Matrix modeling of age- and stage- structured populations in R Chris Free

What is a population model? A model is a representation of reality. A population model is no different! A population model is a simplification of a biological system. Complexity can be incorporated with additional data and programming skills.

Why use population models? Estimate r (intrinsic growth rate) or k (carrying capacity) and other parameters Predict consequences of conservation or management strategies Understand the spread of parasites, viruses, and diseases Compare populations in different locations

A population can be described by its demographic rates Nt+1 = Nt + Bt – Dt + It - Et N = Abundance B = Birth rate D = Death rate I = Immigration rate (individuals entering the population) E = Emigration rate (individuals leaving the population) These rates often vary between age classes and its important to account for these variations in population modeling. This can be done using matrix models

Demographic rates are age-specific

What information is needed to build an age-structured matrix model? Age of females in population Age-specific survivorship (lx, proportion of females surviving to age x) Age-specific fecundity (mx or bx, number of female offspring born to females of age x) Easier for some taxa then others! Must be able to age individuals, follow survival, and measure reproductive output

How to build a life table Age (x) nx lx sx mx lxmx 1000 1.0 0.5 0.0 1 500 0.2 2 100 0.1 5.0 3 50 0.05 9.0 4.5 4 5 --- nx = number of female of age x lx = probability a female newborn attains age x (calculated as nx / n0) sx = age-specific survival - survival between age x to x+1 (calculated as lx+1 / lx) mx = number of female progeny per female lxmx= age-specific fecundity – per capita average number of female progency from a female of age x (calculated as Ix * Mx)

What can you learn from a life table? Age (x) nx lx sx mx lxmx 1000 1.0 0.5 0.0 1 500 0.2 2 100 0.1 5.0 3 50 0.05 9.0 4.5 4 5 --- Net reproductive rate (R0, net female replacement rate): average lifetime number of offspring produced per female Generation time (G): average age of the parents of newborn individuals Population growth rate (r): ln(R0) / G puts you within 10% of the true r OR use the Euler equation:

Life tables help setup matrix models Age (x) nx lx sx mx lxmx 1000 1.0 0.5 0.0 1 500 0.2 2 100 0.1 5.0 3 50 0.05 9.0 4.5 4 5 --- initial abundance (Nx) age-specific survivorship (Px) age-specific fecundity (Fx) Age-structuctured matrix models are powered by Leslie matrices

Leslie matrices Example 1: A Leslie matrix for an organism living 5 years with 2-5 year olds reproducing: Life cycle diagram P1 F2 P2 F3 P3 F4 P4 F5 Age-specific fertilities in the first row Age-specific survival probabilities in the diagonal

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: 6 P5 F6 How should we modify the life cycle diagram? removed P1 P2 F3 P3 F4 P4 F5 P5 F6 What should the Leslie matrix look like?

How does matrix multiplication work? Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: 6 P5 F6 Abundancet+1 Leslie matrix Abundancet N1,t+1 N2,t+1 N3,t+1 N4,t+1 N5,t+1 N6,t+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Note: we have to use matrix multiplication!

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet

N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0 If we do this over and over, we can track age-specific abundance over time. This will allows us to learn things like the finite rate of increase (λ), stable age distribution (SAD), and the reproductive value, sensitivity, and elasticity of each age class.

Leslie matrices Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6 N1,t N2,t N3,t N4,t N5,t N6,t Leslie matrix Abundancet N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6 N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0 N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0 If we do this over and over, we can track age-specific abundance over time. This will allows us to learn things like the finite rate of increase (λ), stable age distribution (SAD), and the reproductive value, sensitivity, and elasticity of each age class.

Leslie matrices Simulate age-specific abundance over time Evaluate population trends under different management strategies Measure important characteristic about the populations dynamics

What can we learn about the population from Leslie matrices? λ, finite rate of increase: the direction and rate of population growth, λ < 1 = decreasing and λ > 1 = increasing population The dominant eigenvalue of the Leslie matrix Stable age distribution: the population structure that would emerge if demographic rates (F, P) don’t change The eigenvector of the dominant eigenvalue of the Leslie matrix Reproductive value: the expected contribution of an individual of each age to present and future reproduction The dominant left eigenvector of the Leslie matrix Sensitivities and elasticities: sensitivities indicate the direct contribution of each transition to λ; elasticities are sensitivities weighted by transition probabilities; more complicated to measure

Advantages/disadvantages of age-structured matrix models Easy to implement Fewer assumptions than models without age structure Don’t need to assume stable age distribution Sensitivity analyses Can be modified to incorporate stochasticity, density dependence, stage structure, etc. Disadvantages Assumptions Sex ration is 1:1 Individuals can be aged Demographic rates are constant in age class No density dependence Survivorship and fecundity can be difficult to derive

Let’s look at an example… The authors wanted to know which of the following stage classes is both an important driver of population growth and responsive to management actions.

They use a stage-structured model driven by Lefkovitch matrices 1 3 2 4 5 6 7 G1 G2 G3 G4 G5 G6 P1 P2 P3 P4 P5 P6 P7 F5 F6 F7 Eggs/ hatchlings Small juveniles Large Sub- adults Novice breeders 1st-year remigrants Mature

Matrix modeling exercise Task #1 Which stage classes should management target? If management actions could increase the survivorship of the following stage categories by 20%, which set of management actions would be most beneficial? eggs, hatchlings (1) small/large juveniles, sub-adults (2-4) novice breeders, 1st-yr remigrants, mature breeders (5-7) Hint: Make a model for each management action where you increase survivorship for the targeted categories by 20% of their current values. Which action best promotes population recovery and how do you know? Task #2 Incorporate stochastic survivorship into your population model. The survival values in the current projection matrix are constant. Incorporate stochasticity (randomness) into your best model from the task above. Hint: This can be done using the rnorm() function to randomly generate a value using a normal distribution with a mean of the constant survival and a variance of 0.03, 0.05, and 0.1. You will have to enforce against negative values using pmax(). How do your results change as survival becomes more variable?