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F. Massol, V. Calcagno, J. Massol. The metapopulation fitness criterion: Proof and perspectives.
Theoretical Population Biology, Elsevier, 2009, 75 (2-3), p. 183 - p. 200. �10.1016/j.tpb.2009.02.005�.
�hal-00457617�
The metapopulation fitness criterion: proof and perspectives
François Massola,b,c,∗ , Vincent Calcagnod and Julien Massole
aCentre d’Ecologie Fonctionnelle et Evolutive (CEFE, CNRS) - UMR 5175, 1919, route de Mende, 34293 Montpellier cedex 5, France
bCentre Alpin de Recherche sur les Réseaux Trophiques des Ecosystèmes Limniques (INRA), 75, avenue de Corzent - BP 511, 74203 Thonon-les-Bains cedex, France
cCEMAGREF - UR HYAX, 3275, route de Cézanne, Le Tholonet, CS 40061, 13182 Aix-en-Provence cedex 5, France
dMcGill University, c/o Biology Department, 1205 av Docteur-Penfield, Montreal, Qc, H3A1B1, Canada
eEcole Normale Supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, France
Abstract
Metapopulation theory has recently been stirred by the development of a metapop- ulation persistence criterion (Rm) quantifying the “lifetime dispersal success” of a newly colonized deme in a sparsely occupied metapopulation. No rigorous proof of this criterion in continuous time was available so far. Here, we show that this crite- rion can be mathematically justified from standard Jacobian arguments. A summary of the key elements of this proof emphasizes the assumptions under which this crite- rion is valid. Examples illustrate how to generally compute metapopulation fitness in continuous time models. The underlying assumptions behind the criterion are
discussed, as well as theoretical puzzles surrounding the concept of metapopulation fitness.
Key words: dispersal, fitness, invasibility, metapopulation, persistence criterion, viability
1 Introduction
A topic of primary interest in population biology is the possibility of persis- tence (May, 1999). A biological system is said persistent when all its com-
2
ponents are protected from deterministic extinction. This means that the boundaries at which one component gets extinct are repellent, or equivalently
4
that each component of the system has positive growth rate when rare (for more details on definitions of persistence, see Jansen and Sigmund, 1998).
6
This “protection” criterion is used extensively in population genetics, popula- tion dynamics and ecology (Roughgarden, 1979; Charlesworth, 1994; Caswell,
8
2001; Chesson, 2000).
For single-component systems, such as a single-species population, it is usu-
10
ally called a viability criterion. The viability of a population is commonly as- sessed by computing the basic reproduction ratio (R0) of the species when it
12
is rare (Diekmann et al., 1990; Metz et al., 1992; Mylius and Diekmann, 1995;
Caswell, 2001). In physiologically structured (e.g. age-structured) populations,
14
∗ Corresponding author; phone: +33442669945; fax: +33442669934 Email addresses: [email protected](François Massol), [email protected](Vincent Calcagno),
[email protected] (Julien Massol).
this computation requires knowledge of the structure of the population (e.g.
the age pyramid) and its fecundity function (the expected rate of offspring
16
sired by one adult in each category).R0 is the fecundity averaged over all cat- egories, weighted according to the population structure (Charlesworth, 1994;
18
Caswell, 2001).
In multiple-component systems, e.g. communities of several species, persis-
20
tence can be studied for each species individually, with the background en- vironment (parameter values in the model) being set by the other species in
22
the community. R0 can be calculated as before but assuming that the focal species has no impact on its environment because it is rare (Diekmann et al.,
24
1998, 2001, 2003; Chesson, 2000). In this scenario, the persistence criterion is called an invasibility criterion, since it indicates whether the focal species
26
(the invader) is able to invade the other species (the residents). This criterion constitutes a useful fitness measure to predict evolutionary trajectories under
28
recurrent substitutions of mutants (Metz et al., 1992). Persistence of the whole community is achieved when each species can invade the others (a condition
30
sometimes called “mutual invasibility”).
A difficulty arises when one wants to apply persistence criteria to spatially
32
structured systems, such as metapopulations (Levins, 1969; Hanski and Gilpin, 1997) and metacommunities (Holt, 1997; Leibold et al., 2004; Holyoak et al.,
34
2005). Such systems are made of local subsystems (demes or patches) con- nected by dispersal. In this case, a raw use of R0 as a persistence criterion
36
leads to tedious computations and often lacks simplicity (e.g. Lebreton, 1996).
Dispersal multiplies the number of categories to keep a census of in order to
38
compute the fecundity function of individuals. In addition, qualifying the rar- ity of a species is more difficult in a spatially structured system because a
40
species can be rare in different ways (for instance it can have very few indi- viduals in multiple demes or have a little more of them in a unique deme). As
42
changes of deme state are not unidirectional (a local population can increase or decrease in abundance regardless of whether it increased or decreased in
44
the recent past), a stable deme state distribution is a bit trickier to compute than a stable age distribution in a closed population (but see Diekmann et al.,
46
1990 for complicated, though spatially unstructured, computations of R0).
The quest for a tractable persistence criterion and its application to metapop-
48
ulations and metacommunities has been a recurrent subject of theoretical research (e.g. Chesson, 1984; Lebreton, 1996; Casagrandi and Gatto, 2002;
50
Hastings and Botsford, 2006). One solution is to study the dynamics of state probabilities (e.g. Chesson, 1984; Metz and Gyllenberg, 2001; Casagrandi and
52
Gatto, 2002) rather than the deterministic dynamics of each deme (as when one uses a logistic model of population growth to predict the trajectory of the
54
mean population abundance in each deme, e.g. Hastings and Botsford, 2006).
This kind of formalism is often employed in statistical physics (van Kampen,
56
2007). Focusing on state probabilities without monitoring the state of each deme is justified as long as the number of demes is large (effectively infinite).
58
Under this assumption, mean values of stochastic variables equal their average values over all the demes studied.
60
In two recent papers, Metz and Gyllenberg introduced the metapopulation closest equivalent to the single-population R0 (Gyllenberg and Metz, 2001;
62
Metz and Gyllenberg, 2001). Their fitness measure, called Rm, measures the lifetime dispersal success of occupied demes in a metapopulation: Rm is com-
64
puted as the average number of dispersers produced by a typical colonized deme, from its colonization by a unique immigrant up to its extinction. The
66
proposed method solves the problem of rarity in a metapopulation by comput- ing the "typical rarity" pattern of a species based on its dispersal parameters.
68
This persistence measure was already mentioned in a seminal paper of Ches- son (1981), in the context of discrete-time metapopulation models, and proved
70
in a following paper (Chesson, 1984). Its application to continuous-time mod- els was proposed almost simultaneously by Metz and Gyllenberg (Gyllenberg
72
and Metz, 2001; Metz and Gyllenberg, 2001) and by Casagrandi and Gatto (2002), albeit in different contexts. Metz and Gyllenberg (2001) emphasized
74
the use ofRm as an invasion criterion, whereas Casagrandi and Gatto (2002) used it as a viability criterion. Interestingly, Casagrandi and Gatto (2002)
76
derived Rm from feasibility constraints (i.e. as a necessary condition for the existence of an interior invariant state of the metapopulation), whereas Metz
78
and Gyllenberg (2001) directly defined it as a persistence property. Casagrandi and Gatto (2002) proved that both conditions (existence of a feasible invariant
80
probability measure and viability of the metapopulation near total extinction) are equivalent for their particular metapopulation model, in which there are
82
no deme extinctions. From Markov chain theory, these conditions are indeed linked: the viability criterion means that the Markov chain is irreducible on
84
non-empty states (i.e. there is a path leading from the state where the fo- cal species is rare to a state where it is not), while the feasibility criterion
86
indicates that the Markov chain admits an invariant measure. The viability criterion thus implies the feasibility criterion, but the reverse should not hold
88
in general. Hence, it is probably more honest to attribute the paternity of the Rm criterion to Chesson (1984) and Metz and Gyllenberg (2001), since the
90
case studied by Casagrandi and Gatto (2002) is a particular case of the one envisaged by Metz and Gyllenberg (2001).
92
Whereas Chesson (1984) proved the validity of the Rm criterion for discrete- time Markovian metapopulation dynamics (under specific assumptions), no
94
such proof has been provided for continuous-time metapopulation models.
Gyllenberg and Metz (2001) seemed to imply that the Rm criterion is always
96
true in continuous-time metapopulation models, and only mentioned that “in most cases of interest, positivity arguments” should suffice to ensure this (Gyl-
98
lenberg and Metz, 2001). However, this has not been checked in the papers presenting the basic metapopulation model (Gyllenberg and Metz, 2001; Metz
100
and Gyllenberg, 2001). And, accordingly, positivity of a matrix is a mathe- matical property with little biological meaning. This article has two primary
102
goals: (i) remedying the absence of proof for the continuous-time version of the criterion and (ii) assessing whether the Rm criterion of Metz and Gyllenberg
104
depends on specific assumptions.
We use a general but sensible model of metapopulation with continuous-time
106
dynamics (similar to the one in Gyllenberg and Metz, 2001; Metz and Gyl- lenberg, 2001; Casagrandi and Gatto, 2002). Classically, persistence analysis
108
of a population or metapopulation studied in continuous time (i.e. through differential equations) is carried out by linearization of the dynamical equa-
110
tions. Persistence holds as soon as the Jacobian matrix qualifying the trivial (extinction) equilibrium admits at least one eigenvalue with positive real part.
112
This method of investigation has solid mathematical background (e.g. Horn and Johnson, 1991).
114
Using these standard Jacobian methods, we find necessary and sufficient condi- tions for persistence. We discuss their biological meaning and their connection
116
to Metz and Gyllenberg’s Rm criterion. Our main finding is that the latter verbal criterion is acceptable in all continuous-time Markov process metapop-
118
ulation models. We provide a recipe to obtain analytical expressions of the criterion for a broad class of models, and illustrate it with examples. Com-
120
bined with Chesson (1984)’s proof for discrete-time Poisson processes, our results suggest that the Rm criterion can be applied to typical “memoryless”
122
metapopulations (i.e. models with no time lag between population abundance and recruitment).
124
Given the similarity between the viability criterion in a metapopulation and the invasion criterion in a metacommunity, we present results for single-species
126
systems (viability) only. The extension of our results to invasion are straight- forward, as discussed in the last section. We first summarize the issue and
128
model chosen by Metz and Gyllenberg (2001), using a very simple structured metapopulation.
130
2 Rationale of the Rm criterion
2.1 A simple toy model
132
Consider a closed spatially implicit metapopulation, formed by an infinity of demes, in which every deme has two potential microsites, and thus a popula-
134
tion size of 0, 1 or 2 individuals. At birth, individuals may choose to disperse and, if they do so, they enter a disperser pool. Only individuals that have
136
settled in microsites can reproduce (asexually) and there are no explicit life stages except the potential disperser stage between birth and settlement.
138
Let µi, λi and γi be the rates of per capita mortality, per capita birth, and deme extinction respectively, in a deme containing i individuals (i equals 1
140
or 2). Newborns in a deme with i settled individuals choose to disperse with probability di. Let δ be the density of dispersers in the disperser pool, and
142
assume that this density is homogeneous in space and can be described using a deterministic equation in continuous time. LetαandµDbe the rates at which
144
dispersers try to settle in a deme and the rate at which they die, respectively.
Dispersers arriving in a deme with i settled individuals stay with probability
146
si. A disperser that does not stay in a deme goes back to the disperser pool.
Intuitively, s2 must be zero so that fully occupied demes cannot receive new
148
migrants, andd2must be one, so that newborns in fully occupied demes always disperse. This model is summarized in Fig. 1.
150
Letpk(t)be the proportion of demes that are occupied bykindividuals at time t. The four quantities, p0, p1, p2 and δ obey the following master equation:
152
dp0
dt = (γ1+µ1)p1+γ2p2−αδs0p0
dp1
dt =αδs0p0+ 2µ2p2−(γ1+µ1+λ1(1−d1) +αδs1)p1
dp2
dt = (αδs1+λ1(1−d1))p1−(γ2+ 2µ2)p2
dδ
dt =λ1d1p1+ 2λ2p2−(µD+αs0p0+αs1p1)δ (1)
or, in matrix form:
dP
dt =G(P)P (2)
where P = [p0, p1, p2, δ]and G(P) is the following transition matrix function:
G(P) =
−αδs0 γ1+µ1 γ2 0
αδs0 −αδs1−γ1−µ1−λ1(1−d1) 2µ2 0
0 αδs1+λ1(1−d1) −γ2−2µ2 0
0 λ1d1 2λ2 −(α(p0s0+p1s1) +µD)
(3)
2.2 Applying the Rm criterion
154
We want to assess whether the metapopulation is viable, i.e. grows when individuals are scarce. Intuitively, when individuals are scarce, a good criterion
156
for viability is obtained by looking at the “disperser output” yielded by a unique colonized deme, from its colonization up to its eventual extinction. The
158
Rm criterion (or metapopulation fitness criterion) can be formulated verbally as (Gyllenberg and Metz, 2001; Metz and Gyllenberg, 2001; Parvinen and
160
Metz, 2008):
Criterion 1 A metapopulation is viable if, and only if, the average number
162
of dispersers (emigrants) produced by a population founded by a typical rare immigrant is greater than 1.
164
Here, we develop this intuitive criterion. Suppose that the metapopulation and the disperser pool are empty. Let us impose an extrinsic flow of immigrants,
166
say at rate ǫ. Each immigrant has probability µ αs0
D+αs0 of surviving dispersal and landing into an empty deme, founding a deme with one individual. Demes
168
occupied by one individual are thus created at rateǫµαs0
D+αs0. We want to mon-
itor the flow of dispersers that will be produced by settled individuals, under
170
the sole effect of the extrinsic flow of immigrants (the density of dispersers is thus kept nil). Let z1 and z2 be the probabilities that a deme contains 1 and
172
2 individuals, respectively. Equ. 1 now applies to the system at the viability boundary:
174
dz1
dt = 2µ2z2−(γ1+µ1 +λ1(1−d1))z1+ǫ αs0 µD +αs0
dz2
dt =λ1(1−d1)z1−(γ2+ 2µ2)z2 (4)
Eq. 4 has an invariant measure Z(ǫ) given by:
z1(ǫ) =ǫ αs0
µD +αs0
γ2+ 2µ2
(γ1+µ1)(γ2+ 2µ2) +γ2λ1(1−d1) z2(ǫ) =ǫ αs0
µD +αs0
λ1(1−d1)
(γ1+µ1)(γ2+ 2µ2) +γ2λ1(1−d1)
z0(ǫ) = 1−z1(ǫ)−z2(ǫ) (5)
The rate at which dispersers will be intrinsically produced from settled indi-
176
viduals is thus [λ1d1,2λ2].[z1(ǫ), z2(ǫ)]. For the metapopulation to be viable (following our heuristic criterion), this intrinsic rate (emigrant flow) must ex-
178
ceed the imposed rate of immigrants (immigrant flow), ǫ, i.e.:
[λ1d1,2λ2].[z1(ǫ), z2(ǫ)]> ǫ (6)
Since the left-hand side of Eq. 6 is linearly dependent onǫ(cf. Eq. 5), the choice
180
of ǫhas no incidence on the viability property. We can rephrase this criterion using symbols from Metz and Gyllenberg (2001). If vector A is defined as
182
A = [0, λ1d1,2λ2] (7)
and vector Z asZ = Z(ǫ)ǫ , then the heuristic Rm viability criterion is
Rm =A.Z >1 (8)
which develops as:
184
Rm = s0αλ1(2λ2(1−d1) +d1(γ2+ 2µ2))
(s0α+µD)(γ2(γ1+µ1+λ1(1−d1)) + 2µ2(γ1+µ1)) (9)
Vector Z actually describes the “typical rarity” of the species, i.e. it contains the weigths that should be given to every patch state. These weights are
186
analogous to the “lx” coefficient used in stage-structured populations (Caswell, 2001), i.e. the proportion of individuals that survive from birth to stage x.
188
Here, zx yields the relative weight of state x in the computation of the Rm, i.e. a measure of the total time passed in state x in the course of a patch
190
lifetime.
IfY is the probabilistic state of a patch that has just received a rare immigrant
192
(in our present example Y = [0,1,0]), then Z = −G−10 .Y, with G0 being G computed without the immigration pressure of the rare species. In terms of
194
dimensions,Y is dimensionless andGis a matrix of (typically negative) rates, so that G−10 and Z are times. Rm is dimensionless since it is the product of
196
vector A (a vector of rates, i.e. inverse of times) and vector Z (eq. 8).
2.3 Proving the criterion mathematically
198
The criterion given in section 2.2 is formulated in biological terms: it addresses the problem of metapopulation viability using a quantity (the “lifetime pro-
200
duction of emigrants” of a patch) with a tangible ecological meaning. The intuitive explanation of this criterion has already been given by Metz and
202
Gyllenberg (2001). This does not constitute a proof of its validity, and the simple formulation retained may not be suitable for the whole class of related
204
metapopulation models.
Assessing whether a metapopulation described by equation 1 is viable is equiv-
206
alent to assessing whether the linearized dynamical system implied by equa- tion 1, when the metapopulation is almost empty, is unstable. Instability may
208
be looked for by studying the eigenvalues of the linearized system: when all eigenvalues have negative real parts, the system is stable; when at least one
210
eigenvalue has positive real part, the system is unstable. The Rm criterion should thus be shown to summarize the behavior of the dominant eigenvalue
212
of the linearized system, using a single number.
That’s what we establish in the next section, providing a more general formu-
214
lation of the Rm criterion, and translating it into biological terms.
3 Mathematical proof of the criterion: a digest
216
The complete proof of theRm criterion can be found in Appendix A. The Ap- pendix is self-sufficient, so that interested readers may read it without coming
218
back incessantly to the main text. Here, we only give a sketch of the proof, highlighting its key issues and assumptions.
220
3.1 General model and assumptions
We consider a closed spatially implicit metapopulation occupied by individ-
222
uals from one species. We note P the vector that describes the state of the metapopulation, i.e. given first by all the probabilities that a sample patch is
224
found containing 0, 1, 2,...,N individuals (possibly structured by classes, sex, etc.), and then by all densities of “free” individuals that do not reside in a
226
patch (e.g. the density of dispersers, if the model uses a disperser pool as in the toy model above). The size of vector P is at least N + 1.
228
We noteG(P)the “transition matrix”, i.e. the matrix function that determines the master equation ofP:
dP
dt =G(P)P (10)
Elements of G correspond to transition rates, i.e. the rates at which popula- tions change in state (upper diagonal block ofG), the rates of free individual
230
production (lower left-hand block), and the mortality and flows among cate- gories of free individuals (lower diagonal block).
232
More specifically, the first column of G (g.,1) represents the rates (i) of tran- sition from an empty population to a population containing at least one indi-
234
vidual, and (ii) of free individual production due to empty populations (that are necessarily 0).
236
In the simplest models, onlyg2,1 is non null (i.e. the only possible transition for an empty patch is to be colonized by one individual). More complicated models
238
can be envisaged in which individuals belong to different classes (e.g. wingless and winged individuals in aphids, or males and hermaphrodites, see our third
240
Example), or in which dispersal occurs in groups of more than one individual (e.g. fruits with multiple seeds). In these cases, more than one component of
242
g.,1 may be positive. Positive components of g.,1 are called colonization rates, and are functions of P. We assume that there are q such colonization rates,
244
noted (Mk)k∈[1;q] .
We are interested in determining whether eq. 1 displays an unstable equilib-
246
rium at P∗ = [1,0,0, ...,0]. This is achieved through the knowledge of the eigenvalues of equation 10, linearized near P∗.
248
3.2 Linearization of the equation, and definition of the initial states
Colonization rates Mk are assumed to obey two simple rules (see Appendix
250
A):
(1) colonization rates are not directly affected by the proportion of empty
252
patches (i.e. ∂Mk/∂p0 = 0 for all k);
(2) when the metapopulation is almost empty, colonization rates increase
254
with an increase of any population state probability (except empty pop- ulation state) and any density of free individuals (i.e. for allk, the vector
256
∂PMk taken at P =P∗ has only non-negative components).
These two conditions allow the following transformation of equation 10 near the equilibriumP∗ (section A.3):
dP
dt ≈G(P∗).P +
Xq k=1
(Ak.P)Yk (11)
where the vectors Yk correspond to the different possible initial states of re-
258
cently colonized patches (formally, Yk =∂Mkg.,1), and the vectors Ak are the first-order production rates of emigrants that will found populations with ini-
260
tial state Yk (formally,Ak =∂PMk).
Following equation 11, the viability problem is equivalent to finding the dom-
262
inant eigenvalue of matrixG+PYkATk at the equilibrium P∗. The eigenvalue corresponding to its first row and first column is always 0 (the empty state is
264
neutrally stable), which simplifies the issue.
From now on, tilde quantities (Xfand so forth) indicate vectors and matrices whose first row/column have been removed. The final eigenvalue problem is reduced to finding the dominant eigenvalue of matrix Jedefined by:
Je=Ge +
Xq k=1
YfkAfkT (12)
(for clarity, the dependence of G onP∗ will now be omitted).
266
3.3 Simplification of the eigenvalue calculation
Finding the dominant eigenvalue of Jeis made easier by following two simpli-
268
cation steps. First, it can be proved that matrix Jeadmits one real dominant eigenvalue (section A.4).
270
Second, some manipulations (sections A.6 and A.7) show that if we consider
the q×q matrix R(λ), defined by its elements rij(λ):
rij(λ) =−Afi.Ge −λI−1Yfj (13) then the problem of finding the greatest real eigenvalue of J,e λ, is turned into finding a matrix R(λ)that admits 1 as an eigenvalue.
272
MatrixR(0) contains the flows of migrant individuals produced, weighted by population lifetimes. Indeed,Yfj is thejth possible initial state of a newly colo-
274
nized patch. −Ge−1Yfj corresponds to the quasi-equilibrium state probabilities for a population that began in state Yfj (cf. the Z vector of section 2). On
276
average, a population that began in state Yfj spends a proportion h−Ge−1Yfji
k
of its lifetime (i.e. up to its eventual extinction) in state k.
278
The product −Afi.Ge−1Yfj thus gives the amount of emigrants that will found new populations starting in state fYi, produced by populations that initially
280
started in state Yfj. For some positive λ, the matrix R(λ) corresponds to the same quantities, but with an artificial increase in the extinction rates of the
282
populations, quantified byλ.
3.4 Final steps and the Rm criterion
284
TheRmcriterion is found by focusing on the dominant eigenvaluem(λ)ofR(λ) (see sections A.7 and A.8). The dominant eigenvalue of R is a non-negative
286
continuous decreasing functions of λ, that converges towards 0 (since all rij
have these properties). Therefore, by checking that the dominant eigenvalue
288
ofR(0) is strictly greater than 1, one effectively checks that there exists some positive λ such that the dominant eigenvalue ofR(λ)is 1. Conversely, if such
290
aλexists, the dominant eigenvalue ofR(0)must be greater than 1, given that
it is a decreasing function ofλ.
292
The mathematical formulation of the Rm criterion is thus:
m(0)>1⇔Sp(J)e ∩R∗
+ 6=∅ (14)
3.5 Biological interpretation of the criterion
Interpreting the Rm criterion in biological terms requires understanding the
294
signification of m(0). As already stated, the matrix R(0)contains the flows of emigrant productions weighted by population lifetimes.
296
Each column of this matrix corresponds to a colonization pathway, i.e. an initial state Yk. The life cycles of patches are thus distinguished in different
298
classes, depending on the initial state in which they began. Each class of life cycle initiates other classes of life cycles (the rows of R(0)) throughout its
300
life.R(0) can thus be seen as a transition matrix between classes of patch life cycles, just as a Lefkovitch matrix is a transition matrix between classes of
302
individuals.
The dominant eigenvalue of this matrix, m(0), can be interpreted as the
304
“metapopulation basic reproduction ratio”, by analogy with the R0 criterion, the basic reproduction ratio of stage-structured populations. If greater than
306
one, the population of patches initially grows (viability), otherwise it will de- cline.
308
When newly colonized patches have only one initial state (Ye, with the corre- sponding emigrant production vector, A),e R reduces to a 1×1 matrix with
310
only one eigenvalue equal to −A.eGe−1Ye (the expression given by Metz and
Gyllenberg, 2001), just like a matrix population model reduces to a scalar
312
population model when individuals are not structured. When there are sev- eral population initial states, the computation of the Rm criterion becomes
314
more complicated, paralleling the complexification of R0 in stage-structured populations (Diekmann et al., 1990).
316
4 A recipe for the analytical computation of Rm, with examples
4.1 The recipe
318
Based on our proof of theRm criterion, we give a simple five-point recipe that helps make no mistake in computing theRm value for a theoretical metapop-
320
ulation model:
(1) clearly state the transition matrixG(P), extract its first columng.,1 when
322
P =P∗, and find the colonization rates Mk;
(2) find the corresponding initial population states Yk =∂Mkg.,1;
324
(3) compute the emigrant production vectors, Ak =∂PMk; (4) compute the inverse of the reduced transition matrix,Ge−1;
326
(5) collect therij(0) =−Afi.Ge−1Yfj in matrix R(0) (tilde versions of matrices and vectors are obtained by removing the first row/column) and compute
328
Rm =ρ[R(0)], the dominant eigenvalue ofR(0).
Obviously this recipe is only efficient to get analytical expressions forRm. For
330
numerical calculations, it is probably better to use an approximated procedure, such as the one provided by Metz and Gyllenberg (2001), instead of computing
332
the inverse of the reduced transition matrix.
We now illustrate our recipe with three examples. We first provide two exam-
334
ples in which the Rm criterion is very simple, and then one example in which the Rm criterion is more complicated to compute. All these models are sim-
336
ple extensions of Metz and Gyllenberg’s model presented in section 2.1, and they retain the same notations and deme size N = 2. These models have not
338
been explicitly designed to answer biological questions (even if they convey some interesting results, assumptions of these models are quite unrealistic),
340
but rather to illustrate how to apply the Rm fitness concept. Each example is presented briefly, then the recipe given in this section is applied to the model,
342
and finally we perform a simplification of the example to get some tractable results.
344
4.2 Example 1: an animal model with mobile adults
4.2.1 The model
346
Consider a metapopulation as in section 2.1, but in which all individuals (adults and juveniles) can disperse (e.g. to escape crowded demes) and enter
348
the disperser pool. Assume for simplicity that after dispersal, adults behave just as juveniles migrants do. As before, individuals that were not accepted
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in a deme get back to the disperser pool, hence the qualifier of active dis- perser pool. We slightly change the parameterization and state that ei is the
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per capita rate of emigration from a deme with i individuals. The model is summarized in Fig. 2.
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4.2.2 Step 1: transition matrix and colonization rates
The G transition matrix (when N = 2) is rewritten as (cf. Casagrandi and
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Gatto, 2002; Parvinen et al., 2003):
G(P) =
−αδs0 γ1+µ1+e1 γ2 0
αδs0 −αδs1−γ1−µ1−λ−e1 2(µ2+e2) 0 0 αδs1+λ −γ2−2(µ2+e2) 0
0 d1 2d2 −(αhsii+µD)
(15)
In this example, the first column of G only depends on one colonization rate:
M =αs0δ (16)
4.2.3 Step 2: initial population states
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The initial state vectorY =∂Mg.,1 is given by:
Y = [−1,1,0,0] (17)
4.2.4 Step 3: emigrant production vectors
The emigrant production vector A=∂PM is given by:
A= [0,0,0, αs0] (18)
4.2.5 Step 4: inverse of the transition matrix
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The reduced transition matrix is:
Ge =
−γ1−µ1−λ−e1 2(µ2+e2) 0
λ −γ2−2(µ2+e2) 0
e1 2e2 −(αs0+µD)
(19)
Its inverse is:
e
G−1= 1 detGe
a2(αs0+µD) 2(µ2+e2) (αs0+µD) 0 λ(αs0+µD) a1(αs0+µD) 0
a2e1+ 2λe2 2µ2e1+ 2(a1+e1)e2 2(µ2+e2)(µ1+γ1+e1) +γ2a1
(20)
where
a1 =γ1+µ1+λ+e1 (21) a2 =γ2 + 2(µ2+e2) (22) detGe =−[2(µ2+e2)(µ1+γ1+e1) +γ2a1] (αs0+µD) (23)
4.2.6 Step 5: R matrix and Rm calculation
The value ofRm is obtained from the four previous steps:
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Rm = s0α(2e2(e1+λ) +e1(γ2+ 2µ2))
(s0α+µD)(γ2λ+ (γ1+µ1+e1)(γ2+ 2(e2 +µ2))) (24) The condition Rm = 1 in the three-parameter space (e1, e2, λ) is represented in Fig. 3. This condition is a surface which separates the region of viability
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(above the surface) from the region of extinction (under the surface). What
we observe is that viability is easier whene1 is low,λ is high ande2 is not too
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low (Fig. 3).
4.2.7 Application
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We note c = αs0µ+µD
D the cost of dispersal. When the species is “environmen- tally blind” (i.e.e cannot be adjusted according to local population size) and parameters are locally population size-independent (i.e. µ1 = µ2 = µ and γ1 =γ2 =γ), we obtain:
Rm = (1−c) [2e(e+λ) +e(γ+ 2µ)]
γλ+ (γ+µ+e) [γ+ 2(e+µ)] (25) so that the condition Rm >1requires:
(1) λ > µ+γ, i.e. births more than compensate deaths and extinctions;
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(2) a low value of c, i.e. the reward of dispersing to other patches is not too low (the maximal value of cis a complicated function of parameters;
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when γ →0, this can be approximated byc <λ−µλ+µ2);
(3) an intermediate value ofe(the maximal and minimal values ofeare com-
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plicated functions, but whenγ →0, this simplifies toλ−µ−c(λ+µ)−√
(1−c)[(λ−µ)2−c(λ+µ)2]
2c <
e < λ−µ−c(λ+µ)+√
(1−c)[(λ−µ)2−c(λ+µ)2]
2c ). This condition is intuitive since low
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e prevents the rescue of extinct patches (due to stochastic deaths or ex- tinction) by occupied patches, and high e diminishes the average occu-
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pancy of patches, and thus lowers the metapopulation reproduction rate.
We can visually check these analytical conditions of viability by looking at
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the intersection of the plane e1 =e2 = e with the surface depicted in Fig. 3.
Viability is easiest to achieve when λ is high and e is intermediate (with the
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parameters of Fig. 3, this occurs fore≈1.7).
