Basic Reproduction Ratio for a Fishery Basic Reproduction Ratio for a Fishery ModelModel

in a Patchy Environmentin a Patchy Environment

A. Moussaoui* A. Moussaoui* , P. Auger, G. , P. Auger, G. Sallet Sallet

* Université de Tlemcen. Algerie* Université de Tlemcen. Algerie

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The complete modelThe complete model

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The matrix A is an irreducible matrix

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Aggregated model

Fast equilibria

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Aggregated Model

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Fishing Free Equilibrium (FFE) :

Stability analysis

There exists a extinction "equilibrium" given by

There exists « predator-free » equilibrium in the positive orthant given by

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The “extinction" equilibrium is always unstable

In a completely analogous way, as in epidemiology, we can define the basic reproduction ratio of the predator".

[van den Driessche and Watmough, 2002]

[Diekmann et al., 1990]

(FFE) is Locally asymptotically stable,

this equilibrium is unstable.

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Global stability of the “Fishery-Free”Equilibrium FFE

Theorem

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Sustainable Fishing Equilibria (SFE)

We consider the face

We have, for the relation

The equilibria has a biological meaning if it is contained in thenonnegative orthant, then we must have

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Eventually by reordering the coordinates, we can assume that

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We can have again sustainable fishing equilibria.

To summarize a SFE exists, if it exists a subset of subscripts J such that

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Stability analysis when

We recall that we have ordered the patches such that

DefinitionA flag in a finite dimensional vector space V is an increasing sequence of subspaces.

The standard flag associated with the canonical basis is the one where the i-th subspace is spanned by the first i vectors of the basis. Analogically we introduce the standard flag manifold of faces by defining in

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Then the flag is composed of the N faces

In each face of this flag a SFE can exist.

Proposition

If R0 > 1 then there exists a SFE in a face F of the standard flag, and noSFE can exist in the faces of the flag containing F.

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Theorem

When R0 > 1, the SFE is globally asymptotically stable on the domain which is the union of the positive orthant and the interior of the face of the SFE.

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Numerical example

1. Two patches

When N = 2 the reduced system is

Assuming the ordering of coordinates

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