\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 76, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/76\hfil Bioeconomical Ricker's model]
{Bioeconomical Ricker's model of marine protected areas}
\author[C. N. Christou, L. V. Idels \hfil EJDE-2012/76\hfilneg]
{Cameron N. Christou, Lev V. Idels} % in alphabetical order
\address{Cameron N. Christou \newline
Department of Mathematics,
University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada}
\email{cnchrist@math.ubc.ca}
\address{Lev V. Idels \newline
Department of Mathematics,
Vancouver Island University\\
900 Fifth St. Nanaimo BC, V9S5S5, Canada}
\email{lev.idels@viu.ca}
\thanks{Submitted November 29, 2011. Published May 14, 2012.}
\subjclass[2000]{34C60, 37N25, 49K15, 92B05}
\keywords{Marine protected areas; source-sink models;
fishery models, \hfill\break\indent
bioeconomical analysis, Pontryagin's maximum principle, nonlinear
differential equations}
\begin{abstract}
Marine protected areas (MPA) become part of modern fishery management
to safeguard marine life and sustain ecosystem processes. Based on a classical
Ricker's model, the deterministic nonlinear sink-source model of MPA is presented.
Sufficient conditions for the existence of positive bounded steady-states are
obtained. The present value of net revenue is maximized subject to
population dynamics in the fishing zone and in the protected area.
The analysis has shown that there is an optimal equilibrium
solution, and the best harvesting policy to attain this equilibrium
position is a bang-bang control effort. It was demonstrated
numerically by comparing the optimal harvesting rate against a
constant harvesting rate, and the fast convergence to the optimal
equilibrium guarantees greater profits under the optimal harvesting strategy.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
Traditional methods of controlling fishing include:
\begin{itemize}
\item limiting the effort (restricting the fishing season);
\item limiting the catch (restricting the number or biomass of fish captured);
\item limiting the fish size (restricting the size of fish captured);
\item limiting the location (restricting the places available for fishing).
\end{itemize}
The last option includes the establishment of marine reserves, areas
permanently closed to all fishing. These can supplement or
potentially replace other methods of regulation. For this reason,
the reserve concept has recently attracted great interest in the
community of fisheries scientists and managers. Marine protected
areas have been promoted as conservation and fishery
management tools, and at present, there are over 1300 marine
reserves in the world
\cite{Ami,Gell,Graf,Ko,Pi,So, Sum}. Since 1999, the number and extent
of, and the amount of research on, MPA has grown rapidly. Areas
protected from fishing provide multiple benefits, including: reduced
likelihood of a stock collapsing, enhanced spawning biomass,
improved recruitment survival of fish to a more mature state,
allowing increased harvest and recovery of the habitat. Much of the
empirical case study research on the ecological impact of MPA
indicate that within the no-take area boundaries, fish both grow
and attain a broader age and size distribution
\cite{Ko,Pi,So}. There is robust demonstration of conservation
benefits of the MPA, but fishery benefits and design of reserves
remain controversial. Most of the researchers suggest that in the
long term no-fishing zones sustain both fisheries and fish
populations. It is also known that highly migratory species are
among the most overexploited finfish species. Many species,
especially adults such as cod, show directed migration on an annual
basis; thus adults could be very vulnerable at certain periods of
the year when most of them would be likely to move outside
of a protected area \cite{Do2,Gell,Graf}.
The biological interconnections between patches may be of many
varieties and we will use a sink-source model based on the
Ricker's growth curve.
In recent publications \cite{Arm,Do,Dub,Kara},
MPAs are modeled by using logistic differential equations. More complex
models were studied in \cite{B1,B2,B3,Je,I}.
Suppose that two regions $R$
(reserve area) and $U$ (unprotected or fishing zone) have habitat
areas: $A_1$ is the area of a protected zone and $A_2$ is the area of the fishing zone.
Let $x(t)$ denotes the biomass of fish in the reserve $R$, and
$y(t)$ is the fish biomass in the unprotected (fishing) area. Assume that
fishing takes
place only in the region $U$, with the region $R$ established as a marine
reserve or protected area. The two equations in model \eqref{rickermodel}
are linked by the movement
terms, which depend on a difference of biomass densities per unit of the
habitat areas. The depletion of fish in region $U$ will generally
decrease the fish density there and cause fish to move away from a
higher density in the protected region $R$.
We introduced the migration rate
\[
d_i = \frac{\sigma}{A_i},\quad \sigma > 0,
\]
where $\sigma$ is a migration coefficient.
Then the coupled system of nonlinear differential equations
\begin{equation} \label{e1}
\begin{gathered}
\frac{dx}{dt} = -m_1 x(t) + d_2 y(t) - d_{1} x(t)
+ F_1 (x(t)), \\
\frac{dy}{dt} = -m_2 y(t) + d_{1} x(t)- d_2 y(t) + F_2 (y(t)) -
E(t)qy(t)
\end{gathered}
\end{equation}
could represent the population dynamics in each region, where $m_i$
is a natural mortality rate,
\[
F_i (x) = p_i xe^{-x},\quad p_i > 0.
\]
is the birth function.
Finally, the system under study has the form
\begin{equation}\label{rickermodel}
\begin{gathered}
\frac{dx}{dt} = -m_1 x + d_2 y - d_{1} x + p_{1}x e^{-x}, \\
\frac{dy}{dt} = -m_2 y + d_{1} x- d_2 y + p_2ye^{-y} - E(t)qy.
\end{gathered}
\end{equation}
In system \eqref{rickermodel} a positive constant $q$ is called a catchability
coefficient, whereas the effort function $E(t)$ is a dynamic variable.
MPA design is a joint economic and biological problem, and MPA
creation alters economic incentives. In fisheries the optimal
control has often been applied to compute bieconomic optima via
locally-optimal linear feedback controls, and mostly the optimal
solutions obtained are bang-bang controls
\cite{Arm,Bea,conrad,Do,Dub,Je,Mch,Ne}.
\section{Equilibrium analysis}
To find all equilibria of system \eqref{rickermodel} we set
\begin{equation}\label{eqsystem}
\begin{gathered}
m_1 x + d_{1} x - d_2 y - p_1 xe^{- x} = 0, \\
m_2 y - d_{1} x + d_2 y + Eqy - p_2 ye^{- y} = 0.
\end{gathered}
\end{equation}
Apart of a trivial solution, system \eqref{eqsystem} has a positive
equilibrium $(x^{*}, y^{*}).$
From system \eqref{eqsystem} we have two curves:
\begin{equation}\label{equilib2}
\begin{gathered}
L_{1}: \quad y =\phi_{1}(x)= \frac{m_{1}+d_1-p_{1}e^{-x} }{d_2} x,\\
L_2: \quad x=\phi_2(y)= \frac{m_2+d_2+Eq-p_2e^{-y}}{d_1} y.
\end{gathered}
\end{equation}
\begin{theorem}\label {thm3.1}
System \eqref{rickermodel} has a unique internal positive equilibrium if
the following two conditions hold:
(i) $m_1+d_{1}\leq p_1$ and
(ii) $m_2+d_2+Eq\leq p_2$.
\end{theorem}
\begin{proof}
To prove the existence of a nontrivial equilibrium,
firstly note that curves $L_1$ and $L_2$ have asymptotes
$$
y=\frac{m_1 +d_1}{d_2}x\quad\text{and}\quad y=\frac{d_1}{m_2+d_2+Eq}x
$$
correspondingly. Clearly,
$$
\frac{m_{1}+d_1}{d_2}>\frac{d_1}{m_2+d_2+Eq} \hspace{1mm},
$$
thus, for sufficiently large $x$, points on the curve $L_1$ lie
above the corresponding points of the curve $L_2$. On the other
hand, in the neighborhood of the origin, each of the conditions
(i)--(ii) guarantees that points on the curve $L_2$ lie above the
points of the curve $L_1$. Therefore a positive internal equilibrium
of system \eqref{rickermodel} exists.
To prove that this equilibrium is a unique point, firstly, we note
that from system \eqref{equilib2}
\begin{equation}\label{divequilib2}
\begin{gathered}
L_1: \quad \frac{dy}{dx} =\frac{y}{x}+
\frac{p_1}{d_2} x \exp(- x )>\frac{y}{x},\\
L_2:\quad \frac{dx}{dy} =\frac{x}{y}+\frac{p_2}{d_1} y
\exp(- y )>\frac{x}{y},
\end{gathered}
\end{equation}
since $x > 0$ on $L_1$ and $ y >0$ on $L_2$. Let $\theta$ be a polar
angle of the point on the curve $L_1$ (with $Ox$ as a polar axis),
$$
\theta=\arctan\frac{y}{x}=\arctan\frac{\phi_1}{x}.
$$
If $x$ moves from $0$ to $\infty$, then
$$
\frac{d\theta}{dx}=\frac{1}{1+(\frac{y}{x})^2}\frac{d}{dx}
\frac{\phi_1}{x}=\frac{x}{x^2+y^2}
\big(\frac{d\phi_1}{dx}-\frac{\phi_1}{x}\big).
$$
The latter equality and inequalities \eqref{divequilib2} guarantee
$\frac{d\theta}{dx}>0$;
thus $\theta(x)$ is a monotone increasing function with
$$
\theta_0 < \theta(x) < \arctan\frac{m_1+d_1}{d_2}.
$$
Similarly, let $\vartheta$ be a polar angle of the point on the
curve $L_2$ with the same polar axis. If $y$ moves from $0$ to
$\infty$, then $\vartheta (y)$ is a monotone decreasing function,
with
$$
\arctan\frac{d_1}{m_2+d_2+Eq}< \vartheta(y) < \vartheta_0.
$$
At the equilibrium point $\theta=\vartheta$, increase of the function
$\theta$ and decrease of the function $\vartheta$
guarantee the uniqueness of a nontrivial equilibrium.
Let none of the conditions (i)-(ii) hold; then for any $x>0$ and
$y>0$,
\begin{equation}\label{impossib}
\begin{gathered}
\arctan\frac{m_1+d_1}{d_2}>\theta \geq \theta_0=\arctan\frac{m_1+d_1-p_1}{d_2}>\\
\vartheta_0=\arctan\frac{d_1}{m_2+d_2+Eq-p_2}\geq \vartheta >
\arctan\frac{d_1}{m_2+d_2+Eq}\nonumber.
\end{gathered}
\end{equation}
Therefore, the equality $\theta=\vartheta$ is impossible, and the
positive equilibrium does not exist. Theorem \ref{thm3.1} is proved.
\end{proof}
\begin{remark} \label{rmk2.1} \rm
Note that nonlinear system \eqref{eqsystem} can not be solved
analytically, the latter fact makes bioeconomical analysis of the
model more difficult than the corresponding analysis of the system
with logistic growth
\begin{gather*}
m_1 x + d_{1} x - d_2 y - p_1 x^{2} = 0, \\
m_2 y - d_{1} x + d_2 y + Eqy - p_2 y^{2} = 0.
\end{gather*}
in \cite{Do,Dub,Graf,Kara}.
\end{remark}
\begin{theorem} \label{thm2.2}
Any solution of system \eqref{rickermodel} with
nonnegative initial functions and positive initial conditions
ultimately enters the square region
$$
K=\{0< x(t) \leq a \text{ and } 0 0$ and $y(t)>0$ for $t>0$, provided that $x(0)> 0$ and
$y(0)> 0$.
Let $ z=x+y $, then from system \eqref{rickermodel} we have
\[
\frac{dz}{dt} =-m_1x-m_2y-Eqy + F_1(x)+F_2(y).
\]
For $0< \epsilon < \min (m_1, m_2 )$ we have
\begin{align*}
\frac{dz}{dt}+\epsilon z
&= -(m_1-\epsilon)x-(m_2-\epsilon)y-Eqy+ F_1(x))+F_2(y) \\
&< F_1(x)+ F_2(y) \leq \max\{F_1,F_2\}\\
&\leq \max \{p_1, p_2\}=W.
\end{align*}
Therefore,
$\frac{dz}{dt}+\epsilon z < W $, or
\begin{equation}\label{eval}
z(t)< \frac{W}{\epsilon}+ \big(z(0)+\frac{W}{\epsilon}\big)
e^{-\epsilon t}.
\end{equation}
Thus all solutions of \eqref{rickermodel} are bounded.
\end{proof}
\section{Optimal harvesting}
In the presence of an MPA, the objective is to optimally exploit the
available resources. The discounted revenue of the fishing industry
is given in \cite{Kot}
\[
J = \int_0^{\infty}e^{-\delta t}\left [pqy(t) - c\right ]E(t) \mathrm{d}t,
\]
where $\delta$ is the instantaneous rate of annual discount, $p$ is
the unit price of the biomass being harvested, and $c$ is the cost
per unit of harvested biomass. The cost of fishing is assumed to
decrease with the size of the stock, i.e.
$c'=\frac{dc}{dy} \leq 0$.
Thus, the objective is to maximize the revenue functional $J$ with
respect to the harvesting rate $E(t)$.
In order to prevent extinction of the species, for the given size of the MPA,
determined by the parameters $A_1$ and $A_2$,
a critical harvesting rate $E_c$ is defined as
\[
E_c = \sup\{ E : m_1 + d_{1} \leq p_1 \text{ or} m_2 + d_2
+ Eq \leq p_2 \}
\]
or
\[
E_c = \sup\{
E:(m_1 + d_{1} - p_1)(m_2 + d_2 + Eq - p_2) < d_{1}d_2\}
\]
Therefore, conditions (i)--(ii) in Theorem \ref{thm3.1} are satisfied.
Note that if condition (i) of Theorem \ref{thm3.1} is satisfied,
then $E_c = \infty$ since
any fishing effort will lead to non-trivial solutions. Also note
that $E_c > 0$, otherwise the fish population will become extinct by
natural causes. Therefore, a maximal harvesting rate is chosen so that
$0 \leq E \leq E_{\rm max} < E_c$. In this manner, the fish population
should be protected from over-exploitation.
Hence the set of allowable harvesting rate functions, $E(t)$, belongs to
the set
\[
E = \{ E(t) : 0 \leq E(t) \leq E_{\rm max} \}.
\]
The objective is to find $\sup_{E} J$ subject to the dynamical system
\eqref{rickermodel}.
An application of the Pontryagin's Maximum Principle yields the
Hamiltonian
\begin{equation}\label{ham1}
\begin{aligned}
H &= e^{-\delta t}(pqy - c)E(t) +
\lambda_1\left(-\left(m_1+d_{1}\right)x +
d_2y + p_1xe^{-x}\right) \\
&\quad + \lambda_2\left(-\left(m_2 + d_2+Eq\right)y +
d_{1}x + p_2ye^{-y} \right).
\end{aligned}
\end{equation}
The PMP also produces the equations of motion for the shadow prices
\begin{equation}\label{costate1}
\begin{gathered}
-\dot{\lambda}_1 = \frac{\partial H}{\partial x} =
\lambda_1\left[-\left(m_1+d_{1}\right)+p_1(1-x)e^{-x}\right]+ \lambda_2d_{1}
\\
-\dot{\lambda}_2=\frac{\partial H}{\partial y} =
e^{-\delta t}\left(pq - c'\right)E(t)
+ \lambda_1d_2 + \lambda_2\left[-\left(m_2 +
d_2+Eq\right)+p_2(1-y)e^{-y}\right].\nonumber
\end{gathered}
\end{equation}
Consider the change of variables:
$\lambda_1 = e^{-\delta t}\Gamma_1$, $\lambda_2=e^{-\delta t}\Gamma_2$.
Substitution of $\Gamma_1$ and $\Gamma_2$ in \eqref{ham1}
yields
\begin{equation}\label{ham2}
\begin{aligned}
H &= e^{-\delta t}\left[(pqy - c)E(t) +
\Gamma_1\left(-\left(m_1+d_{1}\right)x +
d_2y + p_1xe^{-x}\right)\right]
\\
&\quad + e^{-\delta t}\left[ \Gamma_2\left(-\left(m_2 + d_2+Eq\right)y +
d_{1}x + p_2ye^{-y} \right)\right].
\end{aligned}
\end{equation}
Equivalently, substitution of $\Gamma_1$ and $\Gamma_2$ in
\eqref{costate1} produces
\begin{equation}\label{CO1}
\begin{gathered}
-e^{-\delta t}\dot{\Gamma}_1 + \delta e^{-\delta t}\Gamma_1
= e^{-\delta t}\Gamma_1\left[-\left(m_1+d_{1}\right)+p_1(1-x)e^{-x}\right]
+ e^{-\delta t}\Gamma_2d_{1}\\
\Rightarrow \;
\dot{\Gamma}_1 = \Gamma_1\left[\delta
+m_1+d_{1}-p_1(1-x)e^{-x}\right] - \Gamma_2d_{1},
\end{gathered}
\end{equation}
and
\begin{equation} \label{CO2}
\begin{gathered}
\begin{aligned}
-e^{-\delta t}\dot{\Gamma}_2 + \delta e^{-\delta t}\Gamma_2
&= e^{-\delta t}\left(pq - c'\right)E(t)
+ e^{-\delta t}\Gamma_1d_2 \\
+ e^{-\delta t}\Gamma_2
\left[-\left(m_2 + d_2\right)+p_2(1-y)e^{-y}-Eq\right]
\end{aligned} \\
\Rightarrow\; \dot{\Gamma}_2 = (c'-pq)E - \Gamma_1d_2
+ \Gamma_2[\delta +m_2 + d_2- p_2(1-y)e^{-y}+Eq].
\end{gathered}
\end{equation}
Equations \eqref{CO1} and \eqref{CO2} represent the weighted shadow
prices \cite{Kot}.
The objective now is to maximize the Hamiltonian \eqref{ham2} over
the set $E$ subject to equations \eqref{CO1} and \eqref{CO2}.
Note that the Hamiltonian is linearly dependent on $E$; therefore,
the optimal harvesting rate must be
a bang-bang control in the set $E$ . Explicitly, the optimal
harvesting rate is
\[
E_{\rm opt}(t) = \begin{cases}
E_{\rm max} & T> 0\\
0 & T < 0 \\
\tilde{E}& T=0 ,
\end{cases}
\]
where $\tilde{E}$ is a singular harvesting rate and $T=pqy - c - qy\Gamma_2$.
This singular harvesting
rate occurs because the value of $E(t)$ is not determined when $T = 0$.
The possible end-states of the system \eqref{rickermodel} under the
optimal harvesting rate $E_{\rm opt}(t)$ are the same as those
mentioned in \cite{Kara}. These states are
\begin{itemize}
\item No harvesting
\item Maximum harvesting
\item Singular rate harvesting
\item A ``bang-bang'' cycle of harvesting rates.
\end{itemize}
If ``no harvesting'' is the optimal end-state of the system, then it
must be the case that it is never profitable to fish for any biomass
up to the natural carrying capacity of the unprotected fishing zone,
i.e. $pqy - c \leq 0$ for all positive values of $y$ up to the
carrying capacity of the zone. If this was not the case, then there
would exist a bang-bang cycle that would produce a greater profit.
If ``maximum harvesting'' is the optimal end-state of the system;
then it must be the case that fishing at a constant rate of $E(t) =
E_{\rm max}$ drives the system \eqref{rickermodel} to a non-trivial
equilibrium point $(x^*, y^*)$, and that $pqy^* - c(y^*) \geq 0$.
If this was not the case, then the profit would be negative, and
greater profit would be obtained by not harvesting at all.
For the singular harvesting rate i.e. $T=0$, function $\Gamma_2$ satisfies
\begin{equation}\label{SING}
\Gamma_2 = p - \frac{c}{qy}.
\end{equation}
If the end-state is a singular rate, then equation \eqref{SING}
holds for an extended period of time, and
\begin{equation}\label{singdot}
\dot{\Gamma}_2 = \Big(\frac{c'}{qy} - \frac{c}{qy^2}\Big)\dot{y}.
\end{equation}
Equating \eqref{CO2} and \eqref{singdot} produce an expression for function
$\Gamma_1$ which can be solved explicitly.
\begin{equation}\label{singdot2}
\Gamma_1 = \frac{1}{f_{y}}\Big(\delta\big(p-\frac{c}{qy}\big) -
(pq-c')E - \big(p-\frac{c}{qy}\big){g_{y}} +
\frac{c'g}{qy} - \frac{cg}{qy^2}\Big),
\end{equation}
where
\[
f = -m_1 x + d_2 y - d_{1} x + p_1xe^{-x},\quad
g = -m_2 y - d_2 y +d_{1} x + p_2ye^{-y} - Eqy.
\]
Since the system stays in this singular state, function $\Gamma_1$
also satisfies
\begin{equation} \label{gam1}
\begin{split}
\dot{\Gamma}_1
& = -\frac{f_{yy} \dot{y}}{f^{2}_{y}} \Big(\delta\big(p-\frac{c}{qy}\big)
- (pq-c')E - \big(p-\frac{c}{qy}\big)g_{y} +
\frac{c'g}{qy} - \frac{cg}{qy^2}\Big)\\
&\quad +\frac{1}{f_{y}}\Big(\frac{-\delta
c'\dot{y}}{qy} + \frac{\delta c \dot{y}}{qy^2} +
c''E\dot{y}-g_{yy}\dot{y}\big(p-\frac{c}{qy}\big)\\
&\quad +2\frac{c'\dot{y}g_{y}}{qy} -
2\frac{c\dot{y}g_{y}}{qy^2}
+\frac{c''\dot{y}g}{qy} -
2\frac{c'g\dot{y}}{qy^2} + 2\frac{cg\dot{y}}{qy^3}
\Big)
\end{split}
\end{equation}
Equating \eqref{CO1} and \ref{gam1} implicitly define a singular
harvesting rate for every point in the first quadrant of the
$(x,y)$-plane. However, due to the imposed restriction $0 \leq
E(t) \leq E_{\rm max}$, it may not be possible to employ a singular
harvesting rate to certain points in the plane.
If system \eqref{rickermodel} reaches an equilibrium, while fishing
at a singular harvesting rate, $(x^*,y^*)$; then the harvesting rate is
defined by
\begin{equation}\label{equiharvest}
\tilde{E} = \frac{1}{q}\Big( -m_2 - d_2 + d_{1}\frac{x^*}{y^*} +
p_2e^{-y^*}\Big).
\end{equation}
\section{Optimal harvesting: equilibrium solutions}\label{optiharvest}
The objective of this section is to optimize the profit earned while
keeping the total biomass of system \eqref{rickermodel} at
equilibrium. If system \eqref{rickermodel} is in stable equilibrium, then
the harvesting rate must be constant and equal to $\tilde{E}$ defined by
\eqref{equiharvest}.
\textbf{Part 1.}
For a given harvesting rate to be optimal, in the sense that profit is
maximized, the following condition must be satisfied:
\[
\frac{\partial H}{\partial E} = 0,
\]
where $H$ is given by equation \eqref{ham2}. Therefore, equation
\[
\frac{\partial H}{\partial E} = pqy - c - \Gamma_2 qy = 0
\]
must be satisfied and
\begin{equation}\label{gamma2}
\Gamma_2 = p - \frac{c}{qy}.
\end{equation}
Under an equilibrium solution, it is also the case
that $\dot{\Gamma}_1 = \dot{\Gamma}_2 = 0$; i.e.,
\begin{gather}\label{gamma1}
\Gamma_2d_{1}-\Gamma_1 [\delta +m_1+d_{1}-p_1(1-x)e^{-x} ]=0,\\
\label{gamma3} (pq - c')E(t) +
\Gamma_1d_2 - \Gamma_2 [\delta +m_2 + d_2-p_2(1-y)e^{-y}+Eq]=0.
\end{gather}
Substitution of equations \eqref{singdot2}, \eqref{equiharvest} and \eqref{gamma2}
in \eqref{gamma1} and \eqref{gamma3} define
a system of equations that can be solved for the optimal
equilibrium solution, $(x^*, y^*)$.
Alternatively, the optimal equilibrium solution can be found by
locating the critical points of the profit function
\[
(pqy - c)\tilde{E},
\]
where $\tilde{E}$ is given by equations \eqref{equiharvest}, and
\[
y = \frac{m_1x + d_{1}x - p_1xe^{-x}}{d_2}.
\]
The latter is a necessary condition for system \eqref{rickermodel} to be
at the steady state.
Clearly, either of these two methods will produce the same result.
\textbf{Part 2.} The objective now is to reach the optimal equilibrium state in an
optimal way. As shown in the Section 3, this objective can be
achieved by using a bang-bang control policy for the harvesting rate
$E(t)$. Until the system reaches the optimal equilibrium point, we define
\[
E(t) = \begin{cases}
E_{\rm max}, &\text{if } pqy(t) - cy(t) - \Gamma_2qy(t) > 0\\
0, & \text{otherwise }
\end{cases}
\]
and then
$E(t)$ is a constant at the equilibrium harvesting rate. Note that
\begin{equation}
\Gamma_2 = p - \frac{c(y^*)}{qy^*},\nonumber
\end{equation}
where $(x^*, y^*)$ is the optimal equilibrium point.
Alternatively, since an internal equilibrium point of system
\eqref{rickermodel} is globally asymptotically stable under constant
harvesting rates, the optimal equilibrium point can be reached by
employing the constant optimal harvesting rate for all time.
\section{Numerical simulations}
The following simulations were produced using the system parameter
values:
$m_1 = 0.5$, $m_2 = 0.2$, $\sigma = 0.4$, $A_1 = 0.2$,
$A_2=0.8$, $p_1 = p_2 = 1$, $q = 0.1$, $p=7$, $E_{\rm max} = 7$, and
$c = 0.4 + \frac{1}{1+y}$.
\begin{example} \label{examp5.1} \rm
Using the methods described in Section \ref{optiharvest}, the
optimal equilibrium solution is found to be
$(x^*, y^*) = (0.4044, 1.4822)$.
Keeping for all time the system \eqref{rickermodel} at this equilibrium position
produces a maximum profit of 0.17085. This information
is presented in Figure \ref{optequibfig}.
\begin{figure}[ht]
\begin{center}
\includegraphics[trim= 2.5cm 7.5cm 3cm 7.5cm, clip, scale=0.5]{fig1} %OptEquib
\end{center}
\caption{Finding the optimal equilibrium point for maximizing
profit}\label{optequibfig}
\end{figure}
Starting from the initial conditions $(x_0, y_0) = (0,2)$, the
trajectory of the system is shown in Figure \ref{trajectoryfig}.
\begin{figure}[ht]
\begin{center}
\includegraphics[trim= 2.5cm 7.5cm 3cm 7.5cm, clip, scale=0.5]{fig2} % trajectory.eps
\end{center}
\caption{Trajectory of the system given the initial condition
$(0, 2)$}\label{trajectoryfig}
\end{figure}
The full phase portrait of this system is shown in Figure
\eqref{phasefig}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3} %optim
\end{center}
\caption{Phase portrait of the system}\label{phasefig}
\end{figure}
The switching times of the function and the value of the harvesting
rate become apparent in view of the phase portrait. Also, the
time where the singular solution is the optimal harvesting rate is
also visible.
\end{example}
\begin{example} \label{examp5.2} \rm
As a second example of the optimal harvesting strategy, consider the
trajectory followed by the system \eqref{rickermodel} from the
initial position $(x_0, y_0) = (2, 1.2)$. The trajectory is shown
in Figure \eqref{trajec2fig}. The benefits of the optimal
harvesting strategy are clearly demonstrated in this example.
\begin{figure}[ht]
\begin{center}
\includegraphics[trim= 2.5cm 7.5cm 3cm 7.5cm, clip, scale=0.5]{fig4} %trajectory2.eps
\end{center}
\caption{Trajectory of the system given the initial condition
$(2,1.2)$}\label{trajec2fig}
\end{figure}
\end{example}
\section{Conclusions}
It is difficult to solve the marine protected areas problems empirically,
because they require population data of large
spatial and temporal extent. One of the benefits of a theoretical modeling
of MPAs is insight into the type of informative fisheries data that should be
collected in order that the best design can be established.
Based on a classical Ricker's model, the deterministic nonlinear
sink-source model of MPAs is presented. Sufficient conditions for the
existence of the positive bounded steady-states are obtained. The
present value of net revenue is maximized subject to population
dynamics in the fishing zones and in the protected areas.
The impact of MPA can be felt in the critical value of the harvesting rate.
For most parameter choices, it is unreasonable to create an MPA big
enough so that condition (i) of Theorem \ref{thm3.1} is satisfied, and hence
a critical value of the harvesting rate $E$ must not be exceeded in
order to ensure that the fish species does not go extinct. The
analysis has shown that there is an optimal equilibrium solution,
and the best harvesting policy to attain this equilibrium position
is a bang-bang control effort. This was demonstrated numerically by
comparing the optimal harvesting rate against constant harvesting
rates. The convergence to the optimal equilibrium solution was
always faster, which implies greater profits under the optimal
harvesting strategy. Also included is a phase portrait, which shows
a more complete picture of the system being studied.
We have attempted to illustrate the effect of the optimal harvesting strategy
more clearly by picking parameters and initial conditions that better
demonstrate the benefit of the optimal harvesting policy.
Unfortunately, field data required to adequately populate
mathematical models is not classified and scattered in various
scientific papers and reports. Access to the specific fishery data
will help to develop models that can predict the features required
for optimizing the economic benefits of MPA. Contrariwise, one of
the benefits of a theoretical modeling of MPA is insight into the
type of data that should be collected in order that the best harvest
strategies can be established in the fishing zones.
There are many assumptions and simplifications in our model that are open
to objections.
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