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 EndelsLaboratory for Forest, Nature and Landscape Research, KULeuvenVital Decosterstraat 102B-3000 Leuven, BelgiumTel +32(0)16.32.97.69 Fax +32(0)16.32.97.60

whats a matrix projection model?what data is needed?assumptions / restrictionscalculation and interpretation of the projection matrix and its associated valuesapplicationsperturbation analysisRetrospectiveProspective Grimes CRS vs. Silvertowns GLF: a demographic interpretation of functional plant groupsProjectionsStochastic modellingsuggested 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:

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

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?

-> going from life cycle graphs to projection matrices

The projection matrix, A

Projection matrices and population growth rate

Reproductive values

Chart16

0.02448550610.0035739857

0.05781489970.0157874016

0.07877576870.0265717011

0.15167149350.0827986426

0.27393600470.2291473449

0.41331632730.6421209242

alt 00-01

vro 00-01

klassenverd 1_2-5_6-

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klassenver1_2-3_4-

transitionstotals/yearproj matricessensitivitieselasticitiesreprod valuestable stage distrdistr tdistr t+1

lambda2.083

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