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-Patches are large enough to accomodate populations, but not larger -If the assumption that space is discrete can not be defended, then some other approach should be used. -The more discrete and smaller the local breeding populations are, the more useful the metapopulation approach is likely to be
-Migration has no real effect because the rate of migration is very low
Need local density dependence for long term persistence of extinction-prone populations Results in a stationary distribution of population densities. Without density dependence, local populations have such a high extinction rate that the metapopulation can not persist for a long period of time.
Allee Effect: -Females may have lower fecundity because it takes longer to find a mate or they may not be able to find a mate at all - individuals may leave the population to find a larger one -immigrants may avoid small populations Patch Area: assuming larger patches mean a population is larger, expect that probability of extinction decreases with increasing area - The more individuals there are in a population, the more likely it is that some will do relatively well even when most perform poorly
-Metapopulations persist because populations that go extinct are recolonized with migrant from other populations. Therefore, you need a high enough level of migration for the metapopulation to exist -However, migration often entails a cost to migrating individuals, so too much migration may result in quicker metapopulation extinction Why migrate: avoid inbreeding, avoid sib competition, densities are too high (too much resource competition), densities are too low (too hard to find a mate), escape imminent extinction, temporal variance in fitness (spread offspring across lots of patches to hedge bets against a poor year) propagule size:number of immigrants arriving at an empty patch Rescue Effect: immigration decreases the probability of local extinction
Habitat fragmentation results in fewer patches, so the patches are not as connected -harder to recolonize these patches - account for this in the model by reducing the colonization parameter
-This model, in practice, gives good approximations for networks of 100+ patches - local dynamics are independent (extinctions occur independently in different habitat patches) **Since the patches are the same and the setting is stochastic, the populations settle into a quasi-stationary distribution of population sizes, so that on the time scale of extinctions and colonizations, local dynamics (birth rates, death rates, etc) can be ignored! ** -model gives a description of rate of change in metapopulation size based on stochastic local extinctions (rate of change is measured by the fraction of patches occupied at time t)
dP/dt= Rate of change in metapopulation size c= colonization parameter P= fraction of currently occupied patches (source of colonists) 1-P= fraction of currently empty patches (targets of colonization) e= extinction parameter So, as long as e/c is less than 1, the population is expected to persist (because P will be positive)
Ignores differences in patches, local dynamics, etc. In this model, metapopulations survive in a colonization-extinction equilibrium
Used when spatial locations of individuals in populations or local populations in metapopulations seem to greatly influence a phenomena of interest (for example,a species only occurs in a small fraction of suitable habitat) Good for studying general questions about metapopulation dynamics -Includes cellular automata models, interacting particle systems, and the basic lattice models -Assume that local populations are arranged as cells on a regular grid, or lattice -populations are assumed to interact ONLY with populations in the nearby cells- localized interactions can mean that it takes a long time for the metapopulation to settle to a steady state -Migration is distance dependent- takes spatial structure into account if a patch becomes empty due to extinction, there is a probability that cell x will send out a propagule to cell y with a probability Rate of emigration(y-x), which typically depends on the distance between x and y
-Migration is distance dependent- takes spatial structure into account if a patch becomes empty due to extinction, there is a probability that cell x will send out a propagule to cell y with a probability Rate of emigration(y-x), which typically depends on the distance between x and y
-Since each cell on the grid is the same (constant area and constant spacing), mathematical rules that govern behavior are the same from cell to cell, so it is easy to write a computer program to model the dynamics -To describe the state of a metapopulation, you need an entire vector of presences and absences, which requires a lot more computation
Used when you want to make quantitative predictions about real metapopulations -Allow a person to include in the model specific geometry of patch networks (i.e. how many patches are there, how large they are, and where they are located) -randomly scattered cells takes habitat fragmentation into account -patches can differ in area (assumed to affect local extinction probabilities). If the patches are relatively small, it may be advantageous to treat the patches themselves rather than cells -patches have spatial locations (assumed to affect probabilities of recolonization) ** Therefore, there is a patch-specific long-term probability of occupancy**
This model takes this information for every single patch and determines their pressure on patch i by creating a huge complicated formula
-Since you are including specific info, you can make predictions about metapopulation dynamics -These models are more complicated mathematically. -Can begin to model birth, movements, deaths, reproduction, but this does not really take advantage of the metapopulation concept (limited migration, random extinctions and recolonizations, etc)
Habitat patches are of equal size and are equally isolated
Migration has no real effect on dynamics
2 ways to think of migration
5.
Density Dependence
Per capita growth rate depends on past and/or present population densities
not all populations are strongly affected all the time
Morris and Doak, 2002
6.
Extinction
Risk of extinction increases with decreasing population size
Allee Effect-Reduced per capita growth rate at low population density
Habitat fragmentation- reduces patch area
can be easily modeled
7.
Migration
Need a sufficiently high rate of migration for metapopulations to persist as a result of recolonizations
If migration rate is too high, may accelerate metapopulation extinction
Rescue Effect/Propagule Rain
8.
Colonization
Habitat Fragmentation- reduces level of patch connectivity
can be easily modeled
9.
Spatially Implicit (Levins) Model
Assume:
infinite, discrete habitat patches with no variation
stochastic, asynchronous local dynamics
local dynamics be ignored
all patches are equally connected via migration
patches are empty or occupied
Hanski and Gilpin 1997
10.
Levins Model
dP/dt= cP(1-P)-eP
Equilibrium when P hat = 1-e/c
Highlights that recolonization must occur at at a high enough rate to compensate for extinctions
11.
Advantages and Disadvantages of the basic Levins model
+ Easy mathematically and conceptually
- Can only answer a limited number of questions because it ignores so many variables (generally can only be used for metapopulations close to a steady state)
12.
Spatially Explicit Models
Assume:
many patches (arranged as cells in a lattice)
patches may be occupied or empty
no variation in patch size and quality
migration is distance dependent
Hanski, 1999 Fig. 5.3A
13.
Spatially Explicit Model
Probability that a cell will become occupied = γ(y-x)
Where gamma is the rate of emigration and y-x is the distance between x and y
14.
Advantages and Disadvantages
+ Local behavior is same from patch to patch, so dynamics can be easily modeled.
- Can not simply describe the state of the metapopulation by the fraction of patches occupied (need to use a vector- much more complicated)
15.
Spatially Realistic Model
finite number of relatively small patches in comparison with the total landscape
randomly scattered patches
assume real patch attributes (area, location, etc)
patch area and isolation affect extinction and recolonization
occupied patches inflict colonization pressure on all empty patches
declines with distance
Hanski, 1999 Fig. 5.3
16.
Spatially Realistic Model
d p i /dt= C i ( t)[1-p i ]- e i p i
(d p i /dt= rate of change in patch i )
pi= probability that patch i is occupied
C i (t)= colonization rate in patch i, taking connectivity between all patches into account
ei= extinction rate in patch i , which is a function of the area of patch i
17.
Advantages and Disadvantages
+ More realistic; can make quantitative predictions about dynamics
- More complicated; A lot of data has to be assumed; Starts to move away from the metapopulation concept
18.
Works cited
Driscoll, D. 2007. How to find a metapopulation. Can. J. Zool., 85 : 1031-1048.
Hanski, I. 1999. Metapopulation ecology. New York: Oxford University Press.
Hanski, I and Gaggiotti, O (Eds.). 2004. Ecology, genetics, and evolution of metapopulations. New York: Elsevier Academic Press.
Hanski, I and Gilpin, M (Eds.). 1997. Metapopulation biology: Ecology, genetics, and evolution . New York: Academic Press.
Hanski, I and Gilpin, M, 1991. Metapopulation dynamics: brief history and conceptual domain. Biological Journal of the Linnean Society, 42 : 3-16.
Morris, W and Doak, D. 2002. Quantitative Conservation Biology . Sunderland: Sinauer Associates, Inc.
Nie, L and Mei, D. 2007. Fluctuation-enhanced stability of a metapopulation. Physics Letters A, 371 : 111-117.