Population ecology: an introduction to population dynamics Jean-François Le Galliard CNRS – iEES Paris CNRS/Ens – CEREEP/Ecotron IleDeFrance
Introduction
Population ecology: the study of population fluctuations in time and space
Kausrud et al. 2008. Linking climate change to lemming cycles. Nature 456:93-97.
Population ecology: the study of population fluctuations in time and space
Kausrud et al. 2008. Linking climate change to lemming cycles. Nature 456:93-97.
Rodent cath rate (density)
Peak frequency
Population observations against model predictions
Population ecology: characteristic dynamics
Population increase
Population decrease
Population shifting up and down
Population “cycling”
Irregular dynamics (oubreaks)
Population ecology: loss of population size and the biodiversity crisis
Population ecology: resolving mechanisms
Fluctuating population dynamics
Recruitment rate: annual growth rate follows a mean pattern with huge variations around it
Here, immigration is a leading demographic process but that is not a general rule
Saether et al. 2000. Population dynamical consequences of climate change for a small temperate songbird. Science 287:854-856.
Population ecology: spatial ecology and synchrony Sampling of 65 butterfly taxa across a gradient
Time series of four representative taxa from high to low synchrony
Pardikes NA, Harrison JG, Shapiro AM, Forister ML, 2017. Synchronous population dynamics in California butterflies explained by climatic forcing. Royal Society Open Science 4.
Population: the fundamental unit of demography A (biological) population is a group of individuals of the same species from a spatially, genetically and demographically distinct unit than other group of individuals A group of interbreeding organisms found in the same space or area at the same time (Rockwood, 2006) = a closed population A group of individuals of the same species that live together in an area of sufficient size to permit normal dispersal and migration behavior, and in which population changes are largely determined by birth and death processes (Turchin, 2003) Rockwood. Introduction to population ecology. 2006
“Laws of population ecology”
1. Populations have an “intrinsic” tendency to grow exponentially = “exponential law” of population ecology (see this lecture) 2. Populations yet display bounded fluctuations because they also show principles of regulation = “self-limitation law” (see this lecture) 3. Consumer-resource interactions tend to be oscillatory (see lecture on inter-specific interactions)
Turchin PA, 2003. Complex population dynamics: a theoretical-empirical synthesis: Princeton University Press.
Basic principles of population ecology Assuming a discrete population of Nt individuals at time t, the future population size at the next time step (or generation) is given by a simple equation with 4 basic processes
∆Nt = Bt + It − Dt − Et ∆Nt = Bt − Dt ∆Nt = (bt − dt ) × Nt = pgr t × Nt
All populations Closed populations
Per capita expression
Population growth rate rt = Rt when time is number of generations [net reproductive rate] = λt when time is number of time steps
Example of birth and death rates
Rockwood. Introduction to population ecology. 2006
Density-independent growth
Exponential growth dynamic Assuming a constant and homogeneous birth and death rate across all individuals in a closed population (I=E=0) or a migration balance (I=E) then a given population of a species with non-overlapping generations grows according to
∆Nt = (b − d ) × Nt = λ × Nt Nt = λ × N0 t
a density-independent growth trajectory, typical of non regulated populations (also called discrete/geometric population growth). λ is the finite rate of increase
Geometric growth in species with non-overlapping generations
Rockwood. Introduction to population ecology. 2006
Exponential growth dynamic Assuming a constant and homogeneous birth and death rate in a population of a species with overlapping generations and nonseasonal reproduction grows according to a time-continuous process
dN
= r×N
dt rt Nt = e × N0
where r is the intrinsic rate of increase and r=ln(λ)
Exponential growth dynamic: human population
Rockwood. Introduction to population ecology. 2006
Exponential growth dynamic: perennial plant
Begon et al. Population ecology: a unified study of animals and plants. 1996
Exponential growth dynamic: wild bisons in Yellowstone national park
© 2012 Nature Education Adapted from Figure 6.3 in Gates et al. 2010
Population structure and life cycles
Most populations are heterogeneous
Heterogeneity inside natural populations comes from discrete differences among individuals across categories, but also from continuous differences within categories Discrete structure = age, sex, life stage, reproductive state, etc Continuous structure = body size, body condition, physiological stress levels
Example of population heterogeneity
From Fitze & Le Galliard, Ecology Letters, 2008
Consequence of population heterogeneity
Population dynamics cannot be described anymore by average parameters since birth and death rates may vary among individuals Population structure (e.g., age structure, size distribution) is important to track since population growth will depend on individual parameters AND population distribution Individuals must be tracked across all classes and their growth trajectories must be investigated altogether with the population dynamics = life history approach / life cycle representation
Life table analysis of demographic data Static life tables can be obtained from count data on known age individuals at each time step and data on fecundity from each age class
Cohort life tables can be obtained from count data on cohort of individuals born during the same time step 1. Survivorship data obtained from counts of number of individuals through time 2. Fecundity data obtained from counts of seeds on each individual at each age 3. Data can be averaged over several cohorts
Mark-recapture analyses of demographic data
Population heterogeneity and life tables Perennial plant species: simple life cycle with “hidden” population structure and overlapping generations
∆Nt = (b − d ) × Nt = r × Nt b = F × g ×e d = 1− p r = F × g × e − (1 − p )
Begon et al. Population ecology: a unified study of animals and plants. 1996
Population heterogeneity and life tables Annual insect species: simple life cycle with “hidden” population structure and non overlapping generations
∆Nt = (b − d ) × Nt = r × Nt b = F × s1 × s2 × s3 × s4 d =1 r = 2.9 / 2.5
Begon et al. Population ecology: a unified study of animals and plants. 1996
Population heterogeneity and life tables Biennal plant species: age structure with two age classes and a complex “hidden” population structure
Generations are overlapping Each year the population can be described by two age classes “Adults” and “Young adults” contributing differently to the next time step A seed bank is also included in the model, which must be tracked independently from the two age classes to predict population size the next time step
Begon et al. Population ecology: a unified study of animals and plants. 1996
Life cycle graphs: a graph to visualize and compute transition parameters in structured populations Reproduction transition
Stasis and survival
Growth and survival
Shrinkage and survival or Clonal reproduction
Life cycle graph for age-structured populations: example of post-breeding census with 5 age classes
0
1
2
3
Annual survival transitions: S Fertility transitions (assuming reproduction right before sampling): F
4
Leslie projection matrix formulation Population dynamics entirely described by the contribution of each age class to the next time step using a transition matrix (or population projection matrix)
P0 F1 n0 P0 n1 n (t + 1) = 0 2 0 n3 0 n 4
P1 F2
P2 F3
P3 F4
0 P1
0
0 0
0
0 P2
0
0
0 P3
0 n0 0 n1 0 × n2 (t ) = A × n(t ) 0 n3 0 n4
This special age-classified version if often referred to as the Leslie matrix
Life cycle graph for age-structured populations: prebreeding census of the population
1
2
3
Annual survival transitions: S Fertility transitions (assuming reproduction right before sampling): F
4
Leslie projection matrix formulation
F1 P0 n1 P1 n2 n (t + 1) = 0 3 n 0 4
F2 P0 0 P2 0
F3 P0 0 0 P3
F4 P0 n1 0 n2 × (t ) = A × n(t ) 0 n 3 0 n4
Generic properties of Leslie matrices
1. Age-structured populations described by a Leslie matrix are characterized by a stable age distribution: when the age distribution reaches this state, it does not longer change 2. Like other density-independent population model, the Leslie model predicts exponential growth (geometric growth) but only when the population reaches the stable age distribution 3. The exponential growth rate and other asymptotic properties can be derived from an eigendecomposition of the transition matrix
Numerical projections by repeated matrix multiplications: transient dynamics
Simulation of one projection with three age classes including offspring and 2 adult classes and an initial population of one individual in age class 1
Caswell H, 2001. Matrix population models. Sunderland: Sinauer Associates.
Numerical projections by repeated matrix multiplications: transient dynamics
Simulation of 10 projections with randomly selected initial populations of the same total size Caswell H, 2001. Matrix population models. Sunderland: Sinauer Associates.
Asymptotic properties: the characteristic equation 1. We want to find a number (called λ , the eigenvalue) and a vector w (also called the right eigenvector) such that
A × w = λw ⇔ ( A − λI ) × w = 0 where I is the identity matrix.
2. Non-zero solutions for this equation exist if and only if the determinant of the matrix A-λI is zero where the determinant is a scalar function of the matrix giving the volume of the n-dimensional space of column vectors. This determinant can be written like a nth-order polynomial expression of matrix entries and it determines if the matrix is singular (cannot be inverted) or not. When the matrix is non-singular, the characteristic equation has only one solution (the null vector). Otherwise, several non-zero solutions can exist including a “dominant eigenvalue” called λ1. 3. Left eigenvectors are obtained similarly from equation
v T × A = λv T
Asymptotic properties: matrix similarity Lets' call W the matrix of right eigenvectors (on each column), then from previous slide A × W = W × Λ, where Λ is the diagonal matrix of eigenvectors. Thus, we can write A = W × Λ × W -1 , where the inverse matrix W -1 is defined such that W × W -1 = I . This also implies that W -1 × A = Λ × W −1 , indicating from previous slide that W −1 is the matrix of the transposed left eigenvectors (on each row)
Asymptotic properties: exponential growth
n(t ) = A × n(t − 1) = A × n(0) t
A = W × Λ × W −1 n(t ) = W × Λ × W −1 × n(0) t
n
n(t ) = ∑ λ × w i × v i × n(0) i =1
t i
T
lim n(t ) = λ × w 1 × v × n(0) t →∞
t 1
T 1
Asymptotic properties: exponential growth
lim n(t ) = λ × w 1 × v × n(0) t →∞
t 1
T 1
Total population size grows exponentially Stable population structure
Coefficient giving by initial population structure and left eigenvectors
Asymptotic properties: generation time and R We define net reproductive rate R0 like the mean number of offspring by which a newborn will be replaced at the end of its life
∫
w
R 0 = l x × m x dx = 0
∑l
x
× mx =
i −1
∑F ∏P i
i
j
j =1
We then define generation time as the time required for the population to increase by a factor R0 λ1T = R0 ⇒ T =
log R0 log λ1
Worked example: semi-palpated sandpiper Long-term demographic study of shorebird at La Pérouse Bay, Canada by mark-recapture methods of adults at nest and monitoring of nesting success of known-age individuals (3 age classes: yearlings, 2 year-old, > 2 years old). Pre-breeding census and potential for immigration.
1
2
Hitchcock CL, Gratto-Trevor C, 1997. Diagnosing a shorebird local population decline with a stage-structured population model. Ecology 78:522-534.
3
Worked example: mean parameters
Fi = 0.5 × Breeding _ prob × Survival (banding ) × Survival ( juvenile) Si = Survival (adult )
Hitchcock CL, Gratto-Trevor C, 1997. Diagnosing a shorebird local population decline with a stage-structured population model. Ecology 78:522-534.
Worked example: asymptotic properties
λ1 = 0.6389 0.1189 w 1 = 0.1047 0.7764 1.000 v 1 = 1.0973 1.1139
Conclusion: building structured population models from life history data 1. Decide a life history schedule based on prior knowledge of the species/population, best population structure and/or study constraints 2. Estimate vital rates (component traits describing transitions between life history stages) from life table analysis, especially cohort based individual data 3. Dynamical and asymptotic properties of population demography can then be captured in a matrix summarizing properties and values of vital rates
Life history data for age-structured populations: fertility and survivorship Life table analyses allow to calculate age specific survival (Sx) from which cohort survival or survivorship (lx) can be computed. Survivorship curves: average values and patterns of age-dependence vary among species. Death at senescence
Constant death on arithmetic or logarithmic scale
Rockwood. Introduction to population ecology. 2006
Death at juvenile stage
Life history data for age-structured populations: fertility and survivorship
Rockwood. Introduction to population ecology. 2006
Life history data for age-structured populations: fertility and survivorship Life table analyses also allow to calculate age specific fertility (mx). Reproductive patterns are highly variable among species: • iteroparous: several reproduction events during life • semelparous: only one reproduction event, typically at the end of life • the length of pre-reproduction period, reproduction period and postreproduction period (rare) can vary among iteroparous species • the shape of the relationship between fertility and age is also highly variable ranging from saturating functions to bell shaped functions with a peak in prime-age females and a decline in old, senescent females
Life history data for age-structured populations: fertility and survivorship
Rockwood. Introduction to population ecology. 2006 Begon et al. Population ecology: a unified study of animals and plants. 1996
Life history strategies and the pace of life http://images.google.fr
Frequent breeding, iteroparous Flexible clutch size and reproductive effort Short life, early age at sexual maturity Small body size
Income breeding Active foraging strategies Uta stransburiana Egg-laying species, high adult and juvenile mortality Multiple clutches per year (2-3) Variable litter sizes and offspring size Active foragers Small adult body size (< 4g) but relatively large sizeindependent reproductive effort
Infrequent breeding, semelparous Constant clutch size and reproductive effort Long life, late sexual maturation Large body size
Capital breeding Sit-and-wait foraging strategies Eunectes murinus Viviparous species, high adult survival One litter per year or every other year Ambush predators Large adult body size (
