POPULATION ANALYSIS IN WILDLIFE BIOLOGY Stephen J. Dinsmore1 and Douglas H. Johnson2 1Department of Natural Resource Ecology and Management, Iowa State University, Ames, IA and 2USGS- Northern Prairie Wildlife Research Center, St. Paul, MN

Introduction Motivating Questions Animal Populations How many individuals? What are the vital rates of the population of interest? Is the population increasing or decreasing? These concepts lack meaning for lower (individuals) or higher (communities) levels of biological organization. A population is a group of organisms of the same species living in a particular space at a particular time

Population Analysis The study of population dynamics What changes occur over time? What are the causes of those changes? How many individuals are in a population? What is the survival of those individuals? What is their reproductive rate? How do individuals move in and out of the population?

A Theoretical Model of Population Growth The number of individuals (N) in a population at some future time (time t+1) depends on the number of individuals present now (time t) and any gains (births [B] and immigrants [I]) and losses (deaths [D] and emigrants [E]) that occur between times t and t+1: Nt+1 = Nt + Bt + It – Dt - Et

Constraints on Population Growth? What happens when we place constraints on the relationship between Nt and Nt+1? If population growth is unimpeded it is said to be density-independent. Population growth that depends directly on population density is density-dependent. Under each scenario we can also discuss the rate of population growth (λ) from time t to time t+1 using the simple equation λ = Nt+1/Nt.

Population Models Modeling approaches are used to bridge gaps in knowledge. A model is an abstraction of a real system Models can be simple, or complex mathematical exercises A parameter is something that describes a population, e.g., the annual survival rate of male Mallards. Models may be discrete-time (where events such as births only occur at certain times) or continuous-time (these events occur continuously). Some models are deterministic (fixed parameter values through time) while others are stochastic (parameters vary).

Populations with Unimpeded Growth Population changes by a constant ratio, λ, during each unit of time (usually a year). Thus, Nt+1 = λNt. Population is increasing if λ>1 Population is stable if λ=1. Population is decreasing if λ<1. λ is also called the finite rate of population increase. If N0 is the initial population size in some year, then this equation can be rewritten as Nt+1 = λtN0. Another alternative is to replace λ with er (e is the base of natural logarithms and r is the instantaneous rate of increase) and now we have Nt = N0ert.

Populations With Density-dependent Growth It is impossible for any population to grow indefinitely at a constant rate. Most likely, population growth will slow as the population becomes large and some limiting factor exerts an influence. How can density-dependence be included in the model to make it more realistic and useful?

Populations With Density-dependent Growth Continuous-time formulation One approach is to multiply the growth rate by a factor that has a negligible effect when the population is small, but reduces the growth rate to zero as the population approaches some limit, K (which we might call the carrying capacity). The term (K – N)/K does just that. This term is 1 when N is small and converges to 0 as N approaches K.

Populations With Density-dependent Growth Continuous-time formulation, cont. Now, the per capita population growth rate is: 𝑑 𝑁 𝑡 𝑑𝑡 = 𝑟 𝑚 𝑁 𝑡 𝐾− 𝑁 𝑡 𝐾 The solution, known as the logistic equation, is: 𝑁 𝑡 = 𝐾 1+ 𝑒 𝑎− 𝑟 𝑚 𝑡 Here, rm, the maximum growth rate, replaces r, and a measures the size of the population at time 0 relative to asymptotic population size.

Populations With Density-dependent Growth Discrete-time formulation The logistic equation specifically applies to continuously reproducing organisms, although it works for populations with discrete breeding seasons if population size is measured at the same time each year. How can the logistic equation be modified to apply to species with discrete breeding seasons?

Populations With Density-dependent Growth Discrete-time formulation, cont. The discrete counterpart of the logistic equation is: 𝑁 𝑡+1 = 𝑁 𝑡 + 𝑟 𝑚 1− 𝑁 𝑡 𝐾 𝑁 𝑡 This equation has an implicit time delay – the population growth rate at time t+1 depends on population size at time t.

Immigration and Emigration Dispersal Estimation Approaches The movement an animal makes from its point of origin to the place where it reproduces. Immigration (gains) and emigration (losses) differ from dispersal. Observations of marked individuals Genetic markers Studies using radio telemetry or satellite markers Modeling approaches such as robust design or multi-state models

Birth and Death Models Population growth is the net result of births and deaths (ignoring immigration and emigration) Recall the equation Nt+1 = Nt + Bt + It – Dt – Et From this we can define the birth rate (b = Bt/Nt) and death rate (d = Dt/Nt) as simple proportions Distinguish between closed and open populations

Estimating Birth Rates Some Terms Estimation Fertility is the number of live births per unit time (usually a year) Fecundity is the potential level of reproductive performance of a population Recruitment is the addition of new individuals through reproduction Counts of live births, eggs, etc. Age ratios from direct counts Mark-recapture methods Indirect measures such as clutch size and nest success in birds

Estimating Survival Rates Two common measures of survival: True survival – the real probability of living Apparent survival – the product of true survival and fidelity If fidelity is 100%, true survival = apparent survival Relative to true survival, apparent survival will be biased low when emigration is permanent Five general approaches to estimating survival: Observed survival Ratios of population size or indices Change-in-ratio methods Mark-recapture approaches Methods based on tag recoveries

Estimating Survival Rates Observed Survival Ratios of Population Size Difficult to use in practice! Deaths in the population must be known: Captive populations Instances where radio telemetry or other technology provides information on all deaths For a closed population, the mortality between times t and t+1 is the population at time t minus the population at time t+1. If survivors can be distinguished from young, survival can be computed directly. Indices can be substituted for direct counts.

Estimating Survival Rates Change-in-Ratio Methods Is usually applied to estimating population size Can also be used to estimate rate of mortality due to exploitation Requires two distinguishable types of animals (male and female, adult and young, etc.) Can then estimate the proportion of each type in the population before, in, and after the harvest. This approach has strong assumptions!

Estimating Survival Rates Mark-recapture Methods The most widely used approach to estimate survival Many models available, some for survival and others for a combination of survival and other parameters “Recapture” can be physical recapture, live resightings, etc. General Sampling Framework: Consider a study with J occasions on which animals are captured, marked, and returned to the population Assume all animals are alike and have the same probability of capture on each occasion, and that they all have the same probability of surviving between occasions On each occasion animals are captured, of which some are already marked and some will be newly marked

Estimating Survival Rates Definitions: Let Si be the probability of surviving from occasion i to i+1 Define Ni to be the number of animals in the population at occasion i Suppose that Mi of these animals had been previously marked On the ith occasion, ni of these animals are captured, of which mi are already marked and ui are unmarked The survival rate, Si, can be estimated from the equation: 𝑆 𝑖 = 𝑀 𝑖+1 𝑀 𝑖 − 𝑚 𝑖 + 𝑅 𝑖 where 𝑀 𝑖 = 𝑚 𝑖 + 𝑅 𝑖 𝑧 𝑖 𝑟 𝑖

Estimating Survival Rates Equation Terms (from previous slide): Ri is the number of the ni animals that are released after the ith occasion ri is the number of Ri that are released at i and captured again zi is the number of animals that were captured before i, not captured in i, but captured again later

Estimating Survival Rates More Detailed Models: Robust design – a combination of open and closed population models to estimate survival rate and population size in the presence of temporary emigration Reparameterized Jolly-Seber model – can also estimate seniority, recruitment, and rate of population change Multi-state or multi strata models – also useful for estimating survival, but more appropriate for estimating transition probabilities between stages (e.g., life stages, age classes, etc.)

Estimating Survival Rates Methods Based on Tag Recoveries Also called tag or band recovery models, most often used for waterfowl studies General Sampling Framework: Capture and uniquely mark large numbers of individuals each year Birds are harvested by hunters and the band number is reported Many occasions, but only a single recovery is possible for an individual bird

Estimating Survival Rates Methods based on tag recoveries, cont. Band recovery models estimate: Recovery rate – the probability that a bird is shot and its band is reported Survival rate – the probability that a bird survives from the beginning of the first hunting season to the beginning of the second Can also ask questions about effects of age, sex, and other covariates of interest Parameters can be estimated using the recoveries only model in program MARK

Life Tables A summary of the mortality schedule of a population, or the pattern of deaths by age class. Can estimate survival and mortality rates. Example of a life table based on known deaths of 42 gray squirrels born in 1954 (from Downing 1980:256). Age No. in No. of Mortality Survival (years) pop’n deaths rate rate (x) (nx) (dx) (qx) (sx)   0-1 42 22 22/42 = 0.52 20/42 = 0.48 1-2 20 10 10/20 = 0.50 10/20 = 0.50 2-3 10 7 7/10 = 0.70 3/10 = 0.30 3-4 3 2 2/3 = 0.67 1/3 = 0.33 4-5 1 1 1/1 = 1.00 0/1 = 0

Stable Age Distribution The age distribution of a population is the number of individuals of each age class in the population at a particular time. If age-dependent survival and fertility rates remain constant for a long period, the proportion in each age class will stabilize This leads to the stable age distribution The fraction of the population in each age class x is called Cx and is equal to: 𝐶 𝑥 = 𝑒 −𝑟𝑥 𝑙 𝑥 𝑖 𝑒 −𝑟𝑖 𝑙 𝑖

Leslie Matrices A useful way to “project” the population: The population is age structured with M age classes and breeds seasonally Assumes the population has reached a stable age distribution The age-dependent survival and fertility rates are known Let nx,t be the number of individuals of age x in year t The number of 1-year-olds in year t+1 (n1,t+1) will be the number born in year t (n0,t) times the survival rate of 0-year-olds (S0). So, n1,t+1 = S0n0,t

Leslie Matrices Next, consider the age specific births The number of 0-year-olds (births) in year t+1 (n0,t+1) represents the number of 1-year-olds in that year (n1,t+1) times their fertility rate (m1), plus the number of 2-year olds in that year (n2,t+1) times their fertility rate (m2), etc. Thus, n0,t+1 = m1n1,t+1 + m2n2,t+1…mMnM,t+1 If we replace (m1n1,t+1) with (m1S0n0,t) and then use gi = mi+1Si to simplify the notation, we are left with the components of a Leslie Matrix

Leslie Matrices 𝑔 0 𝑔 1 𝑔 𝑀 𝑆 0 0 0 0 𝑆 1 𝑆 𝑀 The Leslie Matrix, L, is then: 𝑔 0 𝑔 1 𝑔 𝑀 𝑆 0 0 0 0 𝑆 1 𝑆 𝑀 We can use this to predict the number of individuals in each age class at time t+1 using matrix multiplication: 𝑛 0 𝑛 1 𝑛 𝑀 𝑡+1 = 𝑔 0 𝑔 1 𝑔 𝑀 𝑆 0 0 0 0 𝑆 1 𝑆 𝑀 × 𝑛 0 𝑛 1 𝑛 𝑀 𝑡 So, nt+1 = L ˣ nt

Parameter Estimation Modeling is an iterative process where multiple models (a model set) to explain the same phenomenon are compared, and then one (or more) are used to describe the process Key steps Develop a priori biological hypotheses about the process; these should strive to answer “Why?” Use computers and approaches like the method of maximum likelihood to estimate parameters of interest (e.g., a survival rate) Use model selection procedures (e.g., AIC model selection) to select a model or models for inference

Modeling Considerations Parameter estimates themselves are important, but biologists should seek to understand why a parameter may vary and what that means. Factors that affect a parameter estimate, called covariates, should be incorporated into an analysis if possible Examples of covariates are gender (male and female), sites (A and B), day of nesting season, measures of weather, or attributes of individual animals (e.g., body condition, genetic characteristics, etc.)

Computer Programs Computer programs to estimate population parameters are becoming more sophisticated. Older programs like JOLLY and SURVIV may be outdated and the software is no longer supported Program MARK is a powerful and flexible program designed to meet the needs of a population analyst: Can model group effects and covariates Model selection by AIC or other approaches Model averaging can be performed Goodness-of-fit testing for some models A Bayesian modeling tool is included

Population Viability Analysis Population Viability Analysis (PVA) seeks to make predictions about future population status using population data. Two general approaches: Estimate the probability that a population of a specified size will persist for a certain time period (PVA) Estimate the Minimum Viable Population (MVP) needed for a population to persist for a specified time period Predictions are made using computer programs such as RAMAS Metapop 5.0 or VORTEX.

Inference Once a population analysis is complete we often desire to draw conclusions from the data. Controlled, manipulative experiments are desirable, but often difficult to implement Increasingly, we model one or more parameters of interest, and different hypotheses about how the population behaves are embedded in different (competing) models. Model selection procedures are used to arrive at conclusions

SUMMARY What is population analysis? Introduction to theoretical models of population growth Exponential and logistic models Birth and death processes Age effects Estimating survival and mortality Making predictions about growth – Leslie matrices General approaches to parameter estimation Predicting the future viability of a population Making inference from a population analysis