Population 3: Matrix population model

Question 1

Simple demographic model

Answer 1

Nt+1/Nt= Lambda

dN/dT= birth - death

These models can be used to predict future popualtion sizes

N1= N0 lambda

Question 2

Problems with simple demographic models

Answer 2

The do not consider density dependence

They do not consider environmental dependece

They do not consider individual heterogeneity

Question 3

Matrices

Answer 3

Matrices can be used to display the heterogeneity of individual in a population.

Question 4

Matrix population models

Answer 4

Matrix population models can be used to model the dynamics of heterogeneous populations, taking into account the different fecundity, birth rate and development of individuals of different ages/stages.

Key points:
- Models that describe populations in discrete time
- Individuals are categorised in terms of traits

Question 5

Leslie matrix

Answer 5

This is an age-based model.

Fecundity: Per capita contribution

Survival: Proportion

This gives number on the sub diagonal (ageing) and the main row (reproducing)

Question 6

Lefkovitch Matrix

Answer 6

This is a stage based model

Fecundity: Per capita contribution

Progression: Probability

Regression: Probability

Question 7

Population projections from age/ stage structured population matrix

Answer 7

If we know Nt (the sum of all individuals in each age) and also the matrix, then we can work out the population size in the future (knowing exactly how many individuals in each stage/ age)

Brute force method:
If this process is repeated over and over the population growth rate (lambda can be calculated)

The dominant eigenvalue:
There is also an analytical method of calculating population growth rate.

Question 8

Sensitivity

Answer 8

How much would the population growth rate change if you varied any of the elements in the matrix

S= change in lambda/ change in specific element

This is repeated for every element to create a sensitivity matrix.

Proportional sensitivity analyses often do not reflect the feasibility and effectiveness of altering particular demographic parameters

Question 9

Elasticity

Answer 9

Elasticity is a sensitivity where the magnitude of the perturbation is relative to how big the thing being perturbed is.

As fecundity values are larger, pertubations have less of an impact leading to lower sensitivity.

Question 10

Example of use of sensitivity analyses

Answer 10

Loggerhead turtle conservation -> which stage to target

Sensitivity analyses showed the key for improving the outlook for these populations is reducing the mortality in the later stages of development (especially of large juveniles)

By increasing the survival of large juveniles a much larger number of turtles are likely to reach maturity

Yangtze River fish
- Compared relative importance of fertility, juvenile survival and adult survival
- Juvenille and adult survival has greater effect of lambda than fertility
- Elasticity of ferility < Adult + Juvenille survival

Question 11

Including economic costs into conservation

Answer 11

Management costs can be included in matrix analyses to find the most suitable conservation mechanisms.

Question 12

Overview

Answer 12

Matrix population models can be used to calculate the population growth, elasticity and sensitivity in heterogenous populations.

The main MPM are the Leslie age-based model and the Lfkovitch stage-based model.

Sensitivity calculations are essential to work out which individuals to prioritise in conservation efforts.

It is also important to consider economic costs when deciding how to prioritise conservation efforts to get the maximum outcome.