Beyond connectivity: how empirical methods can quantify population persistence to
improve marine protected area design
Scott C. Burgess, Kerry J. Nickols, Chris Griesemer, Lewis A.K. Barnett, Allison G. Dedrick,
Erin V. Satterthwaite, Lauren Yamane, Steven G. Morgan, J. Wilson White, Louis W. Botsford
Appendix A: Mathematical description of persistence in spatially structured populations
To determine the information needed to predict whether a network of marine populations
in a system of MPAs will persist we proceed in two steps. First, we consider the information
needed to represent fully the dynamic behavior of such a population to show the logic of
explicitly accounting for the transition of individuals in time (through survival and reproduction)
and through space (through the dispersal of larvae among patches). Second, we examine the
specific characteristics of the model that determine persistence to show the multiple ways to
estimate the essential components of replacement, depending on how the flow of individuals is
partitioned in time and space.
A population model for a marine metapopulation
The dynamics of a spatially structured marine population depend on survival,
reproduction, and movement among patches. We assume that survival depends on age only, so
the number of individuals at location i with age a+1 at time t+1 is the number of individuals at
location i of age a at time t, ni(a, t), multiplied by the age-dependent fraction surviving, s(a),
which can be represented as:
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ni  a 1, t 1  ni  a, t  s  a
(A.1)
Reproduction involves egg production, which we will assume occurs annually and is agedependent. At each location, i, total egg production, ei(t), is the fecundity of an individual at age
a, f(a), multiplied the number of individuals at age a, summed over all ages:
A
ei  t    ni  a, t  f  a
(A.2)
a1
We describe movement in the larval stage using the probability pij, which represents the
proportion of eggs from location j that recruit to location i. Note this fraction is the culmination
of processes influencing larval movement, planktonic mortality, and successful settlement. The
pii and pij terms are often collected into a connectivity matrix, which has origin locations along
the columns and destination locations along the rows (Table 1 and Figure 3 in main text). To
write the population dynamic equation at location i, we need to account for all larvae settling at i,
so we sum the total number of larvae produced at each location j over all values of j. This sum is
the number of larvae dispersing that year to location i from all locations j. That number is
multiplied by the proportion of larvae reaching location i that survive the first year and recruit
into the adult population at age 1, denoted si. This survival rate may be density-dependent
because of limited space for settlement or some other mechanism, but for simplicity we do not
consider that possibility for now (see main text for discussion of density-dependence). For this
density-independent case we write the population dynamics at i for age 1 as:
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J
ni 1, t   si  e j pij
(A.3)
j1
The collection of equations (A.1) to (A.3), for each location i, describe the population dynamics
of the network of populations in terms of continuity of individual movement through age classes
and over space.
What characteristics of the model determine persistence?
To identify the characteristics of such a model that make the modeled metapopulation
persist, we need to understand the logic of replacement: each adult must, on average, replace
itself with one offspring during its lifetime. To determine whether this is occurring, we need to
know the Lifetime Egg Production (LEP), which is R0 (see Table 1 in main text) expressed in
terms of eggs produced, instead of the number of individuals produced that recruit into the first
age class, within a lifetime (if the population is at the stable age distribution, ei = LEP).As
described in the main text, there are two ways a marine metapopulation can persist: 1) selfpersistence, and 2) network persistence.
For self-persistence, if the value of LEP from a single local population is great enough,
and a large enough fraction of the larvae produced by that population also settle and
subsequently survive to maturity in that population, there will be sufficient replacement for that
population to persist. The fraction of eggs produced by a population that also settles at that
population is known as local retention (LR). For location i, the local retention fraction involves
the pii term from the connectivity matrix (Table 1; Figure 3), but can also be thought of as:
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LR 
number of individuals returning to i pii ei
.

ei
total individuals produced at i
(A.4)
Self-persistence then requires that the local retention fraction multiplied by LEP be greater than 1.
Mathematically, self-persistence requires that LEPipii> 1.
Network persistence can occur even when no individual local population can self-persist.
We can begin to understand persistence in the case with more than one population by examining
the case for two populations. The question is what happens when both pii are less than the critical
value needed for self-persistence? Note that in Figure 3, there is an additional replacement loop
in addition to the two loops for local retention: a loop from population 1 to population 2, through
p21, with a return from population 1 to population 2, through p12. We can account for the effect
of that loop on persistence for the case in which neither population 1 nor 2 can persist on their
own (Hastings and Botsford 2006). The expression for persistence of two populations through
network replacement is:
LEP1 p21LEP2 p12
 1.
 LEP1 p11 1  LEP2 p22 1
(A.5a)
or equivalently, since at equilibrium LEPi = ei
e1 p21e2 p12
1
 e1 p11 1  e2 p22 1
(A.5b)
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Equation 5 has the convenient interpretation that the amount of replacement provided by the
shared loop between population 1 and population 2 (i.e., the numerator), must make up for the
combined shortcomings in self-replacement in populations 1 and 2 (i.e., the denominator).
Such network persistence requires closed loops of replacement over multiple patches in addition
to local retention (Hastings and Botsford 2006). Similar expressions and interpretations hold for
cases where the number of patches is greater than 2, although they become more complicated
rapidly (see Hastings and Botsford 2006). The derivation of this condition makes the common
assumption that it is the behavior at low abundance that is important for persistence. For a large
number of patches the requirement can more easily be expressed in terms of the dominant
eigenvalue of a matrix that holds all of the LEPipij values (Hastings and Botsford 2006, White
2010).
Why self-recruitment does not reveal anything about persistence
Self-recruitment is defined as the fraction of recruits arriving at a location that actually
originated at that same location (Table 1). To see how self-recruitment is unrelated to the above
expressions for persistence, we can use the two-patch model as an example and write the
expression for self-recruitment at population 1 as:
SR1 
e1 p11
e1 p11  e2 p12
(A.6)
Cursory examination indicates that equations (A.6) and (A.4) have the same numerator but
different denominators. Equation (A.6) is also dissimilar to equation (A.5b). For example, in
equation (A.5b), only the p’s related to local retention (pii) are in the denominator and only the
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p’s denoting a change in location (pij) are in the numerator (Hastings and Botsford 2006). In
contrast, the p’s denoting a change in location (pij) are in the denominator in equation (A.6).
Consequently, self-recruitment does not provide information related to population persistence. It
does, however, describe the degree of demographic openness (Hixon et al. 2002, Pinsky et al.
2012). As the input of foreign larvae (i.e., the p12 term in the denominator of equation A.6)
increases from 0 to 1, self-recruitment declines from 1 to 0.
Literature cited
Hastings, A., and L. W. Botsford. 2006. Persistence of spatial populations depends on returning
home. Proceedings of the National Academy of Sciences USA 103:6067–6072.
Hixon, M. A., S. W. Pacala, and S. A. Sandin. 2002. Population regulation: historical context and
contemporary challenges of open vs. closed systems. Ecology 83:1490–1508.
Pinsky, M. L., S. R. Palumbi, S. Andréfouët, and S. J. Purkis. 2012. Open and closed seascapes:
where does habitat patchiness create populations with high fractions of self-recruitment?
Ecological Applications 22:1257–1267.
White, J. W. 2010. Adapting the steepness parameter from stock–recruit curves for use in
spatially explicit models. Fisheries Research 102:330–334.
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