Lecture 6: Population modelling 2 Flashcards
modelling populations with greater realism – often models are not realistic :
Conservation and complexity in population models
Adding realism to simple population models:
– density dependence
– stochasticity –random elements in pop. density
Population structure
– spatial structure and metapopulations – aggregation in certain areas
– age, stage and matrices – variations between individuals
– individuals – variations in single individuals
Density dependence
2 types:
– negative (competition) - when pop. Grow theres enough resources (more births than deaths) this decreases when survival rate or birth rate decreases
– positive (“Allee effects”)
Simple model – pop growth rate doesn’t change with density – exponential
(Nt+1=Nte^r)
A population model with Neg. Density dependence changes in increase according to carrying capacity – until pop growth rate = 0 sigmoidal growth rate
(Nt+1=Nte^r)^r(1-Nt/k)
Abundant evidence of NEGATIVE density dependence in a wide variety of taxa:
- linear decline in growth rate vs abundance in Black throated blue warbler
- abundance of wilderbeest up to 10million no effect of abundance on growth rate but sharp drop off after that
*voles small increase in density leads to fast decline and eventual recovery
see graphs in notes
We are fairly certain neg. Density dependence must be considered in pop modelling
Theta logistic added to normal growth model makes estimates more accurate:
Theta = 1 -> normal model (green line) sigmoidal growth curve linear decrease in pop growth rate w/size of pop
When Theta<1 more rodent-like – longer to get to carrying capacity more density dependence at lower pop. density
When Theta >1 more like the wilderbeest curve - initially not much impact rapidly increasing with pop size
SEE: Sibyl et al & Clark et al. - the assumptions we make are not very biologically realistic – may invalidate this approach
Positive density dependence
Literature emphasises negative DD
– fundamental to stability
– necessary for harvesting to work – to identify sustainable harvest limits (I.e. pop kept at 50% of carrying capacity.)
Less attention to positive DD
– “Allee effects” (e.g. Stephens & Sutherland 1999)
– positive relationship between individual fitness and population size or density I.e. intrinsically small pop. May have a neg. DD and therefore prone to extinction
– important impacts on population dynamics, especially in conservation
Positive density dependence:
A range of mechanism examples for Allee effects:
*gannets – mating takes 2! Gathering at breeding grounds to find a mate
*frogs – illustration from “the selfish herd” – aggregating is safer
*guillemots – colonial breeding à predator satiation
*meerkats – collective vigilance
*wolves/musk ox – collaborative hunting
*wild dogs/hyena – collective defence of kills
^all reasons why more conspecifics doesn’t necessarily mean you’re worse off!
^ Example of positive DD:
Gathering at breeding grounds - gannets
Selfish herd – grouping together and cooperating for less chance of being prey + can rely on other individuals for vigilance
in predators larger groups benefit in taking down larger prey and guarding kills
Positive density dependence equations:
modelling DD:
dN/Ndt = r(1-N/K)(N/t-1)
^ the neg DD equation is made positive when N overcomes T becoming a ‘hump’ on the graph
dN/Ndt = r(1-N/K) - a theta/ theta +N
^ using theta to demonstrate real world situations – to id exact density where Allee effects are overcome for more realism
The meaning of K in these equations
K: Different ways you can model it phenomenologically (like the phenomenological model of negative DD). Nothing profound about these – but notice that they usually acknowledge that growth declines at lower population sizes, that it might become negative (i.e., an extinction vortex), and that growth might be depressed across a wide range of population sizes (not just a feature / problem of small populations)
Positive density dependence :
Positive density dependence :
* Difficult to demonstrate
* Must distinguish between component and demographic effects
see in notes Kramer et al 2009 – These are ways that demographic effects on the overall growth of a population have been shown
→ Data support the notion that Allee effects are widespread. They should be considered as important components of demographic models.
Allee effects basically increase extinction risks in small populations and may affect population growth even for larger populations
Population trends in real world data – does it match the model predictions?
see notes for graphs
^ Lande et al 2003
a lot of noise – no smooth curve observable – not following exact lines given in models
Except the Seychelles warbler
Hard to observe whether DD is pos or neg
Huge swings in pop. Size observable
Stochasticity - limitations
Simple models:
– deterministic – given conditions always lead to same outcomes
– unrealistic – allow partial individuals
Real populations subject to stochasticity:
*demographic
*births and deaths happen as integers, not fractions
*sex ratios no. Of breeding females determines no. Of offspring
*environmental
-changing conditions from one year to the
next & catastrophes
Central limit theorem
demographic stochasticity
affects individuals independently and becomes less important as population grows – therefore a problem of small populations
^if sample size is large enough the likeliness that the majority align with the mean is more likely reducing the importance of individual variation
Demographic stochasticity cannot be used for small pop. size
Environmental stochasticity
– affects all individuals together
– not restricted to small populations can be used for small and large pop
Stochasticity: two simple models
Stochasticity: two simple models
– population capped at a “ceiling”, 200
– demographic stochasticity in survival (s = 0.45)
– all individuals breed – but …
– … fecundity differs in the 2 models
Model 1: demographic stochasticity:
each individual produces 1 offspring (with p = 0.5) or 2 offspring (with p = 0.5)
(individual basis)
Model 2: environmental stochasticity:
all individuals produce 1 offspring in bad years and 2 offspring in good years;
good years happen with p = 0.5 (pop wide)
see graphs in notes:
IF populations grow far enough to escape small size, demog stoch ceases to matter. However, environ stoch ALWAYS matters – large numbers don’t insulate you from catastrophes
Environmental stochasticity depresses the median outlook for a population!
– e.g. an analogy with stocks:
– £10,000 investment
* +80% return in 50% of weeks
* -60% return in 50% of weeks
* average return is 10% increase per week
– who wants to invest?
it is a risky gamble 3 out of 4 outcomes results in a loss
Environmental stochasticity also includes catastrophes
– continuum of environmental conditions
– some extreme examples (catastrophes) –
e.g. severe winter + low food availability reduced a population of reindeer on St. Matthew Island from 6000 to 50 individuals!
e.g. a storm that killed 800 seal pups at Donna Nook
Summary
Simple population models ‘have no biology’ (Boyce 1992, p492)
Fundamental additions include:
density dependence -both negative and positive:
positive (Allee effects) may be more common than is easy to demonstrate – should be included in population models
stochasticity:
-both environmental and demographic
-stochasticity leads to extinction in deterministically stable models
-stochasticity depresses median population growth rates (and, for capped populations, mean
growth rates)
