FW662 Lecture 5 – Age-structured models 1Lecture 5. Age- and stage-structured models: Leslie-Lefkovitch Models.Reading:Gotelli, 2001, A Primer of Ecology, Chapter 3, pages 49-80.Noon, B. R. and J. R. Sauer. 1992. Population models for passerine birds: structure,parameterization, and analysis. Pages 441-464 in D. R. McCullough and R. H.Barrett, eds. Wildlife 2001: Populations. Elsevier Applied Science, New York,New York, USA.Optional:Lande, R. 1991. Population dynamics and extinction in heterogeneous environments: theNorthern Spotted Owl. Pages 566-580 in C. M. Perrins, J-D. Lebreton, and G. J.M. Hirons, eds. Bird Population Studies, Oxford, New York, New York, USA.Define age structure dynamics in terms of difference equations.Ni,t is population at time t of age class iTime t is start of biological year, or the time of reproduction.With k age classes, the maximum age an animal can attain is k, so that the survivalrate of animals k years old is 0.Only females are considered in the following example.The modeler must define the anniversary date of the population census. For thefollowing equations, the animal is incremented in age (i.e., mortality takes place),then reproduces. The population census is after the birth-pulse. Define fi as thenumber of young produced by animals of age and si as survival rate of animalsiof age i to end of the year. Animals just born are in the N0 age class.Survival to next age classN1,t+1 = N0,t × s0N2,t+1 = N1,t × s1N3,t+1 = N2,t × s2...Nk,t+1 = Nk-1,t × sk-1ReproductionN0,t+1 = N1,t+1 × f1 + N2,t+1 × f2 + ... + Nk,t+1 × fkHowever, we do not want the equations to refer to the t + 1 populations, but ratherthe t populations. Therefore, we must substitute the survival equations into thereproduction equation:N0,t+1 = N0,t × s0 × f1 + N1,t ×s1 × f2 + ... + Nk!1,t ×sk!1 × fkFW662 Lecture 5 – Age-structured models 2Now, construct the Leslie (1945, 1948) matrix (also known as a projection matrixor transition matrix) based on the above difference equations.Nt+1 = L × NtNt % 1's0f1s1f2þ sk&1fk0s00 0 þ 00 s10 þ 0! ! " " !0 0 0 sk & 10×N0N1N2!NktIn the following equations, reproduction takes place, then mortality. The census of thepopulation is before the birth-pulse. The definitions of the fi and si remain thesame as the after birth-pulse model. However, the interpretation of the Nichanges slightly, because now the population sizes are just prior to reproduction. Hence, there are no newly born animals in this model corresponding to the N0 ofthe after birth-pulse model. Rather, Ni corresponds to animals 2 days youngerthan in the previous model. That is, N1 in the after birth-pulse model was thepopulation of animals aged 1 year + a day. Now, in the before birth-pulse model,N1 is the population of animals aged 1 year - a day, i.e., just before they reach theirfirst birthday. As a result, the N0 age class is no longer being modeled directly,although the survival rate of animals from birth to 1 year of age (s0) is still in themodel.Survival to next age classN2,t+1 = N1,t × s1N3,t+1 = N2,t × s2...Nk,t+1 = Nk-1,t × skReproductionN1,t+1 = N1,t × f1 × s0 + N2,t × f2 × s0 + ... + Nk,t × fk × s0Construct the Leslie matrix.Nt+1 = L × NtFW662 Lecture 5 – Age-structured models 3L 'f0s0f1s0s10Nt % 1'f1s0f2s0f3s0þ fks0s10 0 þ 00 s20 þ 0! ! " " !0 0 0 sk0×N1N2N3!NktCarefully note the differences in the 2 Leslie matrices. In the after birth-pulse matrix, thetop row contains the survival rates of the reproducing animals. In the before birth-pulse matrix, the top row contains the survival rate of new born animals to 1 yearof age. The most common mistake with application of the Leslie notation is thatthe presenter confuses order of the birth and death process. Noon and Saur (1992)discuss how to configure the matrix for use of estimates of survival andreproduction from a population.Benefits of Leslie matrix formulation = eigenvalue of L = rate of population growth, i.e., Nt+1 = Nt.88Equilibrium age ratios are eigenvector of L, i.e., ratio of N1:N2 is the same as theratio of the first 2 values in the eigenvector. The stable age distribution isthe ratio of the various age classes to one another. These ratios are stablein a Leslie matrix projection, regardless of the value of . For , the88' 1age distribution is termed stationary, because the population is stationary.Ease of presentation (if you know matrix algebra!)Construction of a spreadsheet model from this formulation:Provides a simple, available alternative to model a population.Ratio of Nt+1/Nt after a few generations provides .8Ratio of age classes is equivalent to eigenvector after a few generations.Interpretation of age ratios measured in a population: Because the projection matrixgenerates a stable age distribution after effects of initial population size aredepreciated, age ratios tell you nothing about increase or decrease of a population. "To sum up: age ratios cannot be interpreted without a knowledge of rate ofincrease, and if we have an estimate of this rate we do not need age ratios." Caughley (1974).Example of eigenvalue calculation:FW662 Lecture 5 – Age-structured models 40 ' det(L &8I)0 ' det/0000/0000f0s0f1s0s10&8 00 80 ' det/0000/0000f0s0& 8 f1s0s1&80 ' &8f0s0% 82& f1s0s10 ' 82& (f0s0)8 & (f1s0s1)&b ± b2& 4ac2af0s0% f20s20% 4f1s0s12f0s0& f20s20% 4f1s0s12Eigenvalues are values of that solve8where I is the identity matrix.Remember that for the polynomial ax2 + bx + c, the roots areThus, the roots of the characteristic equation areandwith the first root (+) the largest value, or dominant eigenvalue.Typically, the analytical eigenvalues are seldom used, and only theFW662 Lecture 5 – Age-structured models 5s0(Nt) '$0%$1Ntnumerical value provided. Computing the eigenvalues of a matrixis a standard numerical analysis problem.The basic Leslie Matrix formulation is limited because only density-independentpopulation growth with just births and deaths is modeled. The followingexamples (all with the census before the birth-pulse) are some approaches toextend the basic formulation to incorporate additional population processes.Exponential growth is what the above examples portray, i.e., all rates are densityindependent. The matrix can be modified to make and functions offisithe total population size, . Another possibility is to make some age-Ntspecific parameters density-dependent only on the size of a specific ageclass. In both cases, if