Lecture 7: Population modelling 3 Flashcards
Conservation and complexity in population structure models
- Previously, we discussed adding realism
– density dependence (pos and neg)
– stochasticity - But we still considered populations modelled as a single number …
- But what about population structure?
– spatial structure and metapopulations
– age, stage and matrices
– individuals
Spatial modelling
Models can be spatially implicit (more theoretical:
– sites are modelled as separate
– rules govern movement
– specific geographic relationships unimportant
- often grid-based
- require movement rules
- are edge effects interesting or incovenient?
Or models can be spatially explicit (more realistic)
– model a specific landscape
– usually tied to a GIS
– often incorporates much fine-scale detail
see notes for examples
Metapopulations
Linked populations:
– “metapopulation”
– metapopulations have their own dynamics
can be modelled explicitly as the sum of the dynamics of each population
or implicitly, by assigning probabilities of extinction or recolonisation to each patch
conservationists talk about “the rescue effect”
(Brown & Kodric-Brown 1977)
- demographic rescue
- genetic rescue
Connectivity
Arguments for:
- Permit metapopulation processes
- Permit gene flow
- Permit large-scale processes (Rouget et al., 2006)
- Linked populations overcome MVP constraints
Arguments against:
- Put eggs back in one basket – e.g. disease – such as in red squirrel repopulation
- Facilitate predation / expansion of invasives – e.g. tunnels designed for animals to cross roads are used by predators to ambush prey
- Requirements hard to characterise -
Connectivity is Increasingly incorporated in PA design
Aichi target 11
“At least 17 per cent of terrestrial and inland water, and 10 percent of coastal and marine areas conserved through well connected systems of protected areas”
GBF target 2
Ensure that by 2030 at least 30 per cent of areas of degraded terrestrial, inland water, and marine and coastal ecosystems are under effective restoration, in order to enhance biodiversity and ecosystem functions and services, ecological integrity and connectivity.
small scale: under- and over-passes to reduce roads as barriers
e.g. Clevenger & Waltho (2000) & Clevenger et al. (2002)
national scale: protected area networks
e.g. Saura et al (2019)
international scale: transboundary parks: can promote natural processes like migration e.g. See natura 2000 network
Demographic structure
Age/stage matters for modelling
– reproductive behaviour varies
– dispersal behaviour varies
– susceptibility to different environmental forces varies
– hence, survival varies
… and for management – e.g.
– harvesting certain life stages – as in fisheries being only allowed to collect adults
– targeting conservation interventions
Demographic structure modelling
Typically modelled using matrix models
– population broken down as a vector
– vector elements represent different ages (Leslie matrix)
– or stages (Lefkovitch matrix) (more flexible!)
e.g see notes for a Lefkovitch matrix
Matrix models
A simple annual population model:
NH1 = lambda Nt
A matrix model:
Nt+1 = MNt
^ where N, is a vector of numbers in each stage
M is the transition matrix
a typical matrix for 3 life stages
The diagonal tells us about proportions of
individuals that remain in the same stage class
Elements below the diagonal tell us about
proportions of individuals that move
to another stage
Columns tell us about the fate of each life stage (how many survive each stage)
The first row shows - fecundity of each life
stage (or proportion of stage 1 individuals that remain in stage 1)
Rows tell us about contributions to each
life stage
Multiplying a matrix by a vector
- Each element in row 1 by each element in the vector, then sum them -> 1st row of new vector
- ditto row 2, ditto row 3 …
see: https://www.youtube.com/watch?v=hQbIg6bmPhU&t=310s
for guidance in calculations
see notes for examples
Why use matrices?
Expressed in matrix form, population models have a variety of very useful properties
matrix multiplication is simple:
* e.g. a single operation in R – one step in code
* requires no more code to run than a simple population model …
Projecting with matrices
Matrices can project:
– after initial jitters, population settles to a stable stage distribution
– and a stable growth rate
Expressed in matrix form population models have a variety of different properties:
mathematically, several attributes of population dynamics are easily determined from the matrix
* stable stage distribution (SSD)
* asymptotic growth rate (lambda)
* reproductive values of stage classes (V)
we can also easily compute the relative influence of changing different matrix elements
* e.g. does it make more difference to target conservation at increasing
fecundity of adults or increasing juvenile survival?
these parameters are known as
* sensitivity (the effect of a small, absolute change in any matrix element)
* elasticity (the effect of a small, relative change in any matrix element)
What’s wrong with matrix models?
So far, considered only Projection matrices
– projection = what would happen given certain assumptions (e.g. that vital rates remain unchanged)
– forecasting = what will happen
For forecasting, need:
– realism
– complexity (e.g. stochasticity, density dependence)
– to consider individual differences … ?
Matrices or individuals?
- Transition probabilities contingent on behaviour
- Behaviour often contingent on an array of individual attributes:
– local characteristics of habitat
– local characteristics of population
– group characteristics
– past experience
– physiological state
– individual qualities, genetics … personality! (boldness, dominance etc.)
→ Increasing use of Individual Based Models (IBMs)
– e.g. see Feró et al. (2008)
Summary
- Further insights require that we begin to treat components of the population separately
- Spatial structure is one example of such complexity
– can model at the sub-population level (rather than the individual level)
– or at the individual level, modelling each sub-population explicitly - Generalised spatial models have important implications for connectivity of populations
– remains contentious but generally believed to be a good thing
– increasingly incorporated into conservation planning - Might also need to break populations down by demographic structure
– matrix models present a simple and analytically-tractable way to do this
– finer-scale demographic structure can be achieved through IBMs
