(Figure 3).

> decision(species="SMB",R0.vec=runif(1000,1000,10000))Initializing populations ...Time difference of 0.267473 secsScenario: now juv1Calculating population projections ...Time difference of 5.390681 secsScenario: now juv2Calculating population projections ......Scenario: wait allCalculating population projections ...Time difference of 6.116888 secsDrawing decision table>

Figure 2: Screenshot of SMB scenarios.csv, where the only factors that differ among

scenarios is relative effort applied to each of four removal gears (targeting two pre-

recruit stanzas and two methods for recruited animals) and a time delay.

Figure 3: Decision table produced from smallmouth bass removal scenarios shown

in Figure 2. Labels on left refer to scenarios named in column A of ‘SMB

scenarios.csv’. Total cost is net present value to the end of the time series or total

100% eradication (whichever comes first).

ReferencesArreguin-Sanchez, F., 1996. Catchability: assessment a key parameter for fish stock.

Rev. Fish Biol. Fish. 6, 221–242.

Hoenig, J.M., 1983. Empirical use of longevity data to estimate mortality rates. Fish.

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Lorenzen, K., 2000. Allometry of natural mortality as a basis for assessing optimal

release size in fish-stocking programmes. Can. J. Fish. Aquat. Sci. 57, 2374–

2381.

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Walters, C., Ahrens, R., Vanhaverbeke, R., Stone, D., Wilson, W., 2013. An

individual-based model for population viability analysis of humpback chub in

Grand Canyon. North Am. J. Fish. Manag. 33, 626–641.

Ricker, W.E., 1975. Computation and interpretation of biological statistics of fish

populations. Bull. Fish. Res. Board Canada 1913890, 382pp.

Walters, C., Korman, J., 1999. Linking recruitment to trophic factors: revisiting the

Beverton-Holt recruitment model from a life history and multispecies

perspective. Rev. Fish Biol. Fish. 9, 187–202.

Walters, C., Martell, S., 2004. Fisheries ecology and management. Princeton

University Press, Princeton.