Demographic matrix models for structured populations

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Presentation transcript:

Demographic matrix models for structured populations Dominique ALLAINE

Structured populations Vital rates describe the development of individuals through their life cycle (Caswell 1989) Vital rates are : birth, growth, development, reproductive, mortality rates The response of these rates to the environment determines: population dynamics in ecological time the evolution of life histories in evolutionary time

Structured populations Obviously, vital rates differ between individuals. These vital rates may differ according to age, to sex, to size category … Each individual is then described by a state at a given time. For example, an individual may be in the state « yearling female » at time t. Populations are made of individuals that are different. So, we can consider the structure of the population according to sex, to age, to size category, to status … The structure of a population at a given time is then characterised by a distribution function expliciting the number of individuals in each state

Structured-population models General characteristics Models based on the global size of a population does not take into account the differences between individuals. It is important to analyse the response of vital rates in structured populations to take into account differences between individuals Structured-population models aim at including the differences between individuals in vital rates. These models are applied to structured populations.

Structured-population models We will assume that all individuals in a same state experience the same environment and then respond in the same way. In other words, all individuals in a same state will have the same values of vital rates. Structured-population models are mathematical rules that allow to calculate the change over time of the distribution function of the number of individuals in each state. Three types of mathematical approaches: matrix models delayed differential equation models partial differential equation models

Matrix population models General characteristics The different states are discrete This means that individuals are classified into categories. For example, individuals are either males or females, are eitheir juveniles or yearlings or two-years old … When considering size, discrete categories have to be identified. We thus have to recognise categories and at a given time, each individual may be associated unambiguously to a category.

Matrix population models General characteristics The change in the population structure is in discrete time This means that the change of the distribution of the number of individuals in each state is given in discrete time. If we use, for example, an annual time scale, the number of individuals in each state is given for each year.

N(t+1) = M N(t) Matrix population models General characteristics First developped by P.H. Leslie during the 1940’s The number of individuals in each state at time t is given by a vector N(t) This vector is projected from time t to time t+1 by a population projection matrix M N(t+1) = M N(t) (n,1) (n,n) (n,1)

Matrix population models General characteristics Different steps: To determine the projection interval (time scale) To determine the states of importance To identify the vital rates necessary To establish the life-cycle graph

Matrix population models Populations structured in age The projection interval is usually the year in analyses on vertebrates States of importance are often age, sex Vital rates refer to: proportion p of reproductive females per age class fecundity f per age class survival rate s per age and sex classes

Matrix population models Populations structured in age Life cycle in the case of a population structured in age 1 2 3 4

Matrix population models Populations structured in age The matrix model corresponding to the previous life-cycle graph is:  Leslie matrix

Matrix population models Populations structured in stage Life cycle in the case of a population structured in stage E J Ad R1 R2

Matrix population models Populations structured in stage The matrix model corresponding to the previous life-cycle graph is:  Lefkovitch matrix

Matrix population models General characteristics M Types of matrix models: Variable Constant Internal External Both Density- dependent Deterministic stochastic Periodic Aperiodic

Matrix population models General characteristics Types of matrix models: Linear => constant, time-varying Non linear => internally generated variability

N(t+1) = M N(t) Matrix population models Constant linear models (n,1) (n,n) (n,1) In this case, the matrix M has constant coefficients Example: a two age-classes model

Matrix population models Constant linear models The Perron-Frobenius theorem A non negative, irreductible, primitive matrix has 3 properties: the first eigenvalue 1 is unique, real and positive the right eigenvector w1 corresponding to 1 is strictly positive the left eigenvector v1 corresponding to 1 is strictly positive

Matrix population models Constant linear models The Perron-Frobenius theorem Non negative Reductible Irreductibe primitive imprimitive 1 > 0 w1 > 0 v1 > 0 1 = |j| j = 2 …d 1 > |j| j> d 1 ≥ 0 w1 ≥ 0 v1 ≥ 0 1 ≥ |2| 1 > 0 w1 > 0 v1 > 0 1 > |2|

Matrix population models Constant linear models A reductible matrix has some stages that make no contribution to some other stages Example: 1 2 3 4

Matrix population models Constant linear models A matrix is primitive if the greatest common divisor of the lengths of the loops in the life-cycle graph is 1 Example: Primitive Imprimitive 1 2 3 4 1 2 3 4

N(t+1) = M N(t) N(t+1) = M N(t) = Mt N(0) Matrix population models Constant linear models Leslie matrices are often irreductible and primitive N(t+1) = M N(t) N(t+1) = M N(t) = Mt N(0)

  M = WW-1 M2 = WW-1 WW-1 M2 = W2W-1 Mt = WtW-1 Matrix population models Constant linear models Let  an eigenvalue of the matrix M Let N(t) the right eigenvector of the matrix M associated to  Diagonalisation of the matrix M M = WW-1  M2 = WW-1 WW-1  M2 = W2W-1 Mt = WtW-1

Matrix population models Constant linear models Mt = WtW-1 Mt = WtV*

Matrix population models Constant linear models N(t) = Mt N(0) Remember: wi is a column vector and vi* is raw vector so their product is a matrix

Matrix population models Constant linear models It follows that:

Matrix population models Constant linear models Asymptotic results And consequently: The dynamic is driven by the dominant eigenvalue and associated eigenvectors

N(t+1) = M N(t) =  N(t) Matrix population models Constant linear models Asymptotic results It results that asymptotically: N(t+1) = M N(t) =  N(t)

 = er where r is the population growth rate Matrix population models Constant linear models Asymptotic results Asymptotically, the dominant eigenvalue  corresponds to the annual multiplication rate n(t+1) =  n(t)  = er where r is the population growth rate Asymptotically, the right eigenvector associated to the dominant eigenvalue  gives the stable state structure M W =  W

Matrix population models Constant linear models Asymptotic results Asymptotically, the left eigenvector associated to the dominant eigenvalue  gives the stable reproductive value, i.e. the contribution of each state to the population size VM =  V The annual multiplication rate, the stable state structure and the reproductive values depend on the values of vital rates but are independent of initial conditions (ergodicity)

Matrix population models Constant linear models Asymptotic results The damping ratio  expresses the rate of convergence to the stable population structure. In other words, the convergence will be more rapid when the dominant eigenvalue is large relative to the other eigenvalues.

N(t+1) = Mt N(t) Matrix population models Matrix population models Stochastic linear models N(t+1) = Mt N(t) (n,1) (n,n) (n,1) In this case, the matrix M has coefficients varying randomly with time Example: a two age-classes model

Matrix population models Stochastic linear models Growth rate The simple estimation of the multiplication rate, the stable state structure and the reproductive value calculated from eigenvalues and eigenvectors are no longer valid in stochastic linear models The population size at time t is: 

Matrix population models Stochastic linear models Growth rate It has been shown (Furstenberg & Kesten 1960) that: where Lns is the stochastic growth rate

Matrix population models Stochastic linear models Growth rate

Matrix population models Stochastic linear models Growth rate The analytical calculation of Ln s is not easy The stochastic growth rate can be found by Simulation Approximation

å Matrix population models Stochastic linear models Growth rate = - Ln 1. Simulation The stochastic growth rate can be estimated from the average growth rate over a long simulation from the maximum likelihood estimator : An estimation of the stochastic growth rate from a simulation is: å = - i t s Ln n l 1 ) (

Matrix population models Stochastic linear models Growth rate 1. Simulation Usually, the stochastic growth rate is calculated as the average of estimations obtained from several simulations

Stochastic linear models Matrix population models Stochastic linear models Growth rate 1. Simulation: environmental stochasticity P(1) B1(s0,s0) B1(p1,p1) B1(s1,s1) B1(s2,s2)

Matrix population models Stochastic linear models Growth rate 1. Simulation: demographic stochasticity

Matrix population models Stochastic linear models Growth rate 1. Simulation: demographic stochasticity Number of reproductive females : Number of young produced :

Matrix population models Stochastic linear models Growth rate 2. Approximation The stochastic growth rate can be analytically approximated when vital rates vary in a small way that is, their coefficients of variation are much less than 1 If we assume that all distributions are identical and independent, then, the stochastic growth rate is approximated by: where is the mean growth rate

Matrix population models Perturbation analysis From a biological point of view, it is important to know which factor has the greatest effect on  Perturbation analyses allow to predict the consequences of changes in the value of one (or more) vital rate on the value of  Two concepts: Sensitivity Elasticity

Matrix population models Perturbation analysis 1. Sensitivity The sensitivity sij indicates how the value of  is impacted by a modification of the value of the parameter aij  The sensitivity is dependent on the metric of the parameter aij

Matrix population models Perturbation analysis 2. Elasticity The elasticity eij indicates the relative impact on  of a modification of the value of the parameter aij and The elasticity is independent on the metric of the parameter aij

Matrix population models Perturbation analysis 3. Prospective analyses The prospective analyses are done by calculating sensitivities or elasticities

Matrix population models Perturbation analysis 4. Retrospective analyses The objective of a retrospective analysis is to quantify the contribution of each vital rate to the variation in  The variability of  may be due to variations in vital rates in space (between populations) or in time (between years) for example The impact of a given component aij of the projection matrix on the variation of  depends on both the variation of the component and the sensitivity of  to this component

Matrix population models Perturbation analysis 4. Retrospective analyses A component aij will have a small contribution to the variation in  if this component does not change much or if  is not very sensitive to aij or if both are true. The contribution of aij to the variation in  involves the products of sij and the observed variation in aij Two approaches are possible: random effects fixed effects

Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects We want to identify the contribution of vital rates to the variance in  during a time period for example (where time is considered as a random factor) Covariances between parameters are calculated directly from observed matrices Sensitivities are calculated from the mean matrix

Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects The contribution of the vital rate aij can be measured as: It is the sum of contributions including aij

Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects Tuljapurkar showed that: The contribution of the component aij can be measured as:

Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects These formula can also be rewritten as: The contribution of the component aij can be measured as:

Matrix population models Perturbation analysis 4. Retrospective analyses b. Fixed effects We want to identify the contribution of vital rates to the difference in multiplication rates between traitments. To simplify, consider two populations in different environments. These populations then vary according to the values of vital rates and, consequently in projection matrices. To these projection matrices A1 and A2 correspond two multiplication rates respectively equal to 1 and 2

Matrix population models Perturbation analysis 4. Retrospective analyses b. Fixed effects It can be shown that:

Matrix population models Perturbation analysis 4. Retrospective analyses b. Fixed effects The contribution of the vital rate aij to the difference in  is given by: This may be generalised to more than two traitments

Matrix population models Perturbation analysis 4. Retrospective analyses b. Fixed effects This may be generalised to more than two traitments where r is a reference traitment where Cm is the vector of contributions Cm gives the contributions of vital rates to the difference between the traitment m and the reference traitment r

Metapopulation Concept of Metapopulation

Metapopulation Definition A metapopulation is a population of local populations that are susceptible to extinction and that are inter connected by migration. The persistance of the metapopulation depends on a stochastic balance between local extinctions and recolonisation of empty sites.

Metapopulation

Population of populations: diffrent types Metapopulation Population of populations: diffrent types « island/mainland » « source/sink » Metapopulation Isolated

Metapopulation Extinction Stochastic causes Demographic stochasticity Genetic stochasticity Environmental stochasticity Deterministic causes

Metapopulation Extinction General message The risk of local extinction depends on : the patch size the mean growth rate the variability of the growth rate

Metapopulation Migration - Colonisation Two processes: Migration (dispersal) Colonisation

Migration - Colonisation Metapopulation Migration - Colonisation General message The success of dispersal depends on : the distance between patches the realistion of particular conditions (corridors)

Effect of the presence of corridors Metapopulation Migration - Colonisation Effet of the distance of dispersal Effect of the presence of corridors

Migration - Colonisation Metapopulation Migration - Colonisation General message The success of colonisation depends on : the number of immigrants

Metapopulation dynamics The Levins model (continuous time) t1 ti tn

Metapopulation dynamics The Levins model (continuous time)  The metapopulation will persist if c > e Analogy with the logistic model

Metapopulation dynamics The island/mainland model t1 ti tn

The metapopulation will persist as long as Metapopulation dynamics The island/mainland model  The metapopulation will persist as long as colonisation occurs

Metapopulation dynamics Spatially explicit models t 1-pi pi t+1 Ci(t) ei

Metapopulation dynamics Spatially explicit models

Structured populations: constant models Metapopulation dynamics Structured populations: constant models Life cycle in the case of several populations 1 2 3 4 j Sites 1 2 3 4 k

Fragmented populations: deterministic models Metapopulation dynamics Fragmented populations: deterministic models number of newborn females in site j born to a female of age i in site k

Fragmented populations: deterministic models Metapopulation dynamics Fragmented populations: deterministic models proportion of surviving females in site j from those of age i-1 in site k

Fragmented populations: deterministic models Metapopulation dynamics Fragmented populations: deterministic models

Fragmented populations: deterministic models Metapopulation dynamics Fragmented populations: deterministic models N*(t+1) = M N*(t) (np,1) (np,np) (np,1) Det (M – I) = 0   annual multiplication rate of the metapopulation N*(t) right eigenvector  stable age structure per site