Basic Reproduction Ratio for a Fishery Model in a Patchy Environment

Abstract

We present a dynamical model of a multi-site fishery. The fish stock is located on a discrete set of fish habitats where it is catched by the fishing fleet. We assume that fishes remain on fishing habitats while the fishing vessels can move at a fast time scale to visit the different fishing sites. We use the existence of two time scales to reduce the dimension of the model : we build an aggregated model considering the habitat fish densities and the total fishing effort. We explore a regulation procedure, which imposes an average residence time in patches. Several equilibria exist, a Fishery Free Equilibria (FFEs) as well as a Sustainable Fishery Equilibria (SFEs). We show that the dynamics depends on a threshold which is similar to a basic reproduction ratio for the fishery. When the basic reproduction ratio is less or equal to 1, one of the FFEs is globally asymptotically stable (GAS), otherwise one of the SFEs is GAS.

Appendix

This code computes different value linked to the existence of SFE.

We suppose to have parameters of the “vectorialized” system (2) given in the Scilab command window: namely r, K, q, p, c.

In this code we denote by Cost the real c defined by $c = \sum\limits_{i} {c_{i} } v_{i} .,$ where v is the normalized Perron-Frobenius vector of A

This code, in a first step, orders the coordinates such that the inequalities (9) are satisfied. Then rr0 , which is ${c ({\mathcal{R}}_0 -1)}$, is computed. We have to check rr0  > 0 , a necessary condition for the existence of a SFE.

The second inequality, given by relation (18) is critSFE1 which must be positive. The value of critSFE1 , which corresponds to the criteria for the next face in the flag of faces.

function[rr0,Cost,critSFE1,critSFE2,v]=metafish(r,K,A,p,q,c)

// notations are those of the paper

// eigenvectors and eigenvalues of A

[eigenvect,mat]=spec(A)

// Searching the index of the 0 eigenvalue

[m,k]=min(abs(diag(mat)))

// Normalized Perron - Frobenius eigenvector

v=eigenvect(:,k)/sum(eigenvect(:,k))

// ordering the coordinates

rovera=r./(q.*v)

[s,k]=sort(rovera)// sorting in in decreasing order

k=k($:-1:1)// increasing order

P=eye(A)

P=P(:,k)// coordinate change matrix

rovera=rovera(k)

// reordering parameters coordinates

rnew=P’*r;Knew=P’*K;Anew=P’*A*P;pnew=P’*p;

qnew=P’*q;cnew=P’*c;vnew=P’*v;

// first criterion

rr0=sum(pnew.*qnew.*vnew.*Knew-cnew.*vnew)

Cost=sum(cnew.*vnew)

aoverr=(q.*v)./ r;aoverr=aoverr(k)

vect1minus=(ones(r(2:$))-rovera(1)*aoverr(2:$))

SUM=sum(pnew(2:$).*qnew(2:$).*vnew(2:$).*vect1minus

// second criterion

critSFE1=Cost-SUM

Knew(2:$).*vect1minus)

vect1minus2=(ones(r(3:$))-rovera(2)*aoverr(3:$))

SUM2=sum(pnew(3:$).*qnew(3:$).*vnew(3:$).*vect1minus2

// criterion for next face

critSFE2=Cost-SUM2

Knew(3:$).*vect1minus2)

endfunction