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\begin{document}
\title{Transient inconsistency between population density and fisheries yields
under bycatch persistence}
\author[1]{Renfei Chen}%
\affil[1]{Affiliation not available}%
\vspace{-1em}
\date{\today}
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\selectlanguage{english}
\begin{abstract}
Recent studies have demonstrated the great advantages of marine reserves
in solving bycatch problems by maintaining the persistence of endangered
species without sacrificing the fisheries yields of target species.
However, transient phenomena rather than equilibrium states of
population dynamics still require further research. Here, with a simple
and general model, the transient dynamics of strong stock fish species
are investigated under the condition of weak stock persistence. A
surprising and counter-intuitive finding is that fisheries yields can
strongly fluctuate even if population density both inside and outside
marine reserve only slightly varies, leading to transient inconsistency
between the population densities and fisheries yields. This finding
suggests that population density dynamics cannot be used to predict the
transient phenomena of fisheries yields (or vice versa) in fisheries
management. These results will deepen our understanding of the transient
phenomenon in marine ecology.%
\end{abstract}%
\sloppy
\textbf{Introduction}
Marine reserves have been used as an effective tool to protect the
endangered species and achieve sustainable fisheries yields as well as
other benefits (White et al. 2008, Edgar et al. 2014, Herrera et al.
2016, Goetze et al. 2018). To improve the functions of marine reserves,
many marine ecologists have devoted tremendous effort on the design of
marine reserves (Mangel 2000, Gerber et al. 2003, Guichard et al. 2004,
Sanchirico et al. 2006). One of the greatest debates in marine reserve
design is whether permanent marine reserves can improve fisheries yields
while maintaining persistence of the target population (Hastings and
Botsford 1999, Game et al. 2009, Kaplan et al. 2010, Hastings et al.
2017, Hilborn 2017). A simple theoretical model indicated that, without
sacrificing the persistence of the target species, the implementation of
a permanent marine reserve could achieve fisheries yields equivalent to
those obtained with fisheries management methods of harvesting a fixed
amount of fish resources without marine reserves (Hastings and Botsford
1999). However, a general theoretical model of rotational marine
reserves demonstrated that dynamic closures could enhance the mean
herbivore biomass and improve the resilience of coral reef systems under
conditions in which marine protected areas are rotated among a suitable
subset of all coral reef systems (Game et al. 2009, Kaplan et al. 2010).
One of the reasons why such theoretical studies draw different
conclusions is that the assumptions these models based on are different
so that they solve different problems with different purposes. For
example, descriptions of the dynamics of target population density as
well as fisheries yields become complex when the persistence of bycatch
species is taken into consideration. Here, I dedicate this study to
solving some bycatch issues in marine reserves.
The incidental capture of non-target species, which is called
``bycatch'', is a great threat to fisheries sustainability, especially
for some endangered marine species (Komoroske and Lewison 2015, Taylor
et al. 2017, Scales et al. 2018, Welch et al. 2018). To address this
significant global problem, different approaches have been proposed. For
example, these approaches include improving the performance of the
selection devices for fisheries (Santos et al. 2016), refining the
precision of the assessment of species distributions and habitat
preferences (Hazen et al. 2018, Clay et al. 2019, Thorne et al. 2019),
and elucidating the fundamental mechanisms underlying fisheries
interactions with Lagrangian analysis (Scales et al. 2018, Horswill and
Manica 2019). Although the specific details differ among these
approaches, their main goals are the same: i.e., reducing the capture
rate of bycatch species while increasing the capture rate of the target
species. Based on a different ideology, a recent theoretical framework
suggested that marine reserves could improve the fisheries yields of the
strong stock (i.e., the target species) while maintaining the
persistence of the weak stock (the bycatch species), even when both
strong and weak stocks are captured at the same rate (Hastings et al.
2017). However, the research gap is that the transient dynamics of the
two-species system are still not clear, which may become a major barrier
to applying the theoretical framework of marine reserve design to
empirical conservation management.
The application and importance of transient dynamics are emphasized by
the shortcomings of long- term asymptotic population dynamics. Transient
analyses pay attention to dynamical systems over a short time scale
rather than systems in the equilibrium state after a sufficiently long
time (Feng et al. 2019, Mari et al. 2019, Rudolf 2019, Shriver et al.
2019). If the dynamics of a certain system exhibit a strong relationship
with the time scales, then the transient dynamics may be much different
from the dynamics when the system achieves stability. For example, a
two-species system (one native species and an invasive species) may
exhibit the coexistence of the two species for a long time (because of
the long transient times) even though the invasive species will exclude
the native species in the asymptotic equilibrium state (Hastings et al.
2018). Thus, empirical data of such a system monitored on an
intermediate time scale will lead to incorrect judgments and management
in species conservation.
Recent reviews indicate that there are five general scenarios in which
transient phenomena may arise: i.e., ghost, crawl-by, slow-fast systems,
high dimension and stochasticity (Hastings et al. 2018, Morozov et al.
2019). If a bifurcation phenomenon emerges beyond a tipping point of a
system, the dynamics of the system (even if it does not possess an
equilibrium point in the long-term) will mimic a system's dynamics that
have an attractor. Such a case is called ``ghost'' or ``ghost
attractor'' (Hastings et al. 2018). The population dynamics of a system
with a ghost may spend a long time around the ghost, and thus long
transient phenomena may occur. The definitions of the transient dynamics
of a system caused by ''crawl-by'' are very similar to those caused by
''ghost''. The difference between a ghost and crawl-by is that a system
with ghosts may or may not have attracting directions because of the
lack of invariant sets, and the initial state of a ghost system should
be around the ghost while a crawl-by system should have saddles (that is
unstable equilibria for a system) and thus always have attracting
directions (Hastings et al. 2018, Morozov et al. 2019). A fast-slow
system is characterized by different multiple time scales in the system
(Bertram and Rubin 2017). For example, in a resource-consumer system,
insects that feed on trees have a much shorter lifetime than the trees.
Thus, the population dynamics of trees change very slowly (assumed to be
fixed in ideal systems), even when insects have gone through several
dynamical generations (Rinaldi and Scheffer 2000, Hastings et al. 2018).
A high-dimensional system that is close to real-world systems easily
leads to the emergence of transient phenomena such as spatiotemporal
dynamical systems (Hastings and Higgins 1994). However, as a common
property in an ecosystem, time delay in a low-dimensional system may
also result in a similar transient phenomenon in comparison to an
infinite-dimensional instantaneous system(Kuang 1993, Smith 2011).
Transient phenomena caused by stochasticity are much easier to
intuitively understand, and many reasons can lead to stochasticity in
real or natural systems, such as tidal movements, which increase the
stochasticity of larval fish dispersal in a marine ecosystem.
Although transient phenomena have been demonstrated and classified by
increasing evidence both empirically and theoretically (Hastings and
Higgins 1994, Hastings 2001, 2004, Morozov et al. 2019), marine reserve
management and policy-making still often depend on the research on
long-term population dynamics, and the transient research in marine
reserve design is limited. The research in this paper specifically
focuses on the transient dynamics of the discrete model (the definitions
of all the symbols in the models can be seen in Table 1) developed by
Alan et al. (2017). This theoretical framework is based on several main
assumptions. First, this framework assumes that adult fish are
relatively stationary while larvae are mobile and widely distributed.
Second, it assumes that different species considered in the system are
subject to the same fisheries management practices with the same capture
rates outside marine reserves and the same protection inside marine
reserves. Third, no further complex factors, such as time delay and age
structure, are considered in the system. Both analytical and numerical
approaches are used to investigate the transient phenomena that may be
concealed in the target system. With an analytical approach, the
existence of saddle points is discussed for the population density
inside and outside marine reserves at equilibrium. Further, I also
analytically discuss the assumption that the target system is a
fast-slow system (i.e., multiple time scales existing in a system) to
determine whether there is Hopf bifurcation for the fast components of
the system. Last, with the numerical method, simulations are performed
to study the transient dynamics of both population density and fisheries
yields under two scenarios, including the initial population density at
the saddle points and a random initial population density.
\textbf{Model and analysis}
The objective model analysed here was derived from recent research on
bycatch problems (Hastings et al. 2017). We focus on two species, one
called the strong stock (the target species in the fishery) and the
other called the weak stock (an endangered species that could easily
become extinct). To simplify the problem, we use similar approaches to
those used in previous research (Hastings et al. 2017) and only study
the population dynamics of the strong stock under the conditions in
which the weak stock is persistent rather than studying the population
dynamics of both species simultaneously (because, in most cases, the
strong stock is the primary target in a fishery and is expected to
achieve maximum yields while the weak stock species is the secondary
target in the fishery and will much easier to extinct). The population
dynamics of the strong stock are described by keeping track of the
densities inside and outside of marine reserves. Based on the
assumptions that are the same as those in Alan et al. (2017), the
objective system of equations for strong stock is described as follows:
\(n_{t}^{R}a+f(m(cn_{t}^{R}+(1-c)n_{t}^{O}))\)=\(\ n_{t+1}^{R}\ \ \ (1)\)
\begin{equation}
\left[n_{t}^{O}a+f\left(m(cn_{t}^{R}+(1-c)n_{t}^{O})\right)\right]E=\ n_{t+1}^{O}\ \ \ (2)\nonumber \\
\end{equation}
where \(n_{t}^{R}\) and \(n_{t}^{O}\) represent the density of
strong stock inside and outside marine reserves at time \emph{t} ,
respectively. The function\(\ f()\) shows the survival of young
fish individuals until they recruit to the adult population. The
parameters \emph{m} , \emph{a} ,\emph{c} and \emph{E} describe the per
capita fecundity, the survivorship of adults, the fraction of the
coastline in a no-take marine reserve, and the escapement rate
representing the fraction of the fish stock that is left unharvested,
respectively. All the variables and parameters exhibited here are for
the strong stock, which are different from those for the weak stock.
However, for both species in the same system, the marine reserve size
(i.e., the length of marine reserve coastline because the research here
is one dimensional) and the escapement rate are the same. Thus, \emph{c}
and \emph{E} are also used for the weak stock. Accordingly, the
harvested yield of the strong stock produced from such a system (Eqs. 1
and 2) is:
\begin{equation}
Y_{P}=[n_{t}^{O}a+f(m(cn_{t}^{R}+(1-c)n_{t}^{o}))](1-c)(1-E)\ \ \ (3)\nonumber \\
\end{equation}
As for the weak stock, we use the same symbols as those for the strong
stock but add a subscript ``w'' for distinction (please see Table 1 for
the definitions of all the symbols). To achieve the weak stock
persistent condition, let the determinant of the following
matrix\emph{J} be zero:
\(\mathbf{J}=[\par
\begin{matrix}f_{w}^{{}^{\prime}}\left(0\right)m_{w}c+a_{w}-1&f_{w}^{{}^{\prime}}\left(0\right)m_{w}\left(1-c\right)\\
Ef_{w}^{{}^{\prime}}\left(0\right)m_{w}c&E\left[a_{w}+f_{w}^{{}^{\prime}}\left(0\right)m_{w}(1-c)\right]-1\\
\end{matrix}]\) (4)
Then, we have the weak stock persistent condition:
\begin{equation}
E=\frac{a_{w}-1+\alpha_{w}m_{w}c}{\left(a_{w}-1\right)\left(a_{w}+\alpha_{w}m_{w}\right)+\alpha_{w}m_{w}c}\ (5)\nonumber \\
\end{equation}
The calculation of Eq. 5 is based on the Beverton--Holt functional form
for \(f()\ \):
\(f\left(n\right)=\frac{\alpha_{w}n}{1+\frac{n}{\beta_{w}}}=\frac{\alpha_{w}\beta_{w}n}{\beta_{w}+n}\)(6)
where \(\alpha_{w}\) and \(\beta_{w}\) denote the
proliferation rate per generation and carrying capacity, respectively.
\textbf{Analytical solution of equilibrium state for strong stock}
For the objective system consisting of Eqs. 1 and 2, the equilibrium
states occur when \(n_{t+1}^{R}=n_{t}^{R}\), and the following equations can be
achieved with \emph{f} () representing the Beverton--Holt functional
form (\(f\left(n\right)=\selectlanguage{greek}\frac{\text{αn}}\selectlanguage{english}{1+\frac{n}{\beta}}=\selectlanguage{greek}\frac{\text{αβn}}\selectlanguage{english}{\beta+n}\)):
\begin{equation}
m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)\alpha\beta=n_{t}^{R}\left(1-a\right)\left[\beta+m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)\right]\ (7)\nonumber \\
\end{equation}\begin{equation}
m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)\alpha\beta E=n_{t}^{o}\left(1-Ea\right)\left[\beta+m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)\right]\ (8)\nonumber \\
\end{equation}
Assuming\(m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)\neq 0\)(If \(m\left(cn_{t}^{R}+\left(1-c\right)n_{t}^{O}\right)=0\),\(n_{t}^{R}=n_{t}^{O}=0\)),
the quotient of Eq. 7 and Eq. 8 is:
\begin{equation}
\frac{1}{E}=\frac{n_{t}^{R}}{n_{t}^{O}}\frac{1-a}{\left(1-Ea\right)}\ (9)\nonumber \\
\end{equation}
i.e.,
\begin{equation}
n_{t}^{O}=n_{t}^{R}\frac{(1-a)E}{\left(1-Ea\right)}\ (10)\nonumber \\
\end{equation}
Substitute Eq. 10 into Eq. 7:
\begin{equation}
\left(c+\left(1-c\right)\frac{(1-a)E}{\left(1-Ea\right)}\right)m\selectlanguage{greek}\text{αβ}\selectlanguage{english}n_{t}^{R}=n_{t}^{R}\left(1-a\right)\left[\beta+mn_{t}^{R}\left(c+\left(1-c\right)\frac{(1-a)E}{\left(1-Ea\right)}\right)\right]\ (11)\nonumber \\
\end{equation}
Therefore, \(n_{t}^{R}\) has two solutions: \(n_{t}^{R}=0\)
and\(n_{t}^{R}=\selectlanguage{greek}\selectlanguage{english}\frac{\text{αβ}}{1-a}-\frac{\beta\left(1-Ea\right)}{\text{mc}\left(1-Ea\right)+m\left(1-c\right)\left(1-a\right)E}\ \).
According to Eq. 10, \(n_{t}^{O}=0\) and\(n_{t}^{O}=\selectlanguage{greek}\frac{\text{αβE}}\selectlanguage{english}{1-Ea}-\selectlanguage{greek}\frac{\text{βE}\left(1-a\right)}\selectlanguage{english}{mc+m\left(1-a-c\right)E}\). For a
special case of the system (Eq. 1 and Eq. 2), if\(n_{t}^{R}=0\),
then \(n_{t}^{O}=0\). If \(n_{t}^{O}=0\),
then\(n_{t}^{R}=(mc\alpha\beta+(a-1)\beta)/(1-a)mc\) under the condition that \(E=0,a\neq 1,c\neq 0\), or
\(n_{t}^{R}=0\). If\(a=1\), then \(n_{t}^{R}=n_{t}^{O}=0\).
Accordingly, there are three scenarios for the strong stock density at
equilibria:
\textbf{Scenario 1}
\(n_{t}^{R}=0\) \((12)\)
\begin{equation}
n_{t}^{O}=0\ (13)\nonumber \\
\end{equation}
\textbf{Scenario 2}
\begin{equation}
n_{t}^{R}=\frac{mc\alpha\beta+\left(a-1\right)\beta}{\left(1-a\right)\text{mc}}\ (14)\nonumber \\
\end{equation}\begin{equation}
n_{t}^{O}=0\ \left(15\right)\ (E=0,a\neq 1,c\neq 0)\nonumber \\
\end{equation}
\textbf{Scenario 3}
\begin{equation}
n_{t}^{R}=\selectlanguage{greek}\selectlanguage{english}\frac{\text{αβ}}{1-a}-\frac{\beta\left(1-Ea\right)}{\text{mc}\left(1-Ea\right)+m\left(1-c\right)\left(1-a\right)E}\ (16)\ \nonumber \\
\end{equation}\begin{equation}
n_{t}^{O}=\selectlanguage{greek}\frac{\text{αβE}}\selectlanguage{english}{1-Ea}-\selectlanguage{greek}\frac{\text{βE}\left(1-a\right)}\selectlanguage{english}{mc+m\left(1-a-c\right)E}\ \left(17\right)\ (a\neq 1)\nonumber \\
\end{equation}
\textbf{The value ranges for the parameters}
The density inside and outside marine reserves should not be negative
(i.e., \(n_{t}^{R}\geq 0,n_{t}^{O}\geq 0\ \)). Therefore, we have the following inequation
for scenario 2:
\begin{equation}
\frac{mc\alpha\beta+\left(a-1\right)\beta}{\left(1-a\right)\text{mc}}\geq 0\ (18)\nonumber \\
\end{equation}
The condition for scenario 3 is as follows:
\begin{equation}
\selectlanguage{greek}\selectlanguage{english}\frac{\text{αβ}}{1-a}-\frac{\beta\left(1-Ea\right)}{\text{mc}\left(1-Ea\right)+m\left(1-c\right)\left(1-a\right)E}\ \geq 0(19)\ \nonumber \\
\end{equation}\begin{equation}
\selectlanguage{greek}\frac{\text{αβE}}\selectlanguage{english}{1-Ea}-\selectlanguage{greek}\frac{\text{βE}\left(1-a\right)}\selectlanguage{english}{mc+m\left(1-a-c\right)E}\ \geq 0(20)\nonumber \\
\end{equation}
When the escapement rate \emph{E} is 1, there are no fishing activities,
and all the analyses have no biological meaning. Thus,
\(0\leq E<1\). According to the persistent condition of the weak
stock (i.e., Eq. 5), we have:
\begin{equation}
0\leq\frac{a_{w}-1+\alpha_{w}m_{w}c}{\left(a_{w}-1\right)\left(a_{w}+\alpha_{w}m_{w}\right)+\alpha_{w}m_{w}c}<1\ (21)\nonumber \\
\end{equation}
When Eq. 18 is combined with Eq. 21, the condition for scenario 2 is:
\begin{equation}
\selectlanguage{greek}\frac{1-a}{\text{mα}}\selectlanguage{english}\leq c\leq\frac{1-a_{w}}{\alpha_{w}m_{w}}\ (22)\nonumber \\
\end{equation}
The condition for scenario 3 is as follows:
\begin{equation}
\frac{E\left(a-1\right)}{1-E}+\frac{(1-a)(1-Ea)}{m\alpha(1-E)}\leq c\leq\frac{1-a_{w}}{\alpha_{w}m_{w}}\ (23)\nonumber \\
\end{equation}
\textbf{Transient analysis by finding a saddle point}
The Jacobian matrix of the objective system is:
\begin{equation}
J=\begin{bmatrix}a-1+\selectlanguage{greek}\frac{\text{mcα}\beta^{2}}\selectlanguage{english}{\left[\beta+\text{mc}n_{t}^{R}+m\left(1-c\right)n_{t}^{O}\right]^{2}}&\frac{m\left(1-c\right)\alpha\beta^{2}}{\left[\beta+\text{mc}n_{t}^{R}+m\left(1-c\right)n_{t}^{O}\right]^{2}}\\
\selectlanguage{greek}\frac{\text{Emcα}\beta^{2}}\selectlanguage{english}{\left[\beta+\text{mc}n_{t}^{R}+m\left(1-c\right)n_{t}^{O}\right]^{2}}&Ea-1+\frac{\text{Em}\left(1-c\right)\alpha\beta^{2}}{\left[\beta+\text{mc}n_{t}^{R}+m\left(1-c\right)n_{t}^{O}\right]^{2}}\\
\end{bmatrix}\ (24)\nonumber \\
\end{equation}
For scenario 1, \(n_{t}^{R}=n_{t}^{O}=0\). Thus, the Jacobian matrix turns out
to be:
\begin{equation}
J=\begin{bmatrix}a-1+mc\alpha&m\left(1-c\right)\alpha\\
\selectlanguage{greek}\text{Emcα}&Ea-1+Em\left(1-c\right)\alpha\\
\end{bmatrix}\ (25)\nonumber \\
\end{equation}
If there is a saddle point for the system, then one of the eigenvalues
of the Jacobian matrix is positive and the other is negative. Thus:
\begin{equation}
\lambda_{1}\lambda_{2}=\left(1-a\right)\left(1-Ea\right)+m\alpha\left(Ea-E+Ec-c\right)<0\ (26)\nonumber \\
\end{equation}
where \(\lambda_{1}\) and \(\lambda_{2}\) are eigenvalues of the
Jacobian matrix (Eq. 25). Therefore, for scenario 1, the condition that
meets a saddle point is:
\begin{equation}
\selectlanguage{greek}\selectlanguage{english}\frac{\text{mα}}{1-a}>\frac{1-Ea}{c+E(1-a-c)}\ (27)\nonumber \\
\end{equation}
For scenario 2, Eq. 14 and Eq. 15 are substituted into Eq. 24, and the
condition that meets a saddle point is:
\begin{equation}
\lambda_{1}\lambda_{2}=\left(1-a\right)\left(1-Ea\right)-\frac{E\left(1-c\right)\left(1-a\right)^{3}}{mc^{2}\alpha}-\selectlanguage{greek}\frac{\left(1-Ea\right)\left(1-a\right)^{2}}{\text{mcα}}\selectlanguage{english}<0\ (28)\nonumber \\
\end{equation}
Thus,
\begin{equation}
c<\selectlanguage{greek}\frac{1-a}{\text{mα}}\selectlanguage{english}\ (29)\nonumber \\
\end{equation}
These parameters have paradoxes with a suitable value range for the
parameters in this study. Therefore, there is no saddle point for
scenario 2.
For scenario 3, Eq. 16 and Eq. 17 are substituted into Eq. 24, and the
condition that meets a saddle point is:
\begin{equation}
\selectlanguage{greek}\selectlanguage{english}\frac{\text{mα}}{1-a}<\frac{1-Ea}{c+E(1-a-c)}\ (30)\nonumber \\
\end{equation}
These parameters have paradoxes with a suitable value range for the
parameters in this study. Therefore, there is no saddle point for
scenario 3. Accordingly, the saddle point of the system occurs only at
(0,0) under appropriate conditions.
\textbf{Transient analysis by considering fast-slow systems}
If the strong stock species has a much higher growth rate than the weak
stock species, then the objective system can be analysed as a fast-slow
system. Accordingly, we analyse the fast components of the objective
system with a widely accepted method (Rinaldi and Scheffer 2000) to
predict whether there are Hopf bifurcations in the population dynamics
of the strong stock. The equivalent condition of Hopf bifurcations is
that the trace of the Jacobian matrix (Eq. 24) is zero, and the
determinant of this matrix is positive. That is:
\begin{equation}
\text{trace}\left[J\right]=0,\det\left[J\right]>0\ (31)\ \nonumber \\
\end{equation}
Similarly, this part is discussed with three scenarios according to the
density inside and outside marine reserves at equilibria. For scenarios
1, 2 and 3, the calculation results based on Eq. 31 for the condition of
Hopf bifurcations are exhibited as the following three inequations,
respectively:
\begin{equation}
\left(a-1\right)^{2}+mc\alpha a\left(1-E\right)<0\ (32)\nonumber \\
\end{equation}\begin{equation}
1<-\selectlanguage{greek}\frac{a\left(1-E\right)}{\text{mcα}}\selectlanguage{english}\ (33)\nonumber \\
\end{equation}\begin{equation}
\left(1-E\right)c\left(Ea^{2}-1\right)>E\left(1-a\right)^{2}\ (34)\nonumber \\
\end{equation}
all of which are impossible (for example, in scenario 3, the left part
of Eq. 34 is negative while the right part is positive, which is
impossible). Accordingly, when the theoretical framework is regarded as
a fast-slow system, there is no Hopf bifurcation in the strong stock
population dynamics.
\textbf{Two metrics for transient analysis}
Our analytical analysis results indicate that transient phenomena may
occur at the equilibrium of (0,0) for the initial density inside and
outside marine reserves rather than the other two situations (i.e.,
scenarios 2 and 3). Thus, we further study two transient metrics only in
scenario 1 (i.e., the initial density both inside and outside the marine
reserve is zero). By considering the linear form of the objective
system, let \(f\left(n\right)=n\). Thus, the objective system turns out to
be:
\begin{equation}
\mathbf{N}_{\mathbf{t+1}}=\mathbf{\text{AN}}_{\mathbf{t}}\mathbf{\ }(35)\nonumber \\
\end{equation}
where the symbols are defined as the following:
\begin{equation}
\mathbf{N}_{\mathbf{t}}=\begin{bmatrix}n_{t}^{R}\\
n_{t}^{O}\\
\end{bmatrix}\ (36)\nonumber \\
\end{equation}\begin{equation}
\mathbf{N}_{\mathbf{t+1}}=\begin{bmatrix}n_{t+1}^{R}\\
n_{t+1}^{O}\\
\end{bmatrix}\ (37)\nonumber \\
\end{equation}\begin{equation}
\mathbf{A}=\begin{bmatrix}mc+a&m\left(1-c\right)\\
\text{mcE}&\left[m\left(1-c\right)+a\right]E\\
\end{bmatrix}\ (38)\nonumber \\
\end{equation}
The first transient metric is used to calculate the similarity between
the initial conditions and a stable equilibrium state. According to
previous research (White et al. 2013, Kaplan et al. 2019), this metric
can be expressed as an angle \(\theta\) between
vectors\(\mathbf{N}_{\mathbf{0}}\) and \(\mathbf{w}_{\mathbf{1}}\):
\begin{equation}
\theta=\arccos\left(\frac{\mathbf{N}_{\mathbf{0}}\mathbf{\ }\bullet\ \mathbf{w}_{\mathbf{1}}}{|\left|\mathbf{N}_{\mathbf{0}}\right|\mathbf{|\ |}\left|\mathbf{w}_{\mathbf{1}}\right|\mathbf{|\ }}\right)\ (39)\nonumber \\
\end{equation}
where \(\mathbf{N}_{\mathbf{0}}\) represents the initial
density,\(\mathbf{w}_{\mathbf{1}}\) is the dominant right eigenvector of
matrix\textbf{A} , and the double vertical bars denote the vector norm.
The second transient metric is used to show the rate of convergence to
an asymptotic equilibrium state. Similar to previous research (White et
al. 2013, Kaplan et al. 2019), this metric is expressed as
\(\rho\) by approximately calculating the ratio of the first
and second eigenvalues of matrix \(\mathbf{A}\):
\begin{equation}
\rho\approx\frac{\lambda_{3}}{\left|\lambda_{4}\right|}\ (40)\nonumber \\
\end{equation}
where \(\lambda_{3}\) and \(\lambda_{4}\) are the first and
second eigenvalues of \(\mathbf{A}\), respectively. With these
equations, simulations are performed, and the numerical analysis results
are presented in the next section.
\textbf{Population transient dynamics with random initial density}
The random initial density should also be considered rather than only
considering the initial density at the equilibria to investigate the
transient phenomenon in the strong stock population dynamics. First, 500
(enough for the simulation) random numbers are produced for both
population densities inside and outside the marine reserve,
respectively. Second, the transient metric \(\theta\) is
calculated based on the random initial densities. Finally, the random
initial densities that correspond to the maximum value of
\(\theta\) are chosen for further transient dynamic analysis so
that the transient phenomena can be observed.
\textbf{Numerical results}
As the length of the marine reserve coastline becomes larger for the
dynamics starting at the equilibrium (0,0), the population densities and
fisheries yields stay at a low level for a long time and approach zero
both inside and outside the marine reserve (Fig. 1). Correspondingly,
when the length of the marine reserve coastline increases, the fisheries
yields at the final equilibrium state increase for situations 1, 2 and 4
(Fig. 1b, d, h). However, the opposite results occur for situations 3,
5, 6 and 7 (Fig. 1f, j, l, n). Note that the length of the marine
reserve coastline increases from zero (i.e., no marine reserves exist)
to the largest in its value range. Therefore, situations 3, 5, 6 and 7
suggest the advantage of traditional fisheries management without a
marine reserve in improving fisheries yields, which is consistent with
intuition.
At the random initial density, a decrease in the length of the marine
reserve coastline results in increasing fluctuations in the population
density both inside and outside the marine reserve for situations 2 and
3 (Fig. 2c and e). Nevertheless, the length of the marine reserve
coastline has little effect on the perturbation of the population
density for situations 1, 4, 5, 6 and 7 because all of these situations
have weak transient dynamics (Fig. 2a, g, i, k, m). However, for
fisheries yields, the strong transient dynamics can be observed for all
situations except for situation 1 (Fig. 2b, d, f, h, j, l, n). Moreover,
the larger the length of the marine reserve coastline is, the more the
fisheries yields will fluctuate for situations 4, 5, 6 and 7 (Fig. 2h,
j, l, n). Therefore, the transient phenomena for both the population
density and fisheries yields are not consistent.
The robust effect of the length of the marine reserve coastline on the
final fisheries yields at the equilibrium state is demonstrated by the
same conclusion between the analysis with the initial density at (0,0)
and that with a random initial density (i.e., a long marine reserve
coastline increases the fisheries yields at equilibrium for situations
1, 2 and 4 while decreasing those for situations 3, 5, 6 and 7) (Figs. 1
and 2). In other words, the response of the fisheries yields at
equilibrium to the length of the marine reserve coastline is insensitive
to the initial density of the objective population.
Sensitivity analysis shows that the variations in two transient
metrics\(\theta\) and \(\rho\) depend on not only the
variation in the marine reserve size and the escapement rate (strongly
related to fishing effort) but also the life-history parameters for both
the strong and weak stocks (Fig. 3). In general, the marine reserve size
and the escapement rate have a reverse effect on both transient
metrics\(\theta\) and \(\rho\) regardless of the
variations in all other parameters (Fig. 3a, b, g, h; Figs. S1-S6). This
finding is derived from the fact that the population dynamics should be
maintained under the persistence of the weak stock. The sensitivity
analyses based on both the initial density at (0,0) (results not shown)
and a random initial density (Fig. 3; Figs. S1-S6) exhibit similar
phenomena for the two transient metrics responding to different
parameters, which suggests that the robustness of the result is not
affected by the initial density.
\textbf{Discussion}
Permanent marine reserves are an effective way to maintain fisheries
yields without sacrificing the persistence of endangered species and
thus provide new methods for solving an important global issue in
fisheries management (i.e., fisheries bycatch) (Hastings et al. 2017).
However, one of the challenges still faced in fisheries bycatch
assessments is the lack of theoretical predictions in the transient
phenomena, especially when the population dynamics of the target species
are strongly correlated with the time scales. The transient analyses in
this paper show that the transient emergence of fisheries yields of
target species does not depend on the transient emergence of the
population density both inside and outside the marine reserve (Fig. 2).
Thus, the occurrence of transient phenomena in one variable cannot be
used to predict the other. Meanwhile, the simulations show that the
occurrence of transient phenomena strongly depends on the length of the
marine reserve coastline. In addition, the analysis results suggest that
both the advantage of a permanent marine reserve in fisheries management
and the transient occurrence of fisheries yields and population density
are species-specific (Fig. 2).
By increasing the length of the marine reserve coastline, a marine
reserve can improve the fisheries yields of the target species and
maintain the persistence of the weak stock species, which is consistent
with the conclusions of previous research (Hastings et al. 2017).
However, the numerical analysis results also indicate that the advantage
of a marine reserve in improving the yields of the strong stock and
maintaining the persistence of the weak stock are species-specific (Fig.
1). Therefore, the implementation of marine reserves should be based on
enough empirical information about the target species rather than using
the same design method for a marine reserve as a general law.
In comparison with the population density dynamics simulated with a
random initial density (corresponding to the maximum \(\theta\)
among the 500 values), which turns out to be far from (0,0), most
trajectories of the population density show long persistence near (0,0)
for the simulations with an initial density of (0,0). In other words,
the long transients near (0,0) disappear when the initial condition is
far from (0,0) (Figs. 1 and 2). This phenomenon indicates that the
crawl-by transient dynamics that occurred at the saddle (0,0) may also
belong to a ghost. Therefore, the analysis suggests that different types
of transients classified as ''crawl-by'' and ''ghost'' probably occur
simultaneously under certain circumstances because of the similarities
between them (Hastings et al. 2018, Morozov et al. 2019). In addition,
although the population density (especially for the population density
inside the marine reserve) could be sustainable at the asymptotic
equilibrium state, the transient phenomenon of a long stay at (0,0)
during the first stage suggests that fisheries species face a high risk
of going into extinction because the population density decreases to a
very low level. Thus, one may believe that the target fisheries species
are sustainable without conservation, and unsuitable policies may be
developed if fisheries management is based on the information from the
asymptotic state of the population dynamics. This finding again
demonstrates the significant importance of understanding transient
dynamics as well as setting expected timelines for fished population
recovery for natural conservation and fisheries management (Kaplan et
al. 2019).
One of the most important procedures in fisheries management is
collecting information by monitoring the population dynamics of the
target species, and the monitoring information is then used to achieve
the goal of maximizing fisheries yields as well as sustainably
maintaining the species. However, this intuitive management method
(monitoring first and then setting fishing policies based on the
monitoring information) is based on an easily ignored assumption that
there is transient consistency between population dynamics and fisheries
yields. Therefore, it makes sense to use one variable to predict the
other. However, the analysis results in this paper suggest that
fisheries yields are unpredictable even with sufficient monitoring
information about the population dynamics because fisheries yields can
exhibit great variations and are thus unstable even if the population
density dynamics are deterministic and only slightly vary (i.e., the
transient inconsistency between them) (Fig. 2). Thus, the transient
dynamics of the population density of the strong stock cannot be used to
predict the transient dynamics of the fisheries yields of the strong
stock under conditions in which the weak stock is persistent. The
transient dynamics of fisheries yields should be investigated even if
there is no transient phenomenon in the population density dynamics
because of the transient inconsistency between fisheries yields and
population density. This research provides a theoretical understanding
of the ecological transient dynamics of the marine reserve management.
The theoretical framework that is used for transient analysis in this
study does not take time delays into account. Although a time delay
increases the difficulty of studying a specific system, it can also
cause transient phenomena even for some low- dimensional systems such as
a simple two-species model (Hastings et al. 2018, Morozov et al. 2019).
Time delays should be taken into consideration, especially for some very
common cases. For example, the effect of fishing activities on the
population growth rate should be considered when the population density
is very low (in most cases, marine reserves should be established). If
the population density is so high that it exceeds the environmental
carrying capacity, fishing activities can immediately increase the
population growth rate by reducing self-thinning behaviour. In contrast,
a long time is needed to observe the effects of fishing activities, and
thus, time delays should be considered. Another issue that deserves
further attention is the transient dynamics of the weak stock. In this
research, only the transient dynamics of the strong stock are considered
based on the assumption that the weak stock is persistent at the
asymptotic equilibrium state. By considering the transient dynamics of
the weak stock, the weak stock persistent conditions cannot be
calculated from the determinant of the Jacobian matrix at the
equilibrium state. Instead, a dynamical method is necessary to ensure
that the weak stock is persistent.
\textbf{Ethics.} This article has no ethic problems.
\textbf{Data accessibility} . This article has no additional data and
there is no need to deposit data to a public repository.
\textbf{Author contributions.} R.C. did all the work for this paper.
\textbf{Competing interests.} There are no competing interests.
\textbf{Funding.} This research is supported by a foundation from Shanxi
Normal University (0505/02070499).
\textbf{Acknowledgements.} Great thanks to professor Alan Hastings for
his help and supervision when I was studying and working in UC Davis.
\textbf{References}
Bertram, R. and J. E. Rubin. 2017. Multi-timescale systems and fast-slow
analysis. Mathematical Biosciences\textbf{287} :105-121.
Clay, T. A., C. Small, G. N. Tuck, D. Pardo, A. P. Carneiro, A. G. Wood,
J. P. Croxall, G. T. Crossin, and R. A. Phillips. 2019. A comprehensive
large-scale assessment of fisheries bycatch risk to threatened seabird
populations. Journal of Applied Ecology \textbf{56} : 1882-1893.
Edgar, G. J., R. D. Stuart-Smith, T. J. Willis, S. Kininmonth, S. C.
Baker, S. Banks, N. S. Barrett, M. A. Becerro, A. T. Bernard, J.
Berkhout, C. D. Buxton, S. J. Campbell, A. T. Cooper, M. Davey, S. C.
Edgar, G. Forsterra, D. E. Galvan, A. J. Irigoyen, D. J. Kushner, R.
Moura, P. E. Parnell, N. T. Shears, G. Soler, E. M. Strain, and R. J.
Thomson. 2014. Global conservation outcomes depend on marine protected
areas with five key features. Nature\textbf{506} :216-220.
Feng, M., S.-M. Cai, M. Tang, and Y.-C. Lai. 2019. Equivalence and its
invalidation between non-Markovian and Markovian spreading dynamics on
complex networks. Nature communications \textbf{10} :1-10.
Game, E. T., M. Bode, E. McDonald-Madden, H. S. Grantham, and H. P.
Possingham. 2009. Dynamic marine protected areas can improve the
resilience of coral reef systems. Ecology Letters \textbf{12}
:1336-1346.
Gerber, L. R., L. W. Botsford, A. Hastings, H. P. Possingham, S. D.
Gaines, S. R. Palumbi, and S. Andelman. 2003. Population models for
marine reserve design: a retrospective and prospective synthesis.
Ecological Applications\textbf{13} :47-64.
Goetze, J. S., J. Claudet, F. Januchowski-Hartley, T. J. Langlois, S. K.
Wilson, C. White, R. Weeks, and S. D. Jupiter. 2018. Demonstrating
multiple benefits from periodically harvested fisheries closures.
Journal of Applied Ecology\textbf{55} :1102-1113.
Guichard, F., S. A. Levin, A. Hastings, and D. Siegel. 2004. Toward a
dynamic metacommunity approach to marine reserve theory. AIBS Bulletin
\textbf{54} :1003-1011.
Hastings, A. 2001. Transient dynamics and persistence of ecological
systems. Ecology Letters\textbf{4} :215-220.
Hastings, A. 2004. Transients: the key to long-term ecological
understanding? Trends in Ecology \& Evolution \textbf{19} :39-45.
Hastings, A., K. C. Abbott, K. Cuddington, T. Francis, G. Gellner, Y.-C.
Lai, A. Morozov, S. Petrovskii, K. Scranton, and M. L. Zeeman. 2018.
Transient phenomena in ecology. Science \textbf{361} :eaat6412.
Hastings, A. and L. W. Botsford. 1999. Equivalence in Yield from Marine
Reserves and Traditional Fisheries Management. Science \textbf{284}
:1537-1538.
Hastings, A., S. D. Gaines, and C. Costello. 2017. Marine reserves solve
an important bycatch problem in fisheries. Proc Natl Acad Sci U S A
\textbf{114} :8927-8934.
Hastings, A. and K. Higgins. 1994. Persistence of transients in
spatially structured ecological models. Science \textbf{263} :1133-1136.
Hazen, E. L., K. L. Scales, S. M. Maxwell, D. K. Briscoe, H. Welch, S.
J. Bograd, H. Bailey, S. R. Benson, T. Eguchi, and H. Dewar. 2018. A
dynamic ocean management tool to reduce bycatch and support sustainable
fisheries. Science advances\textbf{4} :eaar3001.
Herrera, G. E., H. V. Moeller, and M. G. Neubert. 2016. High-seas fish
wars generate marine reserves. Proceedings of the National Academy of
Sciences \textbf{113} :3767-3772.
Hilborn, R. 2017. Traditional fisheries management is the best way to
manage weak stocks. Proceedings of the National Academy of Sciences
\textbf{114} : E10610-E10610.
Horswill, C. and A. Manica. 2019. California swordfish fishery:
Maximizing the catch rate of a target species simultaneously minimizes
bycatch rates. Proceedings of the National Academy of Sciences
\textbf{116} :7172-7173.
Kaplan, D. M., D. R. Hart, and L. W. Botsford. 2010. Rotating spatial
harvests and fishing effort displacement: a comment on Game et al.
(2009). Ecology Letters\textbf{13} :E10-E12.
Kaplan, K. A., L. Yamane, L. W. Botsford, M. L. Baskett, A. Hastings, S.
Worden, and J. Wilson White. 2019. Setting expected timelines of fished
population recovery for the adaptive management of a marine protected
area network. Ecological Applications \textbf{29} : 1202-1220.
Komoroske, L. M. and R. L. Lewison. 2015. Addressing fisheries bycatch
in a changing world. Frontiers in Marine Science \textbf{2} :1-11.
Kuang, Y. 1993. Delay differential equations: with applications in
population dynamics. Academic press.
Mangel, M. 2000. On the fraction of habitat allocated to marine
reserves. Ecology Letters \textbf{3} :15-22.
Mari, L., R. Casagrandi, E. Bertuzzo, A. Rinaldo, and M. Gatto. 2019.
Conditions for transient epidemics of waterborne disease in spatially
explicit systems. Royal Society open science \textbf{6} :1-16.
Morozov, A., K. Abbott, K. Cuddington, T. Francis, G. Gellner, A.
Hastings, Y.-C. Lai, S. Petrovskii, K. Scranton, and M. L. Zeeman. 2019.
Long transients in ecology: theory and applications. Physics of Life
Reviews.
Rinaldi, S. and M. Scheffer. 2000. Geometric analysis of ecological
models with slow and fast processes. Ecosystems \textbf{3} :507-521.
Rudolf, V. H. 2019. The role of seasonal timing and phenological shifts
for species coexistence. Ecology Letters \textbf{22} :1324-1338.
Sanchirico, J. N., U. Malvadkar, A. Hastings, and J. E. Wilen. 2006.
When are no-take zones an economically optimal fishery management
strategy? Ecological Applications\textbf{16} :1643-1659.
Santos, J., B. Herrmann, B. Mieske, D. Stepputtis, U. Krumme, and H.
Nilsson. 2016. Reducing flatfish bycatch in roundfish fisheries.
Fisheries Research \textbf{184} :64-73.
Scales, K. L., E. L. Hazen, M. G. Jacox, F. Castruccio, S. M. Maxwell,
R. L. Lewison, and S. J. Bograd. 2018. Fisheries bycatch risk to marine
megafauna is intensified in Lagrangian coherent structures. Proceedings
of the National Academy of Sciences \textbf{115} :7362-7367.
Shriver, R. K., C. M. Andrews, R. S. Arkle, D. M. Barnard, M. C.
Duniway, M. J. Germino, D. S. Pilliod, D. A. Pyke, J. L. Welty, and J.
B. Bradford. 2019. Transient population dynamics impede restoration and
may promote ecosystem transformation after disturbance. Ecology Letters
\textbf{22} : 1357-1366.
Smith, H. L. 2011. An introduction to delay differential equations with
applications to the life sciences. Springer New York.
Taylor, B. L., L. Rojas-Bracho, J. Moore, A. Jaramillo-Legorreta, J. M.
Ver Hoef, G. Cardenas-Hinojosa, E. Nieto-Garcia, J. Barlow, T.
Gerrodette, and N. Tregenza. 2017. Extinction is imminent for Mexico's
endemic porpoise unless fishery bycatch is eliminated. Conservation
Letters \textbf{10} :588-595.
Thorne, L. H., R. W. Baird, D. L. Webster, J. E. Stepanuk, and A. J.
Read. 2019. Predicting fisheries bycatch: A case study and field test
for pilot whales in a pelagic longline fishery. Diversity and
Distributions \textbf{25} :909-923.
Welch, H., R. Pressey, and A. Reside. 2018. Using temporally explicit
habitat suitability models to assess threats to mobile species and
evaluate the effectiveness of marine protected areas. Journal for Nature
Conservation \textbf{41} :106-115.
White, C., B. E. Kendall, S. Gaines, D. A. Siegel, and C. Costello.
2008. Marine reserve effects on fishery profit. Ecology Letters
\textbf{11} :370-379.
White, J. W., L. W. Botsford, A. Hastings, M. L. Baskett, D. M. Kaplan,
and L. A. Barnett. 2013. Transient responses of fished populations to
marine reserve establishment. Conservation Letters \textbf{6} :180-191.
\textbf{Figure legends}
\textbf{Figure 1} The population density dynamics and fisheries yields
with the initial density at the equilibrium state (0,0). Different
colours represent different gradients of the length of marine reserve
coastline (i.e., marine reserve size) marked as \(c_{0}\),
\(c_{1}\),\(c_{2}\), \(c_{3}\),
\(c_{m}\) (\(c_{0}\)=0; \(c_{1}\)
\textless{}\(c_{2}\) \textless{} \(c_{3}\);
\(c_{m}\) approaches the maximum value. The legend is shown in
the subplot \textbf{(b)} , which is also suitable for other subplots).
Note that the specific values of \(c_{1}\)
,\(c_{2}\) , \(c_{3}\), and \(c_{m}\) are
different in the different subplots, although \(c_{0}\)=0
remains the same. The solid and dashed lines in the subplots of
population density denote the density inside and outside marine
reserves, respectively. (\textbf{a,b} ) simulation results for situation
1 with \(c_{1}\) = 0.050, \(c_{2}\) =
0.100,\(c_{3}\) = 0.150, and \(c_{m}\) = 0.200
(specific values for the other parameters can be seen in Table 2).
(\textbf{c,d} ) simulation results for situation 2 with
\(c_{1}\) = 0.148, \(c_{2}\) = 0.295,
\(c_{3}\) = 0.443, and \(c_{m}\) = 0.590.
(\textbf{e,f} ) simulation results for situation 3 with
\(c_{1}\) = 0.035, \(c_{2}\) = 0.070,
\(c_{3}\) = 0.105, and \(c_{m}\) = 0.140.
(\textbf{g,h} ) simulation results for situation 4 with
\(c_{1}\) = 0.006, \(c_{2}\) = 0.012,
\(c_{3}\) = 0.018, and \(c_{m}\) = 0.024.
(\textbf{i,j} ) simulation results for situation 5 with
\(c_{1}\) = 0.0003, \(c_{2}\) = 0.0006,
\(c_{3}\) = 0.0009, and \(c_{m}\) = 0.0012.
(\textbf{k,l} ) simulation results for situation 6 with
\(c_{1}\) = 0.0045, \(c_{2}\) = 0.0090,
\(c_{3}\) = 0.0135, and \(c_{m}\) = 0.0180.
(\textbf{m,n} ) simulation results for situation 7 with
\(c_{1}\) = 0.0035, \(c_{2}\) = 0.0070,
\(c_{3}\) = 0.0105, and \(c_{m}\) = 0.0140.
\textbf{Figure 2} The transient phenomena of population density and
fisheries yields with random initial density. Different colours
represent different gradients of the length of marine reserve coastline
(i.e., marine reserve size) marked as \(c_{0}\),
\(c_{1}\), \(c_{2}\), \(c_{3}\),
and\(c_{m}\) whose specific meanings are the same as those
explained in Figure 1. Note that the specific values of
\(c_{1}\) , \(c_{2}\) ,\(c_{3}\), and
\(c_{m}\) are different in the different subplots, although
\(c_{0}\)=0 remains the same. The solid and dashed lines in
the subplots of population density denote the density inside and outside
marine reserves, respectively. (\textbf{a,b} ) simulation results for
situation 1 with \(c_{1}\) = 0.050, \(c_{2}\) =
0.100, \(c_{3}\) = 0.150, and \(c_{m}\) = 0.200
(specific values for other parameters can be seen in Table 2).
(\textbf{c,d} ) simulation results for situation 2 with
\(c_{1}\) = 0.148, \(c_{2}\) = 0.295,
\(c_{3}\) = 0.443, and\(c_{m}\) = 0.590.
(\textbf{e,f} ) simulation results for situation 3 with
\(c_{1}\) = 0.035, \(c_{2}\) = 0.070,
\(c_{3}\) = 0.105, and\(c_{m}\) = 0.140.
(\textbf{g,h} ) simulation results for situation 4 with
\(c_{1}\) = 0.006, \(c_{2}\) = 0.012,
\(c_{3}\) = 0.018, and\(c_{m}\) = 0.024.
(\textbf{i,j} ) simulation results for situation 5 with
\(c_{1}\) = 0.0003, \(c_{2}\) = 0.0006,
\(c_{3}\) = 0.0009, and\(c_{m}\) = 0.0012.
(\textbf{k,l} ) simulation results for situation 6 with
\(c_{1}\) = 0.0045, \(c_{2}\) = 0.0090,
\(c_{3}\) = 0.0135, and\(c_{m}\) = 0.0180.
(\textbf{m,n} ) simulation results for situation 7 with
\(c_{1}\) = 0.0035, \(c_{2}\) = 0.0070,
\(c_{3}\) = 0.0105, and\(c_{m}\) = 0.0140.
\textbf{Figure 3} Sensitivity of two transient metrics to variations in
fisheries management and life-history parameters with a random initial
density for situation 1. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S1} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 2. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S2} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 3. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S3} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 4. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S4} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 5. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S5} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 6. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Figure S6} Sensitivity of two transient metrics to variation in
fisheries management and life-history parameters with random initial
density for situation 7. (\textbf{a-f} ) variations of transient
metric\(\theta\) in response to marine reserve size, escapement
rate, per capita fecundity for the strong stock, adults survivorship for
the strong stock, per capita fecundity for the weak stock, and adults
survivorship for the weak stock, respectively. (\textbf{g-l} )
variations of transient metric \(\rho\) in response to marine
reserve size, escapement rate, per capita fecundity for the strong
stock, adults survivorship for the strong stock, per capita fecundity
for the weak stock, and adults survivorship for the weak stock,
respectively.
\textbf{Table 1} Definitions of the symbols used in this paper.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Symbols & Description\tabularnewline
\midrule
\endhead
a & The survivorship of adults of the strong stock
species.\tabularnewline
m & Per capita fecundity for the strong stock species.\tabularnewline
\(\alpha\) & Proliferation rate per generation in the
Beverton--Holt growth function for the strong stock
species.\tabularnewline
\(\beta\) & Carrying capacity in the Beverton--Holt growth
function for the strong stock species.\tabularnewline
\(a_{w}\) & The survivorship of adults for the weak stock
species.\tabularnewline
\(m_{w}\) & Per capita fecundity for the weak stock
species.\tabularnewline
\(\alpha_{w}\) & Proliferation rate per generation in the
Beverton--Holt growth function for the weak stock
species.\tabularnewline
\(\beta_{w}\) & Carrying capacity in the Beverton--Holt growth
function for the weak stock species.\tabularnewline
\emph{c} & The fraction of the coastline in a no-take marine reserve for
both the strong and weak stock species.\tabularnewline
\emph{E} & The escapement rate representing the fraction of the fish
stock that is left unharvested outside marine reserves for both the
strong and weak stock species.\tabularnewline
\(n_{t}^{R}\) & Population density inside marine reserves at time
\emph{t} for the strong stock species.\tabularnewline
\(n_{t}^{O}\) & Population density outside marine reserves at time
\emph{t} for the strong stock species.\tabularnewline
\(Y_{P}\) & The harvested yield of the strong stock
species.\tabularnewline
\textbf{A} & Population projection matrix of the objective
system.\tabularnewline
\(\mathbf{N}_{\mathbf{t}}\) & \(2\times\)1 vector of population density
at time \emph{t}.\tabularnewline
\(\mathbf{N}_{\mathbf{0}}\) & Initial conditions of the densities both inside
and outside marine reserves.\tabularnewline
\(\mathbf{w}_{\mathbf{1}}\) & The dominant right eigenvector of matrix
\textbf{A}.\tabularnewline
\(\theta\) & The transient metric representing the similarity
between the initial conditions and stable equilibrium
state.\tabularnewline
\(\rho\) & The transient metric representing the rate of
convergence to the asymptotic equilibrium state.\tabularnewline
\(\lambda_{1}\) & One eigenvalue of the Jacobian matrix of the
objective system.\tabularnewline
\(\lambda_{2}\) & The other eigenvalue of the Jacobian matrix of the
objective system.\tabularnewline
\(\lambda_{3}\) & The first eigenvalue of
\(\mathbf{A}\).\tabularnewline
\(\lambda_{4}\) & The second eigenvalue of
\(\mathbf{A}\).\tabularnewline
\bottomrule
\end{longtable}
\textbf{Table 2} Specific parameter values used for the simulation
analyses. By applying the values to different species (including both
strong and weak stocks species) in a natural system, the parameter
values are classified into 7 groups marked as situations 1-7. All the
parameter values are the same as those in previous research that studied
the asymptotic behaviour of the target system (Hastings et al. 2017).\selectlanguage{english}
\begin{longtable}[]{@{}llllllll@{}}
\toprule
Parameters & Situation 1 & Situation 2 & Situation 3 & Situation 4 &
Situation 5 & Situation 6 & Situation 7\tabularnewline
\midrule
\endhead
a & 0.01 & 0.01 & 0.01 & 0.87 & 0.87 & 0.87 & 0.87\tabularnewline
m & 8.5 & 8.5 & 8.5 & 1 & 1 & 1 & 1\tabularnewline
\(\alpha\) & 0.7 & 0.7 & 0.7 & 16 & 16 & 16 & 16\tabularnewline
\(\beta\) & 20 & 20 & 20 & 23799 & 23799 & 23799 &
23799\tabularnewline
\(a_{w}\) & 0.85 & 0.5 & 0.7 & 0.85 & 0.939 & 0.95 &
0.955\tabularnewline
\(m_{w}\) & 1.6 & 1.4 & 2.5 & 1 & 1 & 1 & 1\tabularnewline
\(\alpha_{w}\) & 0.4 & 0.6 & 0.78 & 6.26 & 13.62 & 2.67 &
3.14\tabularnewline
\(\beta_{w}\) & 5 & 5 & 5 & 825.8 & 204.05 & 3495.3 &
72.59\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
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