Pair approximations
in spatial biology
Tim C.D. Lucas†
P
air-approximation methods, as used in ecology, epidemiology and evolutionary biology are examined. A number of different moment closures are
presented, analysed and compared to individual-based simulations. The
closures examined are the unspatial mean-field approximation and then the ordinary
pair approximation, a more generic version of this closure and the Kirkwood closure.
Parallels to the continuous-space literature and the ways discrete models have been
developed and used within the biological literature are reviewed and areas of profitable
research are identified.
1. Introduction
T
he complexity of ecological systems has made them difficult to study with traditional
analytical approaches. The arrival of fast, widely accesible computers has ushered in
an era of individual-based, spatially-explicit ecological models. However, while the
utility of simulations is not in doubt, the results derived from analytically treated models can be
more generic and easier to understand. Therefore, the interface between analytical models and
simulations is an important area of research.
A large proportion of ecological models are formulated under the mean-field assumption;
spatial effects are ignored as the population is assumed to be well-mixed. However, it is clear
that aggregation is a common property of both animal 1–5 and plant populations. 6–8 Whereas
mean-field models assume that individuals are distributed through space randomly, in reality
individual dispersal might be reletively short causing individuals to occur in clusters. Therefore
individuals experience density higher than they would in a well-mixed population. Whenever this
is the case, models should be rooted in a spatial context. Spatial models are, however difficult to
analyse.
Spatial dynamics in ecology are commonly studied using lattice models. 9 This modelling
paradigm treats space as a regular network of spaces. Commonly, networks with a constant
degree of four (i.e. square arrays) are used as this is convenient for computing. As these individual
based simulations are hard to interpret attempts have been made to create analytical models of
these fully specified models. To fully examine the spatial dynamics of populations, one needs to
follow population density (ratio of individuals to lattice spaces) as well as pair density (density
of pairs of individuals within a certain distance from each other) which describes how aggregated
the population is. However, analytical expressions of the density of pairs depend on the density
of triplets which in turn depends on higher order interactions. An intuitive (but incomplete)
way to picture this hierarchy is to consider concentric circles around a focal individual. Only
Supervised by David Murrell and Stephen Baigent.
†Email: timecdlucas@gmail.com.
1
Pair approximations
Tim Lucas
the inner circle directly effects the focal individual, but the state of the inner circle depends
on the circle outside it, whose state is in turn dictated by the circle even further out. This
hierarchy of dependent moments is known as a cascade. One way of dealing with this problem is
to truncate the cascade at a certain level. For example, we can approximate the triplet (and
higher) spatial-correlations as a function of pair density 10 which simplifies spatial dynamics
without completely removing spatial effects. This is an example of a moment closure and the
intended result is a closed system of equations that approximate higher order implementations.
This method is known as pair approximation. 10,11
However, moment closure is not an exact science and there are many ways to close a system
of equations. 12 Many closures are good purely because of their heuristic properties and therefore
searching for closures by considering intuitive approximations is not always effective. Whatever
the closure used 12 the equations should still satisfy a number of conditions; as the spatial scale of
interactions approaches infinity, the equations should reduce to a non-spatial system for example.
Being an approximation method, all solutions will fail at some point. The major trade off between
different moment closures is between simplicity and accuracy. Another consideration is whether
accuracy at steady state is more important than accuracy during the transition towards the
steady state.
I study three different moment closures of a spatial logistic model on a square lattice. The first
closure is the ordinary pair approximation (OPA). 10,13 This closure turns out to be a specific
example of a more general closure which has been considered in continuous-space 14 but not
discrete space. This is the second closure I examine; I refer to this closure as the extended pair
approximation (EPA). Finally, the Kirkwood closure (KPA), 15 which has been rarely used in the
discrete literature (but see Filipe et al. 16 ) is considered.
This report examines some of the details of these closures, specifically identifying when they
are good and when they break down. The paper is organised as follows. First, the ways in
which pair approximation models have been used is reviewed (Section 2). Secondly, a lattice
IBM (Section 3.1) and a pair approximation anologue with different closures (Section 3.2) are
described. Finally a discussion (Section 4) is included.
2. Literature Review
Pair approximation has found applications in many areas of ecology, disease biology and
evolutionary biology. Any process in which interactions occur on a scale smaller than the entire
environment should be considered in a spatial context. As mentioned before, while simulations
are often the only way to incorporate all facets of a system in a model, pair approximation can
provide analytically tractable models which are often easier to interpret and generalise. Ordinary
pair approximation is the most commonly used closure. It rests on the assumption that the
effects of neighbour-of-neighbours is week. Therefore the expected local density of an individual
next to a full-empty pair is assumed to be the same as an individual next to a full lattice space.
Many extensions to the ordinary pair approximation model have been examined. A purely
heuristic solution to the innaccuracies of the ordinary pair approximation, known as the improved
pair approximation, is to estimate from lattice simulations a correction parameter. 13,17 However,
this approach removes some of the advantages of pair approximation. If the point of pair
approximation is to avoid timely simulations, then a closure that requires a simulation to learn
from is not very useful. More importantly, if we assess pair approximation models by comparing
them to simulations, how do we assess the improved pair approximation?
One common extension to the ordinary pair approximation is to blur the boundary between
mean-field and spatial models by allowing a process to occur both on the local scale and on a
global scale. 11,18–20 Steady state results can become very complicated. 11 This model can also be
used to study evolutionary stability of long vs. short dispersers. 20,21 The examination of other
or generalized network topologies is an ongoing research topic 22,23 while using the information
from topological assumptions can also be used to improve moment closures. 22,24–26 Commonly,
2
Pair approximations
Tim Lucas
A)
B)
=
C)
*
*
*
Figure 1. A) Pairs between an individual (red) and any of its pairs (blue) are considered
equal and contribute to ρ11 equally. B) Rotational and reflective symmetry. All reflected
and rotated examples of a configuration between a focal (red) individual, its neighbour
(blue) and neighbour-but-one (light blue) are equal. C) All possible triplet configurations.
In the closures used here, all configurations are considered equal even though starred
triplets are within the red individual’s neighbourhood while the unstarred individuals
are not. Other closures use the information from this geometry to make closures more
accurate, but they are not considered in detail here.
all configurations of triplets are considered equal — on a square lattice, a triplet in a line is
considered the same as an L-shaped triplet. However, individuals of an L-shaped triplet are all
neighbours of each other (in an eight-tile neighbourhood) while the two end members of a linear
triplet are not in each others neighbourhood (see Figure 1).
An obvious extension to the pair approximation model is to truncate the spatial terms at triplet
densities instead of pair densities thus making a triplet-approximation model. 23,25–27 While this
increases accuracy, the expressions become unweildy, although the results can sometimes be
usefully simple, even if their derivations are difficult.
The application of pair approximation to a number of biological topics are reviewed below.
Although modelling different systems, the models are often very similar. For example, examining
the population growth of a forest with suitable and unsuitable habitats, is completely analogous
to the spread of a disease through a population of resistant and non-resistant individuals.
2.1. Ecology. Possibly the most obvious use of pair approximations and the area in which it was
first applied is spatial ecology. While many populations can be considered well-mixed, species
where movement is slow enough and dispersal short enough will have spatial structure. Pair
approximation can add a spatial element to non-spatial models as in the case of the distinction
between mean-field and pair approximations. It is also used to examine explicitely spatial
questions — such as forest gap size — that lose meaning without a spatial model.
As the focus of this paper is to examine the validity of different moment closures, I use a
simple logistic equation. This model was first proposed by Matsuda et al. 10 This most simple
pair approximation model predicts population densities much lower than mean-field models as
individuals constrained in space suffer density dependent death more accutely than individuals
in a well mixed population. Pair approximation models of population growth, as affected by
spatial considerations, have been improved and extended in a number of ways. One common
way to extend the model given here is to allow a proportion of offspring to be dispersed globally
3
Pair approximations
Tim Lucas
instead of locally. 11,19,20 Another approach is to allow density-dependent death and dispersal
to act on different scales 28 (the death rates in this report are considered density independent,
which relates to a density dependent length scale of zero.)
In parallel to the development of pair approximations in discrete spaces, continuous space
models have also been studied. Both the fully defined simulations and analytical approximations
are significantly more complex in continuous space. 14,29 Interesting dynamics have been found
in continuous space logistic models such as extinction of populations with reletively high birth
rates and steady-state population sizes much higher than mean field predictions due to uniform
dispersal. 14 The examination of closures in continuous space seems to be more in-depth. More
closures are analysed, and the relative advantages of each are examined more carefully. 12
The allee effect is any form of negative density dependence that occurs at low population
sizes. Many non-spatial models have been proposed to model the allee effect. 30 Three main
classes of mechanism can be considered: demographic stochasticity and mate finding, inbreeding
and cooperative behaviour. 31 All of these are affected by local as well as global density and
are therefore suitable for study with pair approximation models. Stewart-Cox et al. explicitely
include allee effects into pair approximation models and consider densities of empty, unpollinated
and pollinated spaces and allow pollination to be global or local. 32 If either pollination or seed
dispersal is global, the model becomes equivelent to mean-field models. However, if there is
a local element to pollination or dispersal, allee effects are weaker or absent due to the local
clumping of individuals. Satō reaches similar conclusions 33 by studying a meta-population, in
which each space in the lattice is a small, transient sub-population rather than an individual.
Heterogeneous environments have been modelled by considering densities and pairs of full,
empty and unsuitable sites 20,34,35 which in the simplest case only requires three state variables. 34,35
These models yield results useful for conservation. A reduced proportion of habitable space
reduces the steady state population size, as does reducing environemental correlation i.e. many
small patches can support a smaller population than a few large patches. However, odd results
can occur from the simplest models; Hiebeler found that only the structure of unsuitable habitat,
and not the proportion, changed the population density. 35
2.2. Disease Biology. There is a large literature on epidemiology of infectious diseases that
uses lattice models. Usually, the models are based around the class of SEIRS compartmental
models that were developed under mean-field assumptions. However, when examined with pair
approximation the large number of equations usually restricts analyses to either SI(S) models
and SIR(S) models. Thus, each node in the lattice is either empty or a susceptible, infected or
resistant host.
The first epidemiological pair approximation models came soon after the first ecological
models. 13 These first models obtained results quantitatively different from mean-field models,
such as pathogen driven extinction in large areas of parameter space. While the authors did
not make strong claims that their model reflected reality, it was clear that spatially modelled
disease dynamics were very different to mean-field dynamics and that the spatial dimension of
epidemiology needed to be studied.
STDs have attracted a lot of attention due partially to the rise of high-profile diseases such as
HIV, but also due to the clear cut nature of network edges; edges join individuals in a sexual
relationship. The topology of lattices has been an important area of research with dynamic
lattices 23,36 and heterogenous lattices 23 recieving particular attention. Intuitive results are often
found; short, frequent sexual relationships and concurrent relationships both increase the size of
an epidemic. 23,36,37
3. Methods
3.1. Discrete-space model. I simulated continuous-time spatial ecology on a discrete-space
s × s lattice (see Table 1 for a list of symbols) where each space is either empty (0) or contains an
individual (1). Periodic boundaries prevent edge effects. For comparison with pair approximations
4
Pair approximations
Tim Lucas
q1 1
Ρ1
0.3
0.2
0.2
0.1
0.1
40
80
Time
120
40
(a) Density
Time
120
(b) Local Density
q1 1
Ρ1
æ
æ æææ
ææææ æ
ææææ
ææ ææ
ææ
0.3
0.3
ææ æ
æ
æ
ææææ æ
0.2
ææææ
æ æ
ææ æ æ
æ æ
æ æ
æ
ææææææææææææææææ
1.2
1.4
1.6
0.2
æ
æ
0.1
0.1
1.0
80
b
1.8
æææææææææææææææ
1.0
(c) Density
1.2
1.4
1.6
b
1.8
(d) Local Density
Figure 2.
A–B) Mean trajectories through time from 15 simulations with b = 0.9, 1.1, 1.3, 1.5 and
1.7. A) Density ρ1 . B) Local density q1|1 . Other parameters as z = 8, s = 20. q1|1 becomes
very variable when ρ1 is small. A birth N = 1 → N = 2 causes q1|1 = 0 → q1|1 = 1/z
C–D) Steady state global (C) and local (D) densities for varying birth rates (b). For
b < 1.3 the population goes to extinction so ρ1 , q1|1 = 0.
the density of individuals is the ratio of population size (N ) and available space so that ρ1 = N/s2 .
Each iteration of the simulation entails either a birth or a death. If an individual dies the site it
occupies changes from 1 → 0. If an individual creates an offspring, one of its neighbouring sites
changes from 0 → 1. If the chosen site is already full the birth is aborted, thus creating density
dependance. I use square neighbourhoods with odd number radii. The size of the neighbourhood
is denoted z (z = 8 unless z is the only paremeter being varied). The number of full sites in an
individual’s neighbourhood is zi1 . As the neighbourhood increases, the spatial structure of the
system breaks down and density is explained by a mean-field birth death process.
To speed up simulations, instead of computing aborted births, the likelihood that an individual
will create an offspring is weighted by the proportion of free sites in its neighbourhood. Thus, an
individual and event is randomly selected in one go by choosing from a weighted vector of all
birth and death events. Deaths are scaled to a weighting of 1 making a unit of time equal to 1
generation. The birth weighting wi of an individual is wi = bzi1 /z and the probability of a birth
or death during an iteration is
wi
i=1 (wi + 1)
1
P (i dies) = PN
i=1 (wi + 1)
P (i gives birth) = PN
(1)
(2)
To make time continuous I used a Gillespie algorithm. 38 Unlike discrete time simulations in which
a variable number of events happen with a fixed-length time step, in the Gillespie algorithm,
5
Pair approximations
Tim Lucas
Table 1. List of parameter and variable names.
Symbol Description
ρi
Frequency of sites of type i
qi|j
Conditional probability of a type-i neighbour
given a type-j site
ρij
Frequency of type i - type j pairs
t
Time
b
Birth rate
d
Death rate (scaled to 1)
z
Number of sites in neighbourhood
j
zi
Number of state-j sites in neighbourhood of
individual i
N
Absolute population size
τ
Time-step between events.
wi
Event weighting
s
Length of one side of the lattice
Type
Model
State variable Both
State Variable Both
Variable
Variable
Parameter
Parameter
Parameter
Variable
PA
Both
Both
Both
Both
IBM
Variable
Variable
Variable
Parameter
IBM
IBM
IBM
IBM
one event occurs in each time step, but the length of the time step is variable. The length
of each time step is drawn from an exponential distribution scaled by the summed rate of all
possible events. Thus, in each iteration of the model, one random event is selected. I then draw
a time-step from a probability distribution based on all the possible events. The two possible
events are birth and death and the summed ‘any-event’ rate is given by Equation 3. The time
between events, τ , is by Equation 5. Note that d = 1 due to rescaling and has no coefficient as it
is density independent.
E=
N
X
zi1
b + d)
(3)
R ∼ U(0, 1)
1
τ = − Log(R)
E
(4)
(
i=1
z
(5)
Time-series from this IBM are shown in Figure 2. It can be seen that only when the birth
rate is above about 1.5 does the population avoid extinction. As the population becomes small,
q1|1 becomes very variable. In the limit of a change from a population of 1 to a population of a
11 pair, q1|1 changes from 0 to 1/z. This variability cannot be captured by any of the moment
closures.
3.2. Pair approximation model. We want to examine the density of individuals ρ1 and the
density of pairs ρ11 . However, the final equations are easier in terms of conditional probabilities,
q1|1 . This is the probability that a space is full, given that its neighbour is full and is analogous to
the mean of zi1 for all i. A number of identities connect these quantites. This model is constucted
under the assumption of rotational symmetry (see Figure 1) where a 01 pair and a 10 pair are
equivelent in all contexts; they contribute equally to ρ01 and a 101 triplet and a 110 triplet would
contribute equally to q1|01 . Equation 6 explicitely states this assumption. Equations 7 – 10 are
true by definition.
6
Pair approximations
Tim Lucas
ρ01 ≡ ρ10
(6)
ρ1 ≡ 1 − ρ0
(7)
ρ11 ≡ ρ1 q1|1
(8)
ρ10 ≡ ρ1 q0|1 ≡ ρ0 q1|0
(9)
q1|1 ≡ 1 − q0|1
(10)
We can write a simple birth-death process of the proportion of lattice-space filled, ρ1 with death
rate d and birth rate b (Equation 11). Dividing by d rescales time so that 1 unit of time is 1
generation and reduces the number of parameters to one. In the IBM, density dependance is
included by aborting births that land on a full tile. Births therefore occur at a rate of bzi0 ρ1 .
Thus in the analytical model, births occur at a rate bq0|1 ρ1 (Equation 12) as q0|1 is analogous to
the population mean of zi0 .
dρ1
= −dρ1 + bρ1
dt
dρ1
= −ρ1 + bq0|1 ρ1
dt
(11)
(12)
As ρ1 is dependent on local density q0|1 , we must construct an equation for the second spatial
moment. Even though we want q1|1 as our state variable, we write an equation for ρ1 1 as it is
more intuitive.
dρ11
b
b
= −2ρ11 + 2 ρ10 + 2 (z − 1) q1|01 ρ10
dt
z
z
(13)
where z is neighbourhood size as in the IBM. The first term describes the loss of pairs. Pairs of
full spaces are lost when either of the pairs die i.e. 11 → 10 or 11 → 01; hence the coefficient of 2.
New pairs are created either by an individual creating an offspring next to itself (10 → 11 or
01 → 11) or by a neighbour-but-one individual creating an offspring next to the focal individual
(101 → 111). These pair-birth rates are described by the second and third term in Equation 13
respectively. This last term includes the third spatial moment q1|01 which is, in turn, dependent
on higher order moments. This is a cascade of moments as discussed above and means the system
is not closed and cannot be solved. Thus an approximation is needed to close the system; this is
the role of the closures discussed in Sections 3.3 – 3.6.
We want to examine the state variables ρ1 and q1|1 (equations for q1|1 turn out to be simpler
than for ρ11 ). As the density of pairs is equal to the product of singlets and local pairs (from
ρ11 = ρ1 q1|1 ) we can write an expression for the dq1|1 /dt as in Equation 14.
dq1|1
d(ρ11 /ρ1 )
ρ11 dρ1
1 dρ11
=
=− 2
+
dt
dt
ρ1 dt
ρ1 dt
(14)
For each closure, we find an approximate expression for q1|01 which is substituted into Equation 13.
Equations 12 and 13 are then substitued into Equation 14 to find an equation for dq1|01 /dt in
terms of only state variables and parameters. This and Equation scaled are then a closed system
of ordinary differential equations.
7
0.00
0.15
1
q1
0.15
0.00
ρ1
0.30
Tim Lucas
0.30
Pair approximations
2
4
6
8
2
Radius (r)
4
6
8
Radius (r)
Figure 3. Simulation results (mean after 30 events burn-in) for population density and
spatial second moment for different radii with mean and 95% confidence intervals shown.
Dashed red lines show the predictions from mean field equations. Mean field models the
system well at r = 6. Parameters used: b = 1.1, s = 20, z = 8.
3.3. Mean field approximation. In a mixed population, local density is equal to global density.
Therefore, the mean field assumption can be explicitely stated as
q1|1 ≈ ρ1
(15)
Thus Equation 12 becomes a standard logistic birth-death process with density dependent births
dρ1
= −ρ1 + bρ1 (1 − ρ1 )
dt
(16)
As this equation does not depend on any second-order terms we can neglect equations for
second-order dynamics. Figure 3 shows that this model succesfully predicts population density
for IBMs run with sufficiently large nieghbourhood as such large dispersal breaks all spatial
correlation.
3.4. Ordinary pair approximation closure. The ordinary pair approximation assumption
is that the indirect effect of neighbours-of-neighbours is small and therefore their density be
considered as equal to the density of direct neighbours. Thus
q1|01 ≈ q1|0 =
(1 − q1|1 )ρ1
ρ10
=
ρ0
1 − ρ1
(17)
Identities 7 – 10 can be used to find an exppression for this approximation in terms of triplet
densities as shown below. Either form (but rarely both) are used in the literature. While
Equation 17 is more intuitive, Equation 18 leads more easily to the other approximations used
and so is useful to include.
ρijk ≈
ρij ρjk
ρj
Substituting Equation 17 into Equation 13 yields the approximation
8
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Pair approximations
Tim Lucas
dρ11
ρ10 ρ1
b
b
= −2ρ11 + 2 ρ10 + 2 (z − 1)ρ10
dt
z
z
1 − ρ1
(19)
Substituting Equations 12 and 19, along with identities 7 – 10 gives us a closed set of differential
equations in terms of only the state variables ρ1 and q1|1 and the parameters z and b.
dρ1
= − ρ1 + b(1 − q1|1 )ρ1
dt
h
i
dq1|1
= − q1|1 −1 + b(1 − q1|1 )
dt
"
#
)
(
1
1 (1 − q1|1 )ρ1
+ 2 −q1|1 + b
(1 − q1|1 )
+ 1−
z
z
1 − ρ1
(20)
(21)
These equations give the phase plots in Figure 4. The steady states are given by
ρ∗1 =
ρ∗1
b(z − 1) − z
1
∗
, q1|1
=1−
b(z − 1) − 1
b
∗
q1|1
= 0,
=
2b + z(1 + b) ±
(22)
q
−8b2 z − [2b + z(1 + b)]2
2bz
(23)
If we set ρ∗1 = 0 in Equation 22 and solve for b, we see that this positive steady state becomes zero
when b = z/z − 1. For values of b lower than this, there is only the steady state in Equation 23.
This steady state is charecterised by extinction (ρ1 = 0) but positive q1|1 . This is a failing in
the closure as, from Equation 8, q1|1 = 0 when ρ1 = 0. This relates to the fact that in an empty
lattice, an individual will have no individuals in its neighbourhood. To the best of my knowledge,
this fact is not mentioned in any of the many papers that use this closure. 10,20,34,35,39
The phase plots in Figure 4 show that the positive steady state in (Equation 22) is globally
stable when it exists. Thus whenever birth rate is sufficiently larger than death rate, the
population should persist.
3.5. Extended pair approximation. The ordinary pair approximation is just one of many
possible closures. It is in fact a special form of a more general class of closures that have been
used in the continuous literature. 14
ρijk
1
≈
α+β+γ
ρij ρjk
ρjk ρki
ρki ρij
α
+β
+γ
ρj
ρk
ρi
!
(24)
When i = j = k (i.e. ρ111 or ρ000 ) or β = γ = 0 and α > 0 this closure equals Equation 18. In
this analysis I use ijk = 101 and α = β = γ = 1. However, studies of continuous systems have
heuristically found β = γ = 1, α = 4 to be a good form of this closure. 14 As these parameters
simply define the proportional influence of the difference focal individual in each triplet, setting
one parameter to 1 can always reduce the number of parameters. Getting this into an approximate
value for q1|01 and only in terms of the two state variables and constants, gives
q1|01 ≈
ρ1 + 2q1|1 − 3ρ1 q1|1
3(1 − ρ1 )
Substituting this into Equations 13 and 14 and rearranging gives
9
(25)
Tim Lucas
1.0
1.0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
(a) OPA, b = 1
(b) EPA, b = 1
(c) KPA, b = 1
1.0
1.0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
q1 1
1.0
q1 1
q1 1
q1 1
1.0
q1 1
q1 1
Pair approximations
0.4
0.4
0.2
0.2
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Ρ1
(d) OPA, b = 1.6
(e) EPA, b = 1.6
(f) KPA, b = 1.6
Figure 4. Stream plots for z = 8 and ABC) b = 1, DEF) b = 1.6 using the three closures.
Steady states are marked with a black dot. With b = 1, all three populations yield a
population that goes to extinction. For b = 1.6 there is positive steady states in OPA
and EPA but not the kirkwood closure.
dρ1
= − ρ1 + b(1 − q1|1 )ρ1
dt
dq1|1
1
1 2bρ1 (1 − q1|1 )
= − (ρ1 + b(1 − q1|1 ))q1|1 −
dt
ρ1
ρ1
z
2b(z − 1)ρ1 (1 − q1|1 )(ρ1 + 2q1|1 − 3ρ1 q1|1 )
− 2ρ1 q1|1 +
3z(1 − ρ1 )
(26)
(27)
This system of ODEs yields the steady states
ρ∗1
∗
q1|1
p
2 + b − 2z + 2bz ± 4 + 4b(z − 1)2 + 10z − 5z 2
=
6 − 3z + b(z + 2)
b−1
=
b
(28)
Unlike the other two closures, this system has only two steady states. When b < 1 the steady
states are complex, making this a rather unuseful closure. Furthermore, at b > 1 the steady state
∗ is greater than zero. As mentioned before, this is an inconsistency. However, at
local density q1|1
b = 1 this system has a steady state at ρ1 = 0, q1|1 = 0. This is the only system to have a steady
state at this, true extinction, point. The phase diagrams for this system are shown in Figure 4.
10
Pair approximations
Tim Lucas
3.6. Kirkwood closure. A rarely used closure in discrete space (Filipe et al. 16 apply the closure
to discrete space epidemiology models while Diekmann and Law 12 use the closure in continuous
space) that has been found to have useful properties are Kirkwood style closures. 15 While the
ordinary pair approximation is known as a power-2 closure as it contains pair densities squared,
the Kirkwood closure is a power-3 closure. It takes the form
ρij ρjk ρki
ρi ρj ρk
(29)
q1|1
q1|1
=
1 − ρ1
ρ0
(30)
ρijk ≈
which yields
q1|01 ≈
This can be seen as a corrected form of the ordinary pair approximation. When population
density is high, the probability of have a neighbour-but-one is higher, thus in this closure q1|01
increases as ρ1 increases. Again we substitute this into Equations 13 and 14 and rearrange to
find an equation for the state variables.
h
i
3z(ρ1 − 1)q1|1 + b(q1|1 − 1) z(ρ1 − 3)q1|1 + 2(ρ1 + q1|1 − 1)
dq1|1
=
dt
z(ρ1 − 1)
(31)
This system of ODEs yields the steady states
b−1
1
∗
, q1|1
=
b + z − bz
b
√ √
3bz − 3z − 4b − 3 8bz + 3z 2 − 6z 2 b + 3b2 z 2
∗
∗
ρ1 = 0,
q1|1 =
2(6 + 2b − 3z + bz)
ρ∗1 =
(32)
(33)
These steady states are never positive. Therefore this system never has a positive population
density and cannot describe the lattice model which has stable populations for b > ∼1.3.
4. Discussion
Figure 5 shows the trajectories of all three moment closures as well as simulated time series from
the IBM. As the Kirkwood closure always predicts population extinction, the population density
when b = 1.6 is qualitatively incorrect. The extended pair approximation grossly overestimates
population density at positive population values (Pane B) and underestimates the speed at which
extinction is reached (Pane A). No closures accurately predicts local density during extinction
(Pane C), although the extended pair approximation is probably closest.
The ordinary pair approximation is the best closure, although it overestimates population
density significantly at stable positive populatiaon densities. As discussed, the extinction stable
state is also qualitatively incorrect in its prediction of steady state local density. This can be
seen in the positive value of this in Pane C.
Previous work in the continuous literature has suggested that the extended pair approximation
with different values for α, β and γ might give more accurate approximations. 14 This approximation should be studied in more depth in discrete space. It is internally consistant, unlike the
Kirkwood close, but has more flexibility than the ordinary pair approximation. As mentioned
before, 1 of the 3 parameters defining the relative influence of the three individuals in each triplet
(α, β, and γ) can be set to zero as the closure as a whole is normalised by the 1/(α + β + γ)
11
Pair approximations
Tim Lucas
Ρ1
Ρ1
0.4
0.4
0.2
0.2
0
10
20
30
Time
0
(a) Density. b = 1
20
30
Time
(b) Density. b = 1.6
q1 1
q1 1
0.4
0.4
0.2
0.2
0
10
10
20
30
(c) Local Density. b = 1
Time
0
10
20
30
Time
(d) Local Density. b = 1.6
Figure 5. Plot showing numerical solutions to ODE’s for OPA (blue), EPA (red) and
KPA (purple) as well as an average of 20 simulated time series (black). b = 1.3, z = 8, s =
20.
term. However, this still leaves a large amount of parameter space to search. If these parameters
are set by training them against simulations, we arrive at the same problem of validation as
discussed for the improved pair approximation. Therefore, a number of different parameter sets
should be tried, but training should be avoided.
Although using the geometry of a lattice can make models more accurate when compared with
IBMs on that specific lattice, they suffer from a lack of generality. When accurately describing a
specific simulation is the aim, the motivation for these closures is clear. However when the aim
is to describe generic lattice simulations, continuous-space simulations or real world situations,
these models may reduce accuracy rather than increase it while incurring the cost of increased
complexity. Similarly, the improved pair approximation is ‘improved’ only in the sense that it
closer approximates a specific lattice simulation. This heuristic correction is likely to be more
robust to applications on other geometries and continuous space, but whether it applies to real
life situations is debatable.
The most glaring gap in the literature on pair approximation models is the lack of empirical
tests of the models. However, some examples occur in the epidemiology literature. For example,
Xiao et al. formulated a meta-population model of Salmonella transmission in cattle. 39 In their
model each node can be a susceptible, infected or resistant herd (not individual) of cattle or
a clean or infected environment (a novel way to model free-living bacteria.) The model yields
interesting oscillations as well as useful indications into efficient intervention strategies. Filipe et
al. statistically fitted a pair approximation model to time-series data collected from experimental
radish-Rhizoctonia solani systems. 40 They recover biologically plausible parameter estimates
from the time-series and also obtain qualitatively correct dynamical behaviour. Interestingly,
the parameter estimates with largest standard error are those for primary infection rates. It
12
Pair approximations
Tim Lucas
has been noted that ordinary pair approximation is least accurate at very low densites 25,41 such
as when a new disease is entering a population. Other experiments in mesocosms of benthic
invertebrates, plants or microbial species would likely be reletively easy to perform, if potentially
time consuming.
It is worth remembering that pair approximation is only one approach to studying spatial
systems. For example, Snyder and Nisbet 41 present a model based on the observation that spatial
structure (ρ10 /ρ1 (1 − ρ1 )) increases almost linearly with an increase in population density. They
use this to construct a model that is simpler and more accurate than pair approximations. Their
model is less flexible than pair approximation as the relationship that is the foundation of the
model falls apart if dispersal is based on a more complex kernal than simple nearest neighbour
interactions. A mix of nearest neighbour and global dispersal can however be included.
While pair approximation is a very useful tool, especially at equilibrium, other approaches
should continue to be studied. Similarly, while discrete space, square lattice pair approximation
makes comparison to simulations easy, it is not obvious how results from these models will
compare with continuous space or empirical results. These models are generally formulated
in highly regular spatial arenas, whose geometry is decided based on computational, and not
biological, considerations. Only comparisons to empirical datasets can truly validate these
models.
4.1. Conclusions. I have presented and analysed a suite of moment closures for a pair approximation of the spatial logistic model. I have not found any closures that improve on the ordinary
pair approximation. However, I have highlighted some failings with this closure that have not
been commented on. Further study, especially on the extended pair approximation closure and
in empirical tests of these models, is advised.
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