A Mathematical Model of a Fishery with Variable Market Price: Sustainable Fishery/Over-exploitation

Abstract

We present a mathematical bioeconomic model of a fishery with a variable price. The model describes the time evolution of the resource, the fishing effort and the price which is assumed to vary with respect to supply and demand. The supply is the instantaneous catch while the demand function is assumed to be a monotone decreasing function of price. We show that a generic market price equation (MPE) can be derived and has to be solved to calculate non trivial equilibria of the model. This MPE can have 1, 2 or 3 equilibria. We perform the analysis of local and global stability of equilibria. The MPE is extended to two cases: an age-structured fish population and a fishery with storage of the resource.

Appendices

Appendix 1: Existence Domains for Non-trivial Equilibria (Positive Equilibria)

We determine existence domains for positive equilibria of system (1). Non-trivial equilibria correspond to the solutions of the equation

$$\begin{aligned} D(p^*)=qn^{*}E^{*} \end{aligned}$$

(8)

which can be rewritten

$$\begin{aligned} A -\alpha p^* = f(p^{*}) \end{aligned}$$

(9)

where $f(p)=\frac{rc}{pq}\left( 1- \frac{c}{pqk}\right)$. Solutions correspond to the roots of third degree polynomial

$$\begin{aligned} P_\alpha (p) = p^{3}(\alpha q^{2}k) - p^{2}(Aq^{2}k) + p(rcqk) - rc^{2} \end{aligned}$$

(10)

Because two consecutive coefficients of $P_\alpha$ have opposite signs, real roots are all positive. An equilibrium $\xi ^*$ is then positive if and only if $p^*qk > c$, because $p^*qk < c$ implies $E^*<0$.

Lemma 1

There is a positive equilibrium $\xi ^*$ such that $p^*qk<c$ if and only if $\alpha > qkA/c$. If $p^*$ exists, it is the unique real root of (10).

Proof

$D$ is decreasing, so for $p<c/qk$, $D(p)>D(c/qk)$. If $\alpha \le qkA/c$, $D(c/qk)>0$. Then for $p<c/qk$, $D(p)>0$ and $f(p)<0$. There is no root $p^*$ such that $p^*qk<c$. On the other hand, if $\alpha > qkA/c$, then $D(c/qk)<0$. We have the following properties:

• $D(0)>0>\lim \limits _{p\rightarrow 0} f(p)$;

• if $p>c/qk$, $D(p)<0<f(p)$;

• $D$ is monotonously decreasing, while $f$ is monotonously increasing on $(0,c/qk]$.

We deduce that there exists a unique $p^*$ which verifies $D(p^*)=f(p^*)$. It also verifies $p^* <c/qk$.$\square$

As a consequence, when $\alpha > qkA/c$, there is no positive equilibrium.

We now determine the existence domains of real roots of polynomial $P_\alpha$:

Lemma 2

If $kr<3A$, there is always one real root. If $kr>3A$, there are three domains:

• $\alpha < \alpha ^-$: there is one and only one real root;

• $\alpha ^- < \alpha < \alpha ^+$ : there are three real roots;

• $\alpha ^+ < \alpha$ : there is one and only one real root;

$\alpha ^-$ and $\alpha ^+$ correspond to values for which two real roots merge and vanish, and verify

$$\begin{aligned} \alpha ^-&= q\frac{-2r^2k^2+9rAk-2\sqrt{27kr\left( \frac{kr}{3}-A\right) ^3}}{27rc}\end{aligned}$$

(11)

$$\begin{aligned} \alpha ^+&= q\frac{-2r^2k^2+9rAk+2\sqrt{27kr\left( \frac{kr}{3}-A\right) ^3}}{27rc} \end{aligned}$$

(12)

Proof

The discriminant $\Delta _\alpha$ of polynomial $P_\alpha$ (10) is given by

$$\begin{aligned} \Delta _\alpha = -\frac{R\left( P_\alpha ,P_\alpha '\right) }{\alpha q^2 k} \end{aligned}$$

(13)

where the resultant $R\left( P_\alpha ,P_\alpha '\right)$ of polynomials $P_\alpha$ and its derivated polynomial reads:

$$\begin{aligned} R\left( P_\alpha ,P_\alpha '\right) = \frac{q^6\alpha rc^2}{k^2}g(\alpha ) \end{aligned}$$

(14)

where $g(\alpha )=27\alpha ^2rc^2-18\alpha qrcAk-q^2A^2k^2r+4q^2A^3k+4r^2ck^2q\alpha$. $g$ is a degree 2 polynomial with two roots, $\alpha ^-$ and $\alpha ^+$.

• If $kr<3A$, $g$ has no real roots. We deduce that $\forall \alpha >0$, $g(\alpha )>0$, and $\Delta _\alpha <0$. $P_\alpha$ has exactly one real root.

• If $kr>3A$, for $\alpha <\alpha ^-$ or $\alpha >\alpha ^+$, $g(\alpha )>0$. $P_\alpha$ has exactly one real root. for $\alpha ^-<\alpha <\alpha ^+$, $g(\alpha )<0$. $P_\alpha$ has exactly three real root. For $\alpha = \alpha ^-$ or $\alpha = \alpha ^+$, $\Delta _\alpha = 0$, $P_\alpha$ has real roots with order of multiplicity larger than 1.

$\square$

Appendix 2: Local Stability Positive Equilibria $\xi ^*$

We now determine the stability of positive equilibria of system (1). Let us denote

$$\begin{aligned} p^+= {\frac{kr+\sqrt{rk(rk-3A)^3}}{Akq}c}\; {\text {and}} \; p^-= {\frac{kr-\sqrt{rk(rk-3A)^3}}{Akq}c}. \end{aligned}$$

Lemma 3

If $A>rk/3$, the positive equilibrium $\xi ^*$ is locally asymptotically stable. If $A<rk/3$ , the positive equilibrium $\xi ^*$ is locally asymptotically stable if and only if $p^*<p^-$ or $p^*>p^+$.

Proof

The jacobian matrix of system corresponding to positive equilibria reads:

$$\begin{aligned} J^* = \left[ \begin{array}{ccc} -\frac{rn^*}{k} &{} - qn^* &{} 0\\ p^*qE^* &{} 0 &{} qn^*E^*\\ -\varphi qE^* &{} -\varphi qn^* &{} -\varphi \alpha \end{array} \right] \end{aligned}$$

(15)

The characteristic polynomial is:

$$\begin{aligned} \chi (\lambda )&= det(J^* - \lambda I_3)= \left| \begin{array}{ccc} -\frac{rn^*}{k}-\lambda &{} - qn^* &{} 0\\ p^*qE^* &{} - \lambda &{} qn^*E^*\\ -\varphi qE^* &{} -\varphi qn^* &{} -\varphi \alpha -\lambda \end{array} \right| \\&= \lambda ^3 - \lambda ^2\left( \varphi \alpha + \frac{r}{k}n^*\right) -\lambda \left( \varphi \frac{r\alpha }{k}n^*+\varphi q^2 n^{*2}E^*+p^*q^2n^*E^*\right) \\&- \varphi q^2 n^* E^* \left( \frac{r}{k} n^{*2} +\alpha p^* - q n^* E^*\right) \end{aligned}$$

We now determine the local stability by using Routh-Hurwitz criterion. Let us denote

$$\begin{aligned} \left\{ \begin{array}{lll} a_{3} &=&{} 1\\ a_{2} &=& \varphi \alpha + \frac{r}{k}n^{*}\\ a_{1} &=& \frac{r\alpha }{k}n^* + \varphi q^2n^{*2}E^* + p^*q^2n^*E^*\\ a_{0} &=& \varphi q^2n^*E^*(\frac{r}{k}n^{*2} + \alpha p^* - qn^*E^*) \end{array} \right. \end{aligned}$$

(16)

Equilibrium $\xi ^*$ is stable if and only if $(i)$ $a_i > 0$ for $i\in \{0,\dots ,3\}$ and $(ii)$ $a_2a_1 > a_3a_0$. If $\xi ^*$ is positive, conditions $a_3 > 0$, $a_2 > 0$, $a_1 > 0$ are always verified. We now determine if condition $(ii)$ is satisfied:

$$\begin{aligned} a_2a_1 - a_3a_0&= \left( \varphi \alpha + \frac{r}{k}n^{*}\right) \left( \frac{r\alpha }{k}n^* + q^2n^*E^*\left( \varphi n^* +p^*\right) \right) \\&- \varphi q^2n^*E^*\left( \frac{r}{k}n^{*2} + \alpha p^* - qn^*E^*\right) \\&= \left( \varphi \alpha + \frac{r}{k}n^{*}\right) \frac{r\alpha }{k}n^* +q^2n^*E^*\left( \varphi ^2\alpha n^*+\varphi \alpha p^* + \varphi \frac{r}{k}n^{*2}+\frac{r}{k}n^*p\right) \\&- q^2n^*E^*\left( \varphi \frac{r}{k}n^{*2} + \varphi \alpha p^* - \varphi qn^*E^*\right) \\&= \left( \varphi \alpha + \frac{r}{k}n^{*}\right) \frac{r\alpha }{k}n^* +q^2n^*E^*\left( \varphi ^2\alpha n^*+\frac{r}{k}n^*p^*+\varphi q n^* E^*\right) \\&> 0 \end{aligned}$$

If $\xi ^*$ is positive, condition $(ii)$ is always verified. We now determine the sign of $a_0$. By replacing $n^*$ and $E^*$ by their values, we obtain

$$\begin{aligned} a_{0} = \frac{\varphi rc( p^*qk - c)(\alpha q^2k p^{*3}-rcqkp^*+2rc^2)}{p^{*3}q^3k^2} \end{aligned}$$

(17)

Since $p^*$ is a root of polynomial (10), we have

$$\begin{aligned} a_{0} = \frac{\varphi rc( p^*qk - c)\left( (Aq^2k)p^{*2}-2(rcqk)p^*+3rc^2\right) }{p^{*3}q^3k^2} \end{aligned}$$

(18)

If $A>rk/3$, polynomial $(Aq^2k)p^{*2}-2(rcqk)p^*+3rc^2$ has no real roots and is always positive. If $A<rk/3$, polynomial $(Aq^2k)p^{*2}-2(rcqk)p^*+3rc^2$ has two roots: $p^-$ and $p^+$. Since $\xi ^*$ is positive, $p^*qk>c$. We deduce that $a_0>0$ if and only if $p<p^-$ or $p>p^+$. $\square$

Lemma 4

$p^-$ (resp. $p^+$ ) is the double root of polynomial $P_{\alpha ^+}$ (resp. $P_{\alpha ^-}$ ).

Proof

From Cardano’s formula, we find that double root of polynomial $P_{\alpha ^+}$ reads:

$$\begin{aligned} \frac{3c\left( r^2k^2-3rAk-\sqrt{rk(rk-3A)^3}\right) }{qk\left( 9A^2 -9rAk+2r^2k^2-2\sqrt{rk(rk-A)^3}\right) } \end{aligned}$$

(19)

By simplifying the expression, we obtain that the double root is equal to $p^-$. The same results holds for $P_{\alpha ^-}$ and $p ^+$. $\square$

Appendix 3: Bounded Attractor

We now show that there exists a bounded set in which every trajectories (for system (1)) with a positive initial condition end. It is clear that the set $\varOmega _0$ of the phase space $(n,E,p)$ defined by

$$\begin{aligned} \varOmega _0 = \{(n,E,p)\;|\; 0\le n \le k, p \le A/\alpha \} \end{aligned}$$

(20)

is a forward invariant set for system (1). Furthermore, any trajectory with a positive initial condition has its $\omega$-limit in $\varOmega _0$.

Let us consider the candidate Lyapunov function defined for $n\in \mathbb {R}_+^*$, $E\in \mathbb {R}_+^*$, $p\in \mathbb {R}$:

$$\begin{aligned} V(n,E,p) = pqn+qE-c \ln n-r\left( 1-\frac{n}{k}-\frac{\alpha \varphi }{r}\right) \ln E \end{aligned}$$

(21)

Along the trajectories of system (1), we have

$$\begin{aligned} \dot{V}(n,E,p) = -\alpha \varphi c + n \left( \varphi q A + \frac{r}{k}\left( r\left( 1-\frac{n}{k}\right) -qE \right) \ln E -\varphi q^2 nE \right) \end{aligned}$$

(22)

Note that $\dot{V}(n,E,p)$ does not depend on $p$.

Lemma 5

The set $\varOmega = \{(n,E,p)\;|\; (n,E,p)\in \varOmega _0, \dot{V}(n,E,p)\ge 0\}$ is included in the set $\varSigma \times (-\infty ,A/\alpha ]$, where $\varSigma$ is a compact subset of $(0,k]\times \mathbb {R_+^*} \cup \{(k,0)\}$.

Proof

For $E>1$, $\dot{V}(n,E,p) < n \left( \varphi q A + \frac{r}{k}\left( r-qE\right) \ln E\right)$. The right term tends toward $-\infty$ when $E$ tends toward $+\infty$. We denote

$$\begin{aligned} \varSigma _0=\{(n,E)\;|\;0\le n \le k, E_{min}\le E\} \end{aligned}$$

(23)

where $E_{min}$ is such that $\left( \varphi q A + \frac{r}{k}\left( r-qE_{min}\right) \ln E_{min}\right) <0$. We deduce that

$$\begin{aligned} \forall (n,E) \in \varSigma _0 \Longrightarrow \dot{V}(n,E,p) < 0 \end{aligned}$$

(24)

For $E<1$, $\dot{V}(n,E,p) < n\left( \varphi q A + \frac{r}{k}\left( r\left( 1-\frac{n}{k}\right) -qE \right) \ln E \right)$. We have

$$\begin{aligned} n < f(E) \Longrightarrow \dot{V}(n,E,p) < 0 \end{aligned}$$

(25)

where $f(E) = \frac{k^2\varphi q A}{r^2\ln E} + \frac{k(r-qE)}{r}$. It is easy to see that $f$ is defined on $(0,1)$ and monotonously decreasing, with $\lim \limits _{E\rightarrow 0}f(E)= k$ and $\lim \limits _{E\rightarrow 1}f(E)=-\infty$. We denote $\varSigma _1=\{(n,E)\;|\;0\le n \le k, 0 \le E < f^{-1}(n)\}$. Equation (25) now reads

$$\begin{aligned} (n,E)\in \varSigma _1 \Longrightarrow \dot{V}(n,E,p) < 0 \end{aligned}$$

(26)

Let us consider $E'_{min}=f^{-1}(k/2)$. On the compact set $[0,k/2]\times [E'_{min},E_{min}]$, term $\varphi q A + \frac{r}{k}\left( r\left( 1-\frac{n}{k}\right) -qE \right) \ln E-\varphi q^2 nE$ has a maximum $M$.

Let us denote $\varSigma _2=[0,\alpha \varphi c/M)\times [E'_{min},E_{min}]$. From Eq. (22), we deduce that

$$\begin{aligned} (n,E)\in \varSigma _2 \Longrightarrow \dot{V}(n,E,p) < 0 \end{aligned}$$

(27)

We now define $\varSigma = ([0,k]\times \mathbb {R^+})\backslash (\varSigma _1 \cup \varSigma _2 \cup \varSigma _3)$. $\varSigma$, $\varSigma _0$, $\varSigma _1$ and $\varSigma _2$ are represented in Fig. 5. $\varSigma$ is a compact subset of $(0,k]\times \mathbb {R_+^*} \cup \{(k,0)\}$. Furthermore, $\dot{V}(n,E,p) \ge 0 \Rightarrow (n,E)\in \varSigma$. We deduce that $\varOmega$ is included in $\varSigma \times (-\infty ,A/\alpha ]$. $\square$

Fig. 5

$\dot{V}(n,E,p)$ (black wireframe surface). The white part of the plan ${\dot{V}}=0$ represents the compact set which encompasses the set $\{(n,E)\;|\;\dot{V}>0\}$. Sets $\varSigma$ and $\varSigma _i$ for $i= 1\dots 3$ are represented as grey areas. Parameters are $r=0.9$, $k=3$, $q=0.1$, $c=2$, $A=2$, $\alpha =0.1$ and $\varphi = 0.1$

Lemma 6

In set $\varOmega$, $V$ admits a maximum $V_0$.

Proof

Since $\varSigma$ is a compact set, $V(n,E,A/\alpha )$ admits a maximum $V_0$ on $\varSigma$. From Eq. (21), we deduce that $\forall (n,E,p) \in \varOmega$, $V(n,E,p)\le V(n,E,A/\alpha )$, hence the result. $\square$

Let be $V'_0 \ge V_0$, $\varOmega _{\infty } = \{(n,E,p)\;|\;V(n,E,p)\le V'_0\}$, and $\varOmega _\infty ^+ = \{(n,E,p)\;|\;(n,E,p)\in \varOmega _\infty , p\ge 0\}$. We now consider the flow $\phi$ associated to system (1).

Lemma 7

$\varOmega _\infty$ is forward invariant, and for all $(n,E,p) \in \varOmega _0$ , there exists $t\ge 0$ such that $\phi _t(n,E,p) \in \varOmega _\infty ^+$.

Proof

For $(n,E,p)\in \varOmega _0 \backslash \varOmega _\infty$, $V(n,E,p) > V_0$ and $\dot{V}(n,E,p) < 0$, which means that $\varOmega _\infty$ is forward invariant. Furthermore, it is clear that $\lim \limits _{t\rightarrow +\infty } V(\phi _t(n,E,p))\le V_0$, which means that there exists $t_0$ such that $\phi _{t_0}(n,E,p)\in \varOmega _\infty$.

We now show that there exists $t'_0$ such that $\phi _{t'_0}(n,E,p)\in \varOmega _\infty ^+$. From system (1), we deduce that if $p < 0$, $\dot{E} \le -cE$, and if $p<0$ and $E < A/(2kq)$, $\dot{p}>A/2$. Let us consider $(n,E,p) \in \varOmega _\infty$, and the solution $(n(t),E(t),p(t))=\phi _t(n,E,p)$. We suppose that $\forall t>0$, $p(t)<0$. There exists $t_1$ such that $\forall t>t_1$, $E(t_1)<A/(2kq)$. Then for $t>t_1$, $\dot{p}(t)> A/2$, and $\lim \limits _{t\rightarrow +\infty }p(t) >0$, hence the contradiction. We deduce that there exists $t'_0\ge 0$ such that $p\ge 0$. Since $(n,E,p)\in \varOmega _\infty$ and $\varOmega _\infty$ is forward invariant, $\phi _{t'_0}(n,E,p) \in \varOmega _\infty ^+$, which ends the proof. $\square$

We can now prove Theorem 3.

Theorem 3

There exists a bounded set $\varOmega _\infty ^{++}$ included in $\varOmega _0$ which is forward invariant and such that $\forall (n,E,p), \left\{ t \ge 0 \;|\; \{\phi _t(n,E,p)\}\cap \varOmega _\infty ^{++} \ne \emptyset \right\}$.

Proof

It is easy to deduce from Eq. (21) that $\varOmega _\infty ^{+}$ is a compact set. This is illustrated on Fig. 6.

Let us denote $E_M = \max \{E\;|\;(n,E,0) \in \varOmega _\infty \}$.

For $(n,E,p) \in \varOmega _\infty ^{+}$, we consider the solution $(n(t),E(t),p(t))=\phi _t(n,E,p)$. Let us define $t_m = \inf \{t> 0\;|\;p(t) < 0\}$ and $t_M = \inf \{t> t_m\;|\;p(t) > 0\}$ ($t_m$ and $t_M$ can be equal to $+\infty$). If $t_m = +\infty$, we denote $p_{inf}(n,E,p)=0$, else we denote $p_{inf}(n,E) = \inf \limits _{t\in (t_m,t_M)}p(t)$. This represents the minimal value of $p$ that is reachable when crossing the plan $p=0$ before returning to $\varOmega _\infty ^{+}$. If $t_m < +\infty$, then $\dot{p}(t_m) \le 0$. For all $t\in (t_m,t_M)$, $\dot{E}(t)\le -cE(t)$, and so $E(t) \le E(t_m) e^{-ct}\le E_M e^{-ct}$. Then we have $\dot{p}(t) \ge A - qkE(t) \ge A - qkE_Me^{-ct}$. We deduce that $p(t)$ reaches its minimum before $t'=\ln \left( A/qkE_M\right) /c$. If we denote $p_m =\int _0^{t'} \left( A - qkE_Me^{-ct}\right) dt$, then $p_{inf}(n,E,p)\ge p_m$.

We now define $\varOmega _\infty ^{++}= \{(n,E,p)\in \varOmega _\infty | p \ge p_m\}$. It is clear that $\varOmega _\infty ^{++}$ is bounded, and from the previous demonstration, we deduce that it is forward invariant. since $\varOmega _\infty ^+$ is included in $\varOmega _\infty ^{++}$, we deduce from Lemma (7) that $\forall (n,E,p), \left\{ t \ge 0 \;|\; \{\phi _t(n,E,p)\}\cap \varOmega _\infty ^{++} \ne \emptyset \right\}$. $\square$

Fig. 6

On the surface shown, $V(n,E,p)$ is constant. The space under the surface corresponds to $\varOmega _\infty$. The compact set $\varOmega _\infty ^{+}$ corresponds to the space under the surface and over the plan $p=0$. Parameters are $r=0.9$, $k=3$, $q=0.1$, $c=2$, $A=2$, $\alpha =0.1$ and $\varphi = 0.1$