Introduction .ca



PVR_invas.R User’s GuideBrett van PoortenBritish Columbia Ministry of Environment, Conservation Science Section2202 Main Mall, Vancouver, BC, V6T 1Z4Brett.vanPoorten@gov.bc.caDecember, 2017Table of Contents TOC \o "1-3" Introduction PAGEREF _Toc374390656 \h 3Model description PAGEREF _Toc374390657 \h 3Parameters and controls PAGEREF _Toc374390658 \h 11Running the model PAGEREF _Toc374390659 \h 12Evaluating a single removal strategy PAGEREF _Toc374390660 \h 12Evaluating multiple removal strategies PAGEREF _Toc374390661 \h 14References PAGEREF _Toc374390662 \h 20IntroductionPVA_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 ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Lorenzen", "given" : "K", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Canadian Journal of Fisheries and Aquatic Sciences", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2374-2381", "title" : "Allometry of natural mortality as a basis for assessing optimal release size in fish-stocking programmes", "type" : "article-journal", "volume" : "57" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Lorenzen, 2000)", "manualFormatting" : "(2000)", "plainTextFormattedCitation" : "(Lorenzen, 2000)", "previouslyFormattedCitation" : "(Lorenzen, 2000)" }, "properties" : { }, "schema" : "" }(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 ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.2307/1940001", "ISSN" : "00129658", "author" : [ { "dropping-particle" : "", "family" : "Hoenig", "given" : "John M", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Fishery Bulletin", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "1983", "6" ] ] }, "page" : "898-903", "title" : "Empirical use of longevity data to estimate mortality rates", "type" : "article-journal", "volume" : "82" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Hoenig, 1983)", "manualFormatting" : "(1983; T1.4)", "plainTextFormattedCitation" : "(Hoenig, 1983)", "previouslyFormattedCitation" : "(Hoenig, 1983)" }, "properties" : { }, "schema" : "" }(1983; T1.4). Survival in each time-step follows Lorenzen ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Lorenzen", "given" : "K", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Canadian Journal of Fisheries and Aquatic Sciences", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2374-2381", "title" : "Allometry of natural mortality as a basis for assessing optimal release size in fish-stocking programmes", "type" : "article-journal", "volume" : "57" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Lorenzen, 2000)", "manualFormatting" : "(2000; T1.5)", "plainTextFormattedCitation" : "(Lorenzen, 2000)", "previouslyFormattedCitation" : "(Lorenzen, 2000)" }, "properties" : { }, "schema" : "" }(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 ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Walters", "given" : "Carl", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martell", "given" : "Steven", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2004" ] ] }, "publisher" : "Princeton University Press", "publisher-place" : "Princeton", "title" : "Fisheries ecology and management", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Walters and Martell, 2004)", "plainTextFormattedCitation" : "(Walters and Martell, 2004)", "previouslyFormattedCitation" : "(Walters and Martell, 2004)" }, "properties" : { }, "schema" : "" }(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; ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Walters", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Korman", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Reviews in Fish Biology and Fisheries", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1999" ] ] }, "page" : "187-202", "title" : "Linking recruitment to trophic factors: revisiting the Beverton-Holt recruitment model from a life history and multispecies perspective", "type" : "article-journal", "volume" : "9" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Pine", "given" : "William E. III", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Healy", "given" : "Brian", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Smith", "given" : "Emily Omana", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Trammell", "given" : "Melissa", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Speas", "given" : "Dave", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Valdez", "given" : "Rich", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Yard", "given" : "Mike", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Walters", "given" : "Carl", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Ahrens", "given" : "Rob", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Vanhaverbeke", "given" : "Randy", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Stone", "given" : "Dennis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Wilson", "given" : "Wade", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "North American Journal of Fisheries Management", "id" : "ITEM-2", "issued" : { "date-parts" : [ [ "2013" ] ] }, "page" : "626-641", "title" : "An individual-based model for population viability analysis of humpback chub in Grand Canyon", "type" : "article-journal", "volume" : "33" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Pine et al., 2013; Walters and Korman, 1999)", "manualFormatting" : "Walters and Korman 1999, Pine et al. 2013)", "plainTextFormattedCitation" : "(Pine et al., 2013; Walters and Korman, 1999)", "previouslyFormattedCitation" : "(Pine et al., 2013; Walters and Korman, 1999)" }, "properties" : { }, "schema" : "" }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 ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Walters", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Korman", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Reviews in Fish Biology and Fisheries", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1999" ] ] }, "page" : "187-202", "title" : "Linking recruitment to trophic factors: revisiting the Beverton-Holt recruitment model from a life history and multispecies perspective", "type" : "article-journal", "volume" : "9" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Walters and Korman, 1999)", "plainTextFormattedCitation" : "(Walters and Korman, 1999)", "previouslyFormattedCitation" : "(Walters and Korman, 1999)" }, "properties" : { }, "schema" : "" }(Walters and Korman, 1999)R=N0e-M01+M1M01-e-M0 where e-M0 is maximum survival at low spawning stock and M1M01-e-M0 is the carrying capacity parameter. In the case of cannibalism, M0 can be modified as a function of likely cannibalistic age-classes:M0=ρ+τCTable 1: Model controls, indices, parameters and derived variables used in PVA_invas. Parameter values shown are biological parameters used for the smallmouth bass example.SymbolValueSpreadsheet nameDescriptionControlsdt0.25dtLength of time-step (years)nT50nTNumber of time-stepsns2nSNumber of stanzasAR1ARAge-at-recruitmentng2n.gearNumber of capture gears for recruited animalsnsim1,000n.simNumber of population simulationssampt1samp.tTime-step when each control gear for recruited animals is usedr0.05rStandard economic discount rateG20GHuman generation time (set large to ignore inter-generational discountingqR0.1; 0.1q.RMaximum catchability of each pre-recruit gear (proportion of stanza captured per unit sampling effort at low abundance)qA0.1, 0.1q.AMaximum catchability of each size-selective gear for recruited animalststart1t.startTime-step when sampling begins (allows for delayed start time)EfR,s0; 0E.RSampling effort per time-step for pre-recruit stanzasEfA,g0; 0E.ASampling effort per time-step for recruited animalsva25; 15v.aProportional to rate of increase of ascending limb of dome-shaped selectivity functionvb0.7; 0.5v.bProportional to length at 50% selectivity on ascending limb of selectivity functionvc0; 0.5v.cProportional to rate of decline on descending limb of selectivity function (0 < vc < 1)Cf,R2,000; 2,000C.F.RAnnual fixed cost of using each pre-recruit removal gearCf,A2,000; 2,000C.F.AAnnual fixed cost of using each post-recruit removal gearCE,R100; 100C.E.RCost per unit effort for each pre-recruit removal gearCE,A100; 100C.E.ACost per unit effort for each post-recruit removal gearIndicest{1, 2, …, nT/dt}Time-step (can be sub-annual if dt < 1)a{AR/dt,…,A/dt}dtAge 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 animalsModel parametersR01000R0Unexploited recruitsκ9.24reckCompensation ratio in recruitmentpcann0.2p.canproportion of pre-recruit mortality at equilibrium due to cannibalismK0.25KMetabolic rate parameter of von Bertalanffy functionA15AMaximum age (e.g. probability of survival to age A=1%)af6,000afecFecundity multiplier on weightwm0.4WmatWeight at maturity relative to asymptotic weighttspn0.0; 0.1t.spnStart and end of spawning time as proportion of yearMs*5; 8MsMaximum survival for each pre-recruit stage (relative values)Bs*10; 9BsAvailable habitat for each pre-recruit stage (relative values)V1200V1Initial number of vulnerable animalscanna{5,…,15}cann.aAges that may be cannibalistic on pre-recruits (if pcann>0)σR0.5sd.SStandard deviation in recruitmentDerived variableslaLengthwaWeightM∞Minimum instantaneous mortality faFecundityspntTime-steps when spawning occurs (0/1 flag)spnaAges when spawning occurs (0/1 flag)SaSurvivalva,gSelectivity of gear-glxaSurvivorshipφ0Unexploited eggs-per-recruitα*Maximum survival of Beverton-Holt recruitment functionβ*Carrying capacity parameter of Beverton-Holt recruitment function for each stanza-sαs*Maximum survival of Beverton-Holt recruitment functionβs*Carrying capacity parameter of Beverton-Holt recruitment function for each stanza-sRtAnnual recruitment to age-ARNt,sPopulation abundance for pre-recruited animals in year-t and stanza-sNt,aPopulation abundance for recruited animals in year-t and age-a EtAnnual egg productionCt,gAnnual catch by gearqNDensity-dependent catchabilityFt,aInstantaneous sampling mortality rate by age-classTable 2: Functions used to initialize the population viability analysis modelT2.1la=1-e-KaT2.2wa=la3T2.3fa=afwa-wmT2.4M∞=ln0.01Klnla=AR-lnla=AR+eKA-ARdt-1T2.5Sa=lala+eK?dt-1M∞KT2.6lxa=1a=ARa=ARa-1Saa>ART2.7φ0=a=1Alxafaspna2T2.8αs=κφ0eMsMsT2.9βs=Bs*κ-1R0φ0s'βs'*s''=0s''=s'-1αs''T2.10Ms0=-lnαs=ρs+τsR0a=cannalxaspnaT2.11Ms1=βsMs01-αsT2.12ρs=Ms01-pcannT2.13τs=Ms0pcannR0a=cannalxaspnaT2.14Et=1=i=1Nt=1fawhere 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. V1≥0.9R0lxa), abundance is randomly allocated among age-classes assuming a multinomial distributionNt=1,a~MNV1,lxaeεa; εa~N0,σRwhere random year-class strength is given by εa. 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-classNt,a*=Nt-1,a-1*Sa-1where 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 functionNt,a=1*=Et-1*e-ρ+τCt1+e-ρ+τCtM11-e-ρ+τCt,where ρ=-lnκφ01-pcann,τ=-lnκφ0pcannR0a=cannalxaspna and M1=κ-1lnκφ0R0κ-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 functionNt,s+1~BINNt,s+,e-Fs-Ms,t0+ψt,s1+Ms0Ms,t01-e-Ms,t0; ψt,s=N0,σR,where Ms,t0=ρs+τsa=cannaNt,aand Fs=qsEs;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 Nt+AR-1,a=AR. Survival of each recruited age-class each time-step is given by the Baranov equation ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Ricker", "given" : "W E", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Bulletin of the Fisheries Research Board of Canada", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1975" ] ] }, "page" : "382pp", "title" : "Computation and interpretation of biological statistics of fish populations", "type" : "article-journal", "volume" : "1913890" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Ricker, 1975)", "plainTextFormattedCitation" : "(Ricker, 1975)", "previouslyFormattedCitation" : "(Ricker, 1975)" }, "properties" : { }, "schema" : "" }(Ricker, 1975)Nt+1,a+1~BINNt,a,e-Zt,awhere Zt,a=Ma+Ft,aand Ft,a=g=1ngqN,gEt,gva,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 ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Arreguin-Sanchez", "given" : "F", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Reviews in Fish Biology and Fisheries", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1996" ] ] }, "page" : "221-242", "title" : "Catchability: assessment a key parameter for fish stock", "type" : "article-journal", "volume" : "6" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "(Arreguin-Sanchez, 1996)", "plainTextFormattedCitation" : "(Arreguin-Sanchez, 1996)", "previouslyFormattedCitation" : "(Arreguin-Sanchez, 1996)" }, "properties" : { }, "schema" : "" }(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 AN=γNβ,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 gearqN=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:qN=qN-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 β=1-ln5-ln15ln50-ln200=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 nsim times; The proportion of nsim 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 modelEvaluating 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$datObjectDescriptionA.sStage-specific maximum survival of Beverton-Holt functionB.sStage-specific carrying capacity parameter of Beverton-Holt functioncost.1Annual cost of samplingcost.TTotal cost of sampling up to end of time series or time to 100% eradicationEtMatrix of eggs produced per year for each simulationNPVNet present value of sampling, accounting for intergenerational discountingNtAbundance array (dimensions: time-steps, ages, simulations)NTAbundance in the final time-step (reported as 5th percentile, mean and 95th percentile of distribution) p.extinctVector of proportion of simulations eradiated for each time-stepp.extinct.50Proportion of simulations eradicated by 50th time-stepp.extinct.100Proportion of simulations eradicated by 100th time-stepp.extinct.200Proportion of simulations eradicated by 200th time-stepphieEquilibrium eggs-per-recruitR.ABeverton-Holt maximum survival parameter (stage-independent)R.BBeverton-Holt carrying capacity parameter (stage-independent)runtimeTime to complete simulationt.extinctYears to 100% eradicationVfinVector of total abundance in the final year across simulationsFigure 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). ReferencesADDIN Mendeley Bibliography CSL_BIBLIOGRAPHY Arreguin-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. Bull. 82, 898–903. doi:10.2307/1940001Lorenzen, 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.Pine, W.E.I., Healy, B., Smith, E.O., Trammell, M., Speas, D., Valdez, R., Yard, M., 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. ................
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