n(t + 1) = A n(t) A basic introduction to the construction, analysis and interpretation of matrix projection models (for the evaluation of plant population viability and management) Patrick Endels Laboratory for Forest, Nature and Landscape Research, KULeuven Vital Decosterstraat 102 B-3000 Leuven, Belgium Tel +32(0) Fax +32(0)

what’s a matrix projection model? what data is needed? assumptions / restrictions calculation and interpretation of the projection matrix and its associated values applications –perturbation analysis Retrospective Prospective –Grime’s CRS vs. Silvertown’s GLF: a demographic interpretation of functional plant groups –Projections –Stochastic modelling suggested reading

Method: The linear time-invariant matrix model: n(t + 1) = A n(t) Age vs. Stage classification Determination of the number of life stages: 2 approaches - numerical approach: algortihms for calculating the number of life stages (Moloney, Vandermeer), based on minimising the error due to within cell variance while minimising the error due to small samples - biological approach: relies on field observations of developmental states (Lefkovitch, Werner) => best solution if multiple populations need to be compared Calculating fecundities: - the actual seedling recruitment in year X divided by the number of reprodructive adults in year X-1 - proportional distribution over the different reproductive stages according to fitness characteristics

Assumptions: 1)Discrete model: taking population relevees at pre-determined time intervals results in a realistic impression of (actual) population dynamics? => continuous processes (flowering, growth, etc.) are reduced to discrete events. 2) Transitions (better: transition prob.) between different stages are the same for all the individuals in one life stage 3)Time-invariance (important restriction in projection analysis)

Case study: Primula veris & P. vulgaris Stage classification = a combination of both reproductive and size criteria: Seedlings: individuals developed directly after the germination of seeds, with cotyledons still present and often also one ‘normal’ leaf-pair. Juveniles: immature plants without cotyledons and with only one rosette of leaves. Juveniles can only be distinguished from vegetative adults with one rosette by means of their size: an individual is considered as an adult when its leaf size is comparable to flowering plants in the same population; if leaf size is significantly smaller then the individual is assigned to the juvenile category. Vegetative adults: non-flowering individuals without cotyledons, with one or more rosettes, often showing signs of overwintering leaves. Leaf size is comparable to generative adults which are growing under similar conditions. Reproductive adults: plants baring one ore more flowering stalks, having one or more rosettes and often showing signs of older, overwintering leaves. These flowering adults were divided into three size categories according to the number of rosettes.

P. verisP. vulgaris

Data? Life stage 1999 # rosettes 1999 Life stage 2000 # rosettes 2000 … Plant 1 fl. adult5n-fl. adult10 Plant 2 seedling1juvenile1 Plant 3 juvenile1 Plant 4 n-fl. adult1fl. adult2 Plant 5 seedling1 …

-> going from life cycle graphs to projection matrices SJ NFARA1RA2RA3

YEAR t seedlingjuvenileNF Adult Repr Adult 1 Repr Adult 2 Repr Adult 3 seedling000F1F1 F2F2 F3F3 YEAR t+1 juvenileG 21 L 22 0F4F4 F5F5 F6F6 NF Adult G 31 G 32 L 33 L 34 L 35 L 36 Repr Adult 1 G 41 G 42 G 43 L 44 L 45 L 46 Repr Adult 2 G 51 G 52 G 53 G 54 L 55 L 56 Repr Adult 3 00G 63 G 64 G 62 L 66 The projection matrix, A

SymbolDefinitionDemografic interpretation A a square matrix containing the coefficients that represent proportions of – mostly year-to-year – transitions between life stages. projection matrix a ij the element in row i, collum j of the projection matrix Amatrix element the dominant eigenvalue of Apopulation growth rate  A  =  (the right eigenvector of A associated with ) stable stage distribution  A =  (the left eigenvector of A associated with ) reproductive values s ij  /  a ij (the sensitivity of to changes in matrix element a ij ) sensitivity e ij a ij  /  a ij (the proportional sensitivity of to proportional changes in matrix element a ij ) elasticity  1 / 2 (ratio between the dominant & subdominant eigenvalue) damping ratio

Projection matrices and population growth rate

Reproductive values

Stable stage distribtution

Perturbation analysis: determining the relative importance of changes in vital rates to population growth rate Which of the stages (better: vital rates) is most important to population growth? Prospective (hypothetical changes, sensitivity and elasticity analysis) vs. retrospective (actual changes, LTRE) methods Prospective: Sensitivity (additive) vs. Elasticity (proportional) –E: takes the actual life cycle into account –S: appropriate technique for evolutionary questions retrospective: aims at quantifying the contribution of each of the vital rates to the variability of in different situations Applications: –Life history theory: relationship between changes in vital rates and fitness –Conservation biology: relationship between changes in vital rates and population growth rate –Ecotoxicology,…..

YEAR t seedlingjuvenileNF Adult Repr Adult 1 Repr Adult 2 Repr Adult 3 seedling000F1F1 F2F2 F3F3 YEAR t+1 juvenileG 21 L 22 0F4F4 F5F5 F6F6 NF Adult G 31 G 32 L 33 L 34 L 35 L 36 Repr Adult 1 G 41 G 42 G 43 L 44 L 45 L 46 Repr Adult 2 G 51 G 52 G 53 G 54 L 55 L 56 Repr Adult 3 00G 63 G 64 G 62 L 66 Life table response experiments (LTRE) vs. Sensitivity / elasticity analysis Perturbation analysis: determining the relative importance of changes in vital rates on population growth rate YEAR t seedlingjuvenileNF Adult Repr Adult 1 Repr Adult 2 Repr Adult 3 seedling000F1F1 F2F2 F3F3 YEAR t+1 juvenileG 21 L 22 0F4F4 F5F5 F6F6 NF Adult G 31 G 32 L 33 L 34 L 35 L 36 Repr Adult 1 G 41 G 42 G 43 L 44 L 45 L 46 Repr Adult 2 G 51 G 52 G 53 G 54 L 55 L 56 Repr Adult 3 00G 63 G 64 G 62 L 66 A 1 => S 1, E 1 A 2 => S 2, E 2

grassland restoration site forested site sensitivities

Elasticities Sum to unity => comparison of species or individual populations of the same species Stages with highest mortality (‘bottlenecks’) are not neccessarely those with the highest elasticity (sensitivity) values G-L-F approach: demographic interpretation of Grime’s C-S-R system Differences in G-L-F can be interpreted as trade-offs among life history parameters ( Shea et al.) G-l-f depends on the number of stages YEAR t seedlingjuvenileNF Adult Repr Adult 1 Repr Adult 2 Repr Adult 3 seedling000F1F1 F2F2 F3F3 YEAR t+1 juvenileG 21 L 22 0F4F4 F5F5 F6F6 NF Adult G 31 G 32 L 33 L 34 L 35 L 36 Repr Adult 1 G 41 G 42 G 43 L 44 L 45 L 46 Repr Adult 2 G 51 G 52 G 53 G 54 L 55 L 56 Repr Adult 3 00G 63 G 64 G 62 L 66 E

C ≈C ≈ ≈ S R ≈R ≈ Grime’s CSR vs. Silvertown’s GLF:

CL CL FOR FOR REF99-00 REF00-01 Fitting individual populations into the successional trajectory

LTRE aims at quantifying the contribution of each of the vital rates to the variability of Demographic alternative for ANOVA Fixed & random designs => Factorial designs: –Main effects: combination of differences in individual matrix entries and sensitivity of that entry, calculated from a “mean” matrix –Interactions

Projections & stochastic modelling projections vs. predictions Incorporating stochasticity - Mean matrix = deterministic behaviour - Stdev = stochastic component Several model runs with different settings, based on the actual projection matrix and a series of stdev (popproj, ramas,…) => PVA, extinction probability Applications: guidelines for management (disturbance- recovery cycles,…), extinction risk assessment…

Conclusions Time-consuming & not possible for every (plant) species Very easy to compare the demographical behaviour of populations / species More reliable in combination with long term (less-detailed) population studies Management actions need to be based on a combination of sensitivity analysis and common sense Projections vs predictions vs modelling

Suggested reading: Matrix projection models, theory and methods - van Groenendael, J., de Kroon, H., & Caswell, H. (1988) Projection matrices in population biology. TREE. 3(10), Tuljapurkar, S. & Caswell, H. (1997) Structured population models in Marine, terrestrial, and freshwater systems. Chapman & Hall, New York. 643p. -Caswell, H. (2001) Matrix population models - construction, analysis, and interpretation. Sinauer, Sunderland, MA. 722p. Applications: conservation and management of plant populatons - Valverde, T. & Silvertown, J. (1998) Variation in the demography of a woodland understorey herb (Primula vulgaris) along the forest regeneration cycle projection matrix analysis. Journal of Ecology. 86, Fiedler, P. L., Knapp, B. E., & Fredericks, N. (1997) Rare plant demography: lessons from the Mariposa lilies (Calochortus Liliaceae). p in: Conservation biology for the coming decade (eds. Fiedler P. L. and Karieva P. M.) Chapman & Hall, New York. Algortihms for calculating the number of life stages - Vandermeer, J. (1978) Choosing category size in a stage population matrix. Oecologia. 32, Moloney, K. A. (1986) A generalized algorithm for determining category size. Oecologia. 69,

Suggested reading (continued): Sensitivity and elasticity analysis - de Kroon, H., Plaisier, A., van Groenendael, J., & Caswell, H.(1986) Elasticity, the relative contribution of demographic parameters to population growth rate. Ecology. 67, Silvertown, J., Franco, M., & McConway, K. (1992) A demographic interpretation of Grime's triangle. Functional Ecology. 6, Silvertown, J., Franco, M., & Menges, E.( 1996) Interpretation of elasticity matrices as an aid to the management of plant populations for conservation. Conservation Biology. 10, LTRE - Caswell, H. (1989) Analysis of life table response experiments I. Decomposition of effects on population growth rate. Ecological Modelling. 46, Ehrlén, J. & van Groenendael, J. (1998) Direct perturbation analysis for better conservation. Conservation Biology. 12, Caswell, H. (2000) Prospective and retrospective perturbation analyses their roles in conservation biology. Ecology. 81, More on the demography of P. veris in Voeren - Endels, P., Jacquemyn, H., Brys, R., Hermy, M. (2002) Response of Primula veris populations to ecological restoration: linking fitness-related characteristics with demography (subm. J. App. Ecol.)