Marine reserves can enhance ecological resilience
Lewis A.K. Barnett, Marissa L. Baskett
APPENDIX S1: PARAMETER DETAILS & ADDITIONAL FIGURES AND TABLES
Parameterization Details
In the following text, we explain how the parameters in Table 1 were derived (for those that were
not taken directly from the literature). To calculate the predator’s base birth rate (realized fecundity),
we solved the continuous-time Euler equation for intrinsic growth rate, given estimates of
survivorship and fecundity from the literature. Because such biological estimates were not present
for the species represented by our competitor, we arbitrarily chose a value slightly less than that of
the predator given that fecundity increases with body size and the predator reaches larger size than
does the competitor.
To calculate the conversion efficiency from predation to reproductive output, we assume
10% of consumption is available for all processes: maintenance, growth, and reproduction. From
this, we estimate the proportion of the energetic gain that goes to reproduction from the allometric
relationships predicting the gonadosomatic index from Charnov et al. (2001). To specify carrying
capacities, we use empirical estimates of rockfish density from in situ surveys off the coast of central
California (Yoklavich et al. 2000) and Oregon (Stein et al. 1992) and expand these to the area of the
model domain (chosen to approximately represent the size of the central California coastal habitat,
965.6 km2). By leaving out competition coefficients, we are implicitly assuming that the competitor
and juvenile predator are equally aggressive in competing for space. However, the greater measured
carrying capacity of the competitor provides this species with competitive superiority because it can
use space more efficiently than the juvenile predator.
1
To determine attack rates, we assume that cannibalism (the attack rate on the juvenile
predator) occurs at a lower rate than predation on the competitor. Given this assumption, we
specify the attack rates to be similar to those empirically estimated by Johnson (2006) for other
rockfishes, by expanding the values given in terms of predator density to the model’s predator
density, given the area of the model domain. Finally, maturation rate was calculated as 1/(age at
maturity).
In all simulations, we integrated the ordinary differential equations that define the
continuous-time dynamics of our system using the FORTRAN ODE solver lsoda, implemented by
interfacing through the package deSolve (Soetaert et al. 2010) in R (R Core Team 2015).
2
References cited in appendix
1.
Charnov, E.L., Turner, T.F. & Winemiller, K.O. (2001). Reproductive constraints and the evolution
of life histories with indeterminate growth. Proc Natl Acad Sci U S A, 98, 9460-9464.
2.
Johnson, D.W. (2006). Predation, habitat complexity, and variation in density-dependent mortality
of temperate reef fishes. Ecology, 87, 1179-1188.
3.
R Core Team (2015). R: A language and environment for statistical computing, R Foundation for
Statistical Computing, Vienna, Austria. R Foundation for Statistical Computing Vienna, Austria.
4.
Soetaert, K., Petzoldt, T. & Setzer, R.W. (2010). Solving differential equations in R: package deSolve.
Journal of Statistical Software, 33.
5.
Stein, D.L., Tissot, B.N., Hixon, M.A. & Barss, W. (1992). Fish-habitat associations on a deep reef at
the edge of the Oregon continental shelf. Fish Bull, 90, 540-551.
6.
Yoklavich, M.M., Greene, G.H., Sullivan, D.E., Lea, R.N. & Love, M.S. (2000). Habitat associations
of deep-water rockfishes in a submarine canyon: an example of a natural refuge. Fish Bull, 98, 625641.
3
Table S1. Parameter estimates used in sensitivity analysis shown in Fig. S6, where values of each
parameter were varied independently while all others remained at their base value (Table 1). Note: 𝐶 ,
competitors; 𝐽, juvenile predators; 𝑃, adult predators.
Symbol
𝐻𝐶
Description
Units
𝐶 ∙ 𝐽−1
Value
Source
[0.0001, 0.01]
Arbitrary: order of
magnitude from base
𝐶 ∙ 𝑦 −1
[0.249, 0.448]
Anderson 1984;
𝑏𝑃
𝐽 ∙ 𝑦 −1
[0.249, 0.448]
Love et al. 2002
𝑀𝐽
𝑦 −1
[0.03, 0.44]
Beckmann et al. 1998;
𝑦 −1
[0.03, 0.44]
Spencer & Rooper 2010
𝑦 −1
[0.05, 0.5]
Love et al. 2002
consumption efficiencies
𝐻𝐽
𝑏𝐶
𝐽 ∙ 𝐽−1
fecundity (discrete birth rate)
natural mortality rates
𝑀𝐶
𝛾
maturation rate
4
Figure S1. Example of how abundances change over time in each patch during a single run of the
model when variable recruitment is included and the predator dominates. Panels show abundances
of the predator (a,b), juvenile (c,d), and competitor (e,f) in the reserve (a,c,e) and unprotected (b,d,f)
patch. In this example the proportion of reserve coverage is 0.1, 𝐹 = 0.13, and the standard
deviation of recruitment is 1.0.
5
Figure S2. Example of how abundances change over time in each patch during a single run of the
model when variable recruitment is included and the competitor dominates. Panels show
abundances of the predator (a,b), juvenile (c,d), and competitor (e,f) in the reserve (a,c,e) and
unprotected (b,d,f) patch. In this example the proportion of reserve coverage is 0.1, 𝐹 = 0.13, and
the standard deviation of recruitment is 1.0.
6
7
Figure S3. Change in the coefficient of variation (CV) of catch with proportional increase in reserve
coverage (triangles) or the equivalent spatially-averaged fishing mortality without reserves (squares)
with iterations resulting in 0 catch removed. Columns represent results with differing recruitment
variability: low variability (𝜎𝜌 = 0.5) and high variability (𝜎𝜌 = 1.0). Rows represent the different
values for the maximum fishing mortality used in this study, with low fishing (a-b; 𝐹 = 0.10),
moderate fishing (c-d; 𝐹 = 0.13), and high fishing (e-f: 𝐹 = 0.16). For missing values in panels (e)
and (f), the mean catch is 0.
8
Figure S4. Change in ecological resilience (a-c), catch (d; scaled to the maximum of all values), and
the coefficient of variation (CV) of catch (e, f) with proportional increase in reserve coverage
(triangles) or the equivalent spatially-averaged fishing mortality without reserves (squares) for a lower
value of fishing mortality (𝐹 = 0.1) than shown in Figure 4. Columns represent results with
differing recruitment variability: deterministic (𝜎𝜌 = 0), low variability (𝜎𝜌 = 0.5), high variability
(𝜎𝜌 = 1.0). In the deterministic case resilience is defined as the proportion of initial abundances that
leads to the predator-dominated equilibrium whereas in the stochastic cases resilience is defined as
the likelihood of reaching the predator dominated state given random initial abundances. Variances
around mean values of resilience and catch CV in the stochastic case are too small to be visible.
9
Figure S5. Change in ecological resilience (a-c), catch (d; scaled to the maximum of all values), and
the coefficient of variation (CV) of catch (e, f) with proportional increase in reserve coverage
(triangles) or the equivalent spatially-averaged fishing mortality without reserves (squares) for a
higher value of fishing mortality (𝐹 = 0.16) than shown in Figure 4. Columns represent results with
differing recruitment variability: deterministic (𝜎𝜌 = 0), low variability (𝜎𝜌 = 0.5), high variability
(𝜎𝜌 = 1.0). In the deterministic case resilience is defined as the proportion of initial abundances that
leads to the predator-dominated equilibrium whereas in the stochastic cases resilience is defined as
the likelihood of reaching the predator dominated state given random initial abundances. For
missing values in panels (e) and (f), the mean catch is 0. Variances around mean values of resilience
and catch CV in the stochastic case are too small to be visible.
10
11
Figure S6. Sensitivity of deterministic resilience and the presence of alternative stable states to
specific model parameters, given proportional increases in reserve coverage (triangles) or the
equivalent spatially-averaged fishing mortality without reserves (squares). The parameter being varied
is indicated to the right of each row and the resulting resilience is shown given extreme high or low
values for that parameter (compare columns; values given in Table S1), whereas all other parameters
remain at their base values (see Table 1 of the main text). For some parameters, the bounds of the
values exceed those that produce alternative stable states. For these parameters, the biologicallyrealistic ranges that result in alternative stable states given conventional fisheries management at the
unrestricted rate of fishing mortality (𝐹 = 0.13) are: maturation rate 𝛾 (c,d): 0.05-0.31 𝑦 −1 ,
competitor fecundity 𝑏𝐶 (g,h): 0.30-0.448 𝐶 ∙ 𝑦 −1 , competitor natural mortality 𝑀𝐶 (k,l): 0.03-0.30
𝑦 −1 .
12
Figure S7. Effect of external recruitment on coexistence and community dynamics. Simulations are
as in Fig. 2, but with coexistence of predator and competitor caused by inclusion of a small amount
of external recruitment for each species (constant influx of 500 setters per year from unmodeled
source habitats, where the predator or competitor can dominate due to differences in habitat
characteristics and fishing effort across space). The ternary plots display the trajectories of change
over time in the relative abundances of the competitors, adult predators, and juvenile predators (with
relative abundances normalized to sum to one in each ternary plot). Points represent different
combinations of initial abundances from which the trajectories begin. Trajectories (lines) are colored
according to which stable equilibrium (large, solid black dots) they lead to, the competitor13
dominated equilibrium (red) or predator-dominated equilibrium (blue). Note that stable equilibria
are not quite on the edge of the ternary plot boundary, reflecting at least a small positive abundance
of the non-dominant group. Results are for (a) spatially uniform harvest and (b,c) spatiallyheterogeneous harvest at 20% reserve coverage with effort displacement. The spatially-averaged
fishing mortality is equivalent between the two scenarios (𝐹̃ = 0.104). With reserve management (b)
represents the protected patch and (c) represents the unprotected patch. Parameters are as in Table
1.
14