Basic Reproduction Ratio for a Fishery Model in a Patchy Environment
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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
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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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