Space, Islands, and Metapopulations
Until this point, we have largely considered the interactions and
dynamics of species in a single location, making the tacit assumption
that all individuals have the potential to interact, and, even further,
all individuals interact with one another with the same effect
(i.e., \(\alpha_{ij}\) is a constant).
In this chapter, we will at least extend things out into space to
consider how population size (or density) changes across different
spatial dimensions. For instance, we can consider
- \(P(x)\): number of individuals at
single point in one dimensional space
- \(P(x,y)\): number of individuals
at single point in two dimensional space
- \(P_i\): number of individuals
within some confined area \(i\)
For many applications, we don’t even necessarily care about density,
but simply if the species is present in a given location. These binary
data are incredibly useful for estimating things like
occupancy, which is the fraction of cells or patches that are
occupied by a species across the landscape.
Spatial population dynamics are typically considered in one of two
ways. First, habitat patches of different areas are connected via
dispersal pathways, such that there is some habitat which is not
occupied, and species move between habitat patches that can be occupied.
Second, we divide the landscape into a lattice (a grid of equal area
cells) and estimate density in each cell. There are some important
assumptions about both methods.
What are they?
A two-patch model of competition
Theoretical ecology in this space loves two-patch systems. So much of
metapopulation theory was developed considering just two habitat
patches, connected by varying levels of dispersal. Case sets up the two
patch problem as an interspecific competition problem. Species A is in
one patch, species B is in the other. Both species have Lotka-Volterra
type dynamics. Patches are functionally distinct in that there is no
dispersal to begin with, then we slowly add dispersal connections. What
happens (Figure 16.3)? Before, we had one value for each species for an
interior equilibrium, but the connections between patches mean that we
now have an interior equilibrium of 4 points (one for each speices-patch
combination). This influences the zero net growth isoclines of species
across space (now considering the phase space as defined by the density
of the same species, but at the two different patches; Figure
16.4-16.6). That is, we now have to consider dispersal \(\mu\) in estimates of equilibrium
population density.
Another important conclusion of the two-patch system is that two
species that cannot coexist in a single patch are able to coexist in a
two-patch system with limited dispersal. If dispersal becomes too high,
the two patches operate more like a single patch, and that coexistence
is lost.
A two-patch predator-prey model
We can also consider a predator and a prey in a spatial two-patch
landscape.
How might space influence predator-prey dynamics?
Let’s start with the Lotka-Volterra model of predator (\(P\)) and prey (\(V\)) dynamics.
\[ \frac{dV}{dt} = V(b-aP) \] \[ \frac{dP}{dt} = P(-d + kaV) \]
Recall what the phase space looks like for this model, and the
dynamics of this system in a single patch system (Figure 16.8).
In the absence of the predator, the prey population,\(V\), grows exponentially with an intrinsic
growth rate, b. The prey death rate increases linearly with the number
of predators. Predators have a type 1 functional response, \(kaV\), with an encounter rate parameter,
\(a\). The zero-isocline is depicted in
Figure 16.8. Trajectories spiral around the equilibrium point, but the
cycle is neutrally stable and determined by the initial conditions. Now
imagine two patches that are adjacent to one another. The predator and
prey occur in each patch, and individuals of both species can
potentially move between the two patches.
So now we have 4 differential equations, just as in the two-patch
competition model above (see Equation 16.2). Let’s consider that first
equation describing prey populations in patch A and see if we can find
the equilibrium point.
\[ \frac{dV_A}{dt} = V_A(b_A-a_AP_A) -
\mu_{V,A}V_A + \mu_{V,B}V_B \]
where \(\mu\) terms describe the
movement of prey between patches. Solving for \(\frac{dV_A}{dt}\) leads to
\[ 0 = b_A - a_AP_{A}^{*} - \mu_{V,A} +
\mu_{V,B}\frac{V_B^*}{V_A^*} \]
So we have not managed to isolate \(V_A\) (which is fine), but we also can see
that this expression will be non-linear, as changes to \(P_A\) fundamentally change the ratio \(\frac{V_B^*}{V_A^*}\). So an already
coupled system of predator prey is now also influenced by dispersal.
But this equation also shows us something else interesting. Assume
for a moment that the two patches are identical and movement between
patches is the same (\(\mu_{V,A} =
\mu_{V,B}\)). This equation then reduces down to
\[ 0 = b-aP_A^* \]
which does not contain the influence of dispersal. This
equilibrium value is the same as for a single patch system. That is, if
movement is equal and patches are equal in their ‘quality’ (i.e.,
there’s no difference in species demographic parameters between
patches), then predator prey dynamics in both patches will be the same
as in the single patch situation. It is only when we add differential
movement or differences in demographic rates across the two patches that
we reveal different dynamics.
We can see what happens when we make patch B slightly ‘better’ for
predators (slightly lower death rate) and prey (slightly higher birth
rate) (Figure 16.9 with no movement), and how this changes when we
consider predator and prey to move with different rates (Figure
16.10).
Vandermeer et al. (1980) performed an experiment to test these ideas.
The “patches” were piles of compost. In the experimental piles,
one-quarter of each compost pile was split off and the portions were
transferred to each of the other piles twice a week. The control group
was identical but no transfers took place. The result was that, while
the average species number per patch was similar in both groups, the
overall number of species in the entire collection of patches was about
50% less in the experimental group with enhanced dispersal. Thus species
composition differed more from pile to pile in the control group than in
the experimental “mixed-up” group of populations. Piles were not
experimentally isolated, so it is unknown whether the diversity
reduction that resulted from one-quarter transfers could be duplicated
by experimentally enforcing less colonization than existed in the
controls.
Larger spatial arenas
Two-patch systems are incredibly useful, but we often have more than
two patches, or (worse yet) we have a continuous landscape. In this
case, we can imagine dividing the two patches in half to create a four
patch system, and dividing them further and further until we approximate
a continuous system in which we consider species dynamics at a single
point in two dimensional space. The result is the arrival at the
reaction-diffusion equation described at the top of page 390.
These bizarre dynamics are not simply theoretical constructs without
bearing on the real world. Maron and Harrison(1997) described stable
patches of tussock moth caterpillars feeding on their host plant, bush
lupine, which is continuously distributed. Moth eggs and larvae that are
experimentally introduced outside these patches do fine, so it seemed a
mystery why the patches of caterpillars did not expand over time. The
tussock moth females are wingless, so dispersal is limited. They are
also predated and parasitized by several other species with greater
mobility. Reaction-diffusion models of this interaction predict stable
patches. This occurs because predators(or parasites) spill over the
edges of prey patches, creating zones on the boundary where predator to
prey ratios are elevated. Farther away from existing tussock moth
patches, these ratios are lower and the prey population can increase
(Brodmann et al. 1997). This suggests that new stable patches could be
created if large enough experimental introductions are made.
Colonization and extinction dynamics of a single patch
We are going to pivot away from thinking about estimating density
across space here to focusing a bit more on presence-absence of species
across space. This will set up the underlying theory of
metapopulations as they were originally described. Consider a
large habitat patch with variable habitat, some favorable and some less
so. We may want to simply describe the occupancy dynamics in this single
patch. But we can’t simply look at a single point in time and act as if
the system is not dynamic. So we must model the colonization and
extinction dynamics in each part of the large patch. Here, we care about
the fraction of occupied patches \(J\).
\(\frac{dJ}{dt}\) = gains through
colonization - losses due to extinction
We consider losses due to extinction to be dependent on the fraction
of patches currently occupied \(J\),
such that extinction is modeled as \(eJ\). Colonization is a bit trickier, as
there are at least two ways to define colonization in this simple
model.
Levins model
We’ll start with how colonization is defined in a foundational
metapopulation model; the Levins’ model. Levins created a simple model
focused solely on patch occupancy (i.e., is the species present or
absent) as a way to mathematically assess the proportion of occupied
patches by a species given minimal demographic information. In this
case, local habitat patches are either occupied or unoccupied, and both
patch number and the spatial orientation of patches are undescribed.
Dispersal among habitat patches can rescue patches from extinction, or
allow for the recolonization of extinct patches. All patches are treated
as equal, so that any patch is suitable for a species, and (as a
simplifying assumption) all habitat patches can be reached from all
other patches. This simplified representation treats space as implicit,
and patch quality and size as constant; rather than an explicit
population size, patch occupancy is just a 0 or 1 state.
\[\begin{equation}
\frac{dJ}{dt} = cJ(1-J) - eJ
\end{equation}\]
where the change in the number of occupied sites (\(J\)) by a species is a function of
colonization rate \(c\) and extinction
rate \(e\).
The equilibrium fraction of patches that should be occupied via
colonization and extinction rates is
\[\begin{equation}
J^* = 1 - \frac{e}{c}
\end{equation}\]
Further, this model can be used to generate a threshold condition for
metapopulation persistence, which relates to the balance between
colonization and extinction rates, and is analagous to population growth
rate in the logistic model. That is, a metapoulation will persist if
\[\begin{equation}
\frac{e}{c} < 1
\end{equation}\]
That is, when extinction rate becomes larger than colonization, the
metapopulation will not persist. This shows that even a metapopulation
in equilibrium is still in a constant state of patch-level flux. In real
applications, this implies that just because a patch of habitat is
empty, that may not imply it is uninhabitable; and similarly, just
because a population goes extinct, it may not be indicative of broader
declines or instability.
Mainland-island systems
Colonization comes from a single source, and isn’t dependent on the
fraction of occupied patches. This basically assumes the idea of
“propagule rain”, that a constant supply of immigrants are provided and
patches become colonized from this mainland source.
\[\begin{equation}
\frac{dJ}{dt} = c(1-J) - eJ
\end{equation}\]
This changes our equilibrium fraction of occupied patches though.
Making this assumption shifts the metapopulation \(J^*\) to
\[\begin{equation}
J^* = \frac{c}{c+e} = \frac{1}{1 + \frac{e}{c}}
\end{equation}\]
Note here the effects of the propagule rain. Across a large range of
extinction rates (\(e\)), \(K\) still may be relative unaffected. That
is, colonization processes become far more important here, as the
fraction of occupied patches at equilibrium is now basically the
fraction of colonization relative to extinction. This also brings up the
existence of sources and sinks. The mainland is
assumed to be a source here, defined as those patches with
positive growth rates even in the presence of emigration (these patches
are creating a bunch of individuals and then those individuals are
dispersing). Sink populations are those that persist, but have
a negative population growth rate, such that they are only maintained
via immigration of individuals from other patches.
The effects of island area and isolation
The approach above is spatially agnostic, as we only cared
about the fraction of occupied patches, not really paying attention to
the distribution of patches across the landscape. We’ll remedy this now,
by explicitly considering colonization to be a function of distance
between patches.
\[ c(d) = \omega exp(\dfrac{-d}{d_0})
\]
where \(\omega\) and \(d_0\) are species-specific parameters
related to their dispersal kernel. We can explore how these change the
shape of the kernel (Figure 16.11)
distance <- seq(0,20, length.out=1000)
d00 <- 5
d01 <- 10
omega0 <- 1
omega1 <- 2
layout(matrix(c(1,2),ncol=2))
par(mar=c(4,4,0.5,0.5))
plot(distance, omega0*exp(-distance/d00),
type='l', xlab='Distance', las=1,
ylab='Colonization rate (c(d))',
main='Changes to d0')
lines(distance, omega0*exp(-distance/d01), col='dodgerblue')
legend('topright', pch=16, col=c(1, 'dodgerblue'), bty='n',
c('d0 = 5', 'd0 = 10'))
par(mar=c(4,4,0.5,0.5))
plot(distance, omega0*exp(-distance/d00),
type='l', xlab='Distance', las=1,
ylab='Colonization rate (c(d))',
main='Changes to omega')
lines(distance, omega1*exp(-distance/d00), col='dodgerblue')
legend('topright', pch=16, col=c(1, 'dodgerblue'), bty='n',
c(expression(omega~"="~1), expression(omega~"="~2)))
Extinction will likely not vary as a function of distance between
patches (though we will revisit a related idea), but island area may
influence population size and availability of habitat, such that
extinction is likely related to patch area.
\[ e(A) = \dfrac{\epsilon}{A}
\]
where \(\epsilon\) is some constant
estimated from the species data. We can now substitute these functions
for \(c(d)\) and \(e(A)\) into our estimate of \(J^*\).
\[ J^{*} = \dfrac{1}{1+
\frac{\epsilon}{A\omega exp(-d/d_0)}} \]
we can see how new formulation influences the equilibrium fraction of
occupied patches (\(J^*\)) in Figures
16.12 and 16.13.
The number of species on islands
The balance between extinctions and colonizations of single species
influence metapopulation dynamics. This same idea can be applied to
communities to predict the equilibrium number of species on islands.
This is the heart of the theory of biogeography (Figure
16.14)
Potential effects of competition on extinction rates
Extinction may increase not only as a function of area, but also
based on the number of competing species. That is,
\[ e(A,S) = \dfrac{\epsilon S^{x}}{A}
\]
where extinction \(e\) is a function
of area (\(A\)) and species richness
(\(S\)), where \(x\) and \(\epsilon\) are species-specific parameters
estimated from data. Generalizing to all species in the community, we
can estimate the extinction function for all species as
\[ E'(S) = \dfrac{\epsilon S^{x+1}}{A}
\]
Interspecific competition could also effect the shape of the
colonization curve. Case (1990) simulated colonists arriving in model
Lotka-Volterra competition communities of different species numbers. The
result was that the success of these invaders, evaluated as their
ability to increase when rare in the presence of the residents (see
Chapter 15), declined, on average, with increases in species number
(Figure 16.19).
Differences among species
Species will differ in their colonization and extinction
probabilities, which could be modeled by making \(c\), \(\omega\), and \(d_0\) species-specific estimates.
\[ c_i(d) = \omega_i exp(\dfrac{-d}{d_0i})
\] and
\[ e_i(A) = \dfrac{\epsilon_i}{A}
\]
The distributions for these three specific biogeographic rate
parameters, \(\omega\), \(d_0\), and \(e\), are usually not known. However, the
mere fact that they differ from species to species allows for some
partial conclusions based on the following logic. Some species will be
good colonists; they will tend to be the first to arrive on islands and
the most likely to be present when \(S\) is low. Other species will be poor
colonists and will tend to arrive late; they will thus usually be
restricted to islands with high \(S\).
These factors will qualitatively affect the curves \(C'(S)\) and \(E'(S)\), as shown in Figure 16.20.
How does the equilibrium theory of island biogeography account for
differences among species in colonization and extinction rates?
Rescue effects
Above, we treated the probability of extinction as independent from
the fraction of occupied patches. However, what if local patch-level
extinction probability (\(e\)) was a
function of the fraction of occupied patches(\(N\))? Dispersal individuals from occupied
sites serve not only to (re)colonize habitat patches, but also to
provide individuals to other already established populations. Thus, a
population that may have gone extinct due to small population sizes or
demographic/environmental stochasticity now will not go extinct due to
this extra boost from nearby populations. This boost is the rescue
effect, and was incorporated into the Levins model by Hanski in
1982.
Let’s consider the probability of extinction to depend inversely on
the fraction of occupied patches (i.e., more patches occupied means
fewer extinctions are going to occur). We now consider \(e\) to be similar to the colonization rate,
which depends on the fraction of occupied patches and the availability
of unoccupied patches.
This changes the classic Levins model
\[\begin{equation}
\frac{dN}{dt} = cN(1-N) - eN
\end{equation}\]
by making \(e\) a function of \(N\), and results in the following
\[\begin{equation}
\frac{dN}{dt} = cN(1-N) - eN(1-N)
\end{equation}\]
which doesn’t change the persistence conditions for the
metapopulation as described above or the fraction of occupied patches at
equilibrium.
\[ \dfrac{c_1 - e_1}{c_1} \]
Multi-species extensions
As we saw in the two-patch simple model earlier, species which could
not coexist in the same patch can coexist in a
metapopulation/metacommunity. Community composition in a single patch
varies, as species differ in their ability to colonize patches. Species
may be really good colonizers but terrible competitors (this is often
found in natural systems, and is commonly called a ). Case refers to as
those with high colonization but poor competitive ability.
We can go back to modeling two-species coexistence in the framework
of the Levins model, where we care about the equilibrium fraction of
occupied patches \(J^*\). We can
consider the dominant competitor to have ‘normal’ dynamics (unaffected
by the inferior competitor).
\[ \frac{dJ_1}{dt} = c_1(1-J_1)J_1 -
e_1J_1 \]
and the inferior competitor having dynamics like (Equation 16.19)
\[ \frac{dJ_2}{dt} = c_2(1-J_2-J_1)J_2 -
e_2J_2 - c_1J_2J_1 \]
Solving this system of equations, we arrive at the equilibrium
fraction of occupied patches for species 2 \(J^*_2\) as
\[ J^*_{2} = \left( 1 - \frac{e_2}{c_2}
\right) - J^*_1 \left( 1 + \frac{c_1}{c_2} \right) \]
because recall that \(J_1^* = \frac{c_1 -
e_1}{c_1}\)
This can be reduced to a threshold criteria for species 2 persistence
as
\[ c_2 > \dfrac{c_1(c_1 + e_2 -
e_1)}{e_1} \]
which can be further simplified if we assume that extinction rates
for both species are the same (\(e_1 =
e_2\)) to
\[ \frac{c_2}{c_1} >
\frac{c_1}{e}\]
Essentially, the inferior competitor species 2 must have a greater
colonization rate than species 1, and species 1 should have an
extinction rate less than or equal to it’s colonization rate (which
isn’t too big of an ask).
Another useful generalization is to consider the criterion for
coexistence in terms of the amount of open space left in the
metapopulation with just the superior competitor present:
\[ S_1 = 1 - J_1^* = 1 - \dfrac{c_1 -
e_1}{c_1} \]
Assuming equal extinction rates, we can explore the persistence of
this same inferior competitor, which occurs when
\[ S_1 > \frac{c_1}{c_2} \]
The greater the amount of open space and the greater the colonization
rate of species 2 relative to species 1, the easier it will be for
species 2 to invade.
Adding neighborhood dispersal
The dispersal between any pair of patches is assumed to be an
exponentially declining function of their distance apart. Also since
bigger patches hold more individuals and thus throw out dispersers at a
greater rate, emigration from bigger patches should be higher than for
smaller patches. This influences colonization of patch \(j\) in the following way.
\[ c_j = k \sum_{i=1} A_i e^{-d_{ij}/d_0}
\]
where \(d_{ij}\) is the distance
between patch \(i\) and \(j\), \(A_i\) is the area of patch \(i\), and \(d_0\) and \(k\) are constants. This also allows us to
calculate the extinction probability of a given patch \(j\) as
\[ e_j = \dfrac{1}{A_j} \]
How important is each patch to the persistence of the entire
metapopulation? An answer can be obtained by comparing the persistence
time of the metapopulation with and without each patch. There has been
some development on this front, but we will not go over it, as I am not
entirely sure we’ll make it this far in two lectures.