Can Fishing Pressure Invert the Outcome of Interspecific Competition? The Case of the Thiof and of the Octopus Along the Senegalese Coast

Abstract

We present a mathematical model of two competing marine species that are harvested. We consider three models according to different levels of complexity, without and with species refuge and density-independent and density-dependent species movement between fishing area and refuge. We particularly study the effects of the fishing pressure on the outcome of the competition. We focus on conditions that allow an inferior competitor to invade as a result of fishing pressure. The model is discussed in relationship to the case of the thiof and the octopus along the Atlantic West African coast. At the origin, the thiof was abundant and the octopus scarce in that region. Since, the fishing pressure has strongly increased in some fishing areas leading to the depletion of the thiof and the invasion of its competitor, the octopus.

Appendix: Equilibria and Local Stability Analysis of Aggregated Model

1.1 Rewrite Model

To simplify, we denote by $f(n_1,n_2)$ the function $A - Bn_1 - \dfrac{C}{H(n_1)}n_2$ and by $g(n_1,n_2)$ the function $M - Nn_1 - \dfrac{P}{H(n_1)}n_2$. The model is therefore written as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dn_{1}}{dt} = n_1f(n_1,n_2)\\ \dfrac{dn_{2}}{dt} = \dfrac{n_2}{H(n_1)}g(n_1,n_2) \end{array}\right. } \end{aligned}$$

(3.1)

1.2 Jacobian Matrix

$$\begin{aligned} J(n_{1}, n_{2}) &= \begin{pmatrix} f + n_1f^{\prime}_{n_1} & n_1f^{\prime}_{n_2}\\ \dfrac{n_2}{H(n_1)}g^{\prime}_{n_1} - \dfrac{n_2g\alpha v_1^*}{H^2(n_1)} & \dfrac{g}{H(n_1)} + \dfrac{n_2}{H(n_1)}g^{\prime}_{n_2} \end{pmatrix} \end{aligned}$$

(3.2)

1.3 Equilibria and Stability

• At (0, 0):

  $$\begin{aligned} J(0,0)& = \begin{pmatrix} A & 0\\ 0 & \dfrac{M}{\alpha _0 + m} \end{pmatrix} \end{aligned}$$

  The matrix has two eigenvalues A and $M/(\alpha _0 +m)$. Therefore, (0, 0) stable if and only if $M,A < 0$.

• At $\left( \dfrac{A}{B},0\right)$ (the condition for its existence is that $A>0$)

  $$\begin{aligned} J\left( \dfrac{A}{B},0\right) &= \begin{pmatrix} -A & -\dfrac{AC}{BH(\dfrac{A}{B})}\\ \\ 0 & \dfrac{M - N\dfrac{A}{B}}{\alpha _0 + m} \end{pmatrix} \end{aligned}$$

  The matrix has two eigenvalues: $\lambda _1=-A <0$ and $\lambda _2= (MB - NA)/(B(\alpha _0 + m))$. Thus, (A / B, 0) is stable if and only if $MB - NA< 0$.

• At $\left( 0,MH(0)/P\right)$ (the condition for its existence is that $M>0$)

  $$\begin{aligned} J(0, MH(0)/P) &= \begin{pmatrix} \dfrac{PA-MC}{P} & 0\\ \\ \dfrac{MH(0)/P}{H(0)}g^{\prime}_{n_1} - \dfrac{MH(0)/Pg\alpha v_1^*}{H^2(0)} & -\dfrac{M}{H(0)} \end{pmatrix} \end{aligned}$$

  The matrix has two eigenvalues: $\lambda _1=-M/H(0) <0$ and $\lambda _2= (PA - MC)/P$. Thus, (0, MH(0) / P) is stable if and only if $PA - MC< 0$.

• At $(n_1^*,n_2^*)$ where $n_1^* = (PA - MC)/(PB - NC)$, $n_2^* = (A - Bn_1^*)H(n_1^*)/C$ (the condition for its existence is that $0< (PA - MC)/(PB - NC) < A/B$). The Jacobian matrix reads as follows:

  $$\begin{aligned} J(n_1^*,n_2^*)& = \begin{pmatrix} n_1^*f^{\prime}_{n_1^*} & n_1^*f^{\prime}_{n_2^*}\\ \dfrac{n_2^*g^{\prime}_{n_1^*}}{H(n_1^*)} & \dfrac{n_2^*g^{\prime}_{n_2^*}}{H(n_1^*)} \end{pmatrix} \end{aligned}$$

A straightforward calculation leads to the following two possibilities (see Fig. 6 for more information). The first one, corresponding to conditions $PA-MC>0$ and $MB-NA>0$, is the case where $(n_1^*,n_2^*)$ is a stable point. The second one, corresponding to conditions $PA-MC<0$ and $MB-NA<0$, is the case where $(n_1^*,n_2^*)$ is a saddle point.

Fig. 6

Two cases where there exists a strictly positive equilibrium. a is related to the case where it is stable, (b) is related to the case where it is saddle