On the stock estimation for some ﬁshery systems

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On the stock estimation for some fishery systems

A. Guiro

^a,b

, A. Iggidr

^b∗

, D. Ngom

^b,c

, and H. Tour´ e

^a

a

LAME, URF/SEA

Universit´ e de Ouagadougou B.P: 7021 Ouagadougou, Burkina Faso.

b

INRIA-Lorraine and University Paul Verlaine-Metz LMAM-CNRS UMR 7122

ISGMP Bat. A, Ile du Saulcy 57045 Metz Cedex 01, France.

c

LANI, UFR/SAT

Universit´ e Gaston Berger. B.P. 234 Saint-Louis, S´ en´ egal.

Abstract

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called ”observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. More precisely we built a dynamical system that uses the available data (the total of caught fish) and which produces a dynamical estimation of the stock state.

Keywords: Fishery model, Stage-structured population models, Estimation, Har- vested Fish Population, Observers.

1 Introduction

The stock estimation is one of the most important problem in fishery science. One can quote J.A. Gulland [7]: A major emphasis in fishery science has been on the problems of estimating current and past level using catch levels and fishing effort data.

To make a policy decision about the exploitation of renewable ressources, it is nec- essary to take into account the state of the resource stocks. This implies the need of a good estimate of the available resource. Mathematical models are more and more used to describe the evolution of biological systems. Here, we consider two mathematical models for fishery resources. The first one is a ”stage structured” model [13] that describes the dynamics of a population divided in stage-classes (according to age, length or weight) and submitted to the fishing action. The second model is a ”global” model that describes the evolution of a fish population that can move between an area where it can be harvested and a reserve area where no fishing is allowed [2]. Both models are given by systems of differential equations of the form

∗Corresponding author: e-mail: [email protected], [email protected]

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˙

x=f(x, E), (1)

whereE is the fishing effort (it can be seen as a control or an input) andx(t) is the state of the system at time t. The state variable x(t) represents the density of the population or the number of individuals by stage.

2 Problem Statement

We consider first the following mathematical model describing the dynamical evolution of a fish population submitted to fishing. The population is structured by age class (see for instance [13, 12]):











X˙₀(t) = −α₀X₀(t) +

2

X

i=1

f_il_iX_i(t)−

2

X

i=1

p_iX_i(t)X₀(t)−p₀X₀²(t) X˙₁(t) = αX₀(t)−(α₁+q₁E)X₁(t)

X˙₂(t) = αX₁(t)−(α₂+q₂E)X₂(t).

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X_i: the number of fish in the stage i.

α : linear aging coefficient (in time⁻¹)

m_i : natural mortality rate (in time⁻¹)

α_i =m_i+α (in time⁻¹)

p₀: juvenile competition parameter (in time⁻¹.number⁻¹)

f_i: fecundity rate of class (no dimension)

l_i: reproduction efficiency of class i (in time⁻¹) p_i: predation rate of class i on class 0 (time⁻¹.num⁻¹) qi: capturability coefficient of class i (in unit effort⁻¹) E: instantaneous fishing effort. (in unit effort×time⁻¹) In the second model, we consider the dynamics of a fish population moving between two zones (see [2]). The first zone is a free fishing area, and the second zone is a reserve area where no fishing is allowed.











X˙1 =r1X1

1− X₁ K₁

−m12X1+m21X2−qEX1

X˙2 =r2X2

1− X₂ K₂

+m12X1−m21X2.

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x₁(t) is the biomass density at time t of the fish population in the free fishing area and x₂(t) is the biomass density at a time t of the fish population in the reserved areas. For (i, j)∈ {1,2}², we denote bymij the migration rate from the zoneito the zonej. In the free fishing area, the total fishing effort is denoted byE. The growth of the two sub-population in each zone follows logistic model. r₁ and r₂ represent the intrinsic growth of each fish sub-population, respectively, K1 and K2 are the carrying capacities of fish species in the unreserved and reserved areas, respectively;

qis the catchability coefficient of fish species in the unreserved area. The parameters r1, r2,q,m12, m21, K1 and K2 are positives constants.

To the system (3) we associate the capture (i.e. the output)y=qEx₁ (the total of caught fish in the unreserved area).

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3 Observability and Observer for fishery model

3.1 What is an observateur?

Let an auxiliary dynamical system which uses the information y(t) provided by the system (1). This dynamical system is of the form

˙ˆ

x=g(ˆx, E, y), (4)

where the function g has to be determined in such a way that the solutions of (1) and (4) satisfy x(t)− x(t)ˆ → 0 as t → +∞ regardless of the respective initial conditions of (1) and (4). A dynamical system (4) satisfying this conditions is called an ”observer” for system (1).

3.2 Observability

We use the notation x(x₀, u(.), t) to denote the solution of the differential equa- tion (??) corresponding to the admissible controlu(.) and with initial condition x₀, and The corresponding output is denotedy(x₀, u(.), t) = h(x(x₀, u(.), t), u(t)).

The system (1) is said to be observable if for any pair of different initial states (x₀, x₁)x₀ 6=x₁, there exists an admissible control u(.) such that the outputs corresponding to those initial conditions are not identically equals; that is, there exists a time t≥0 such that : y(x₀, u(.), t)6=y(x₁, u(.), t)

The system (1) is said to be uniformly input observable if for any input u(.) and for any (x₀, x₁), x₀ 6= x₁, there exists a time t ≥ 0 such that : y(x₀, u(.), t) 6=

y(x₁, u(.), t)

The observer is such way thaty(x₀, u(.), t) =y(x₁, u(.), t) ∀t≥0 if x₀ =x₁

3.3 High gain observer for the fishery system

LetY(t) =h(X(t)) =q₂EX₂(t),

f(X) =







−α₀X₀(t) +

2

X

i=1

f_il_iX_i(t)−

2

X

i=1

p_iX_i(t)X₀(t)−p₀X₀²(t) αX₀(t)−α₁X₁(t)

αX₁(t)−(α₂ +q₂E)X₂(t)







and Φ :R³ →R³

defined by :

Φ(X) =







h(X) L_f(h(X)) L²_f(h(X))







=







q₂EX₂

αq₂EX₁−q₂E(α₁+q₂E)X₂ α²q₂EX₀+q₂E(α₂+q₂E)²)X₂

−(αα₁q₂E+αq₂E(α₂+q₂E))X₁







L_f denote the Lie derivative operator with respect to the vector field f, and his the output function. Φ is a diffeomorphism of the state space R³ to its image Φ(R³).

and then the system (2) is uniformly observable (see [3]). The function f is smooth on the compact set D= [a₀, b₀]×[a₁, b₁]×[a₂, b₂]. Hence, it is globally Lipschitz on D. Therefore it can be extended by ˜f, a Lipschitz function on R³

S(θ) is the solution of 0 = −θS(θ)−A^tS(θ)−S(θ)A^t+C^tC. where A is an anti shift matrix and C= [1,0,· · · ,0]

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For θ large enough the Observer for system (2) can be expressed by the following formula:

X˙ˆ = ˜f( ˆX) + dΦ

dx −1

X= ˆX

S(θ)⁻¹C^t(y−h( ˆX)).

Explicitly, the restriction of the estimation system (observer) to the domain D is the following system











X˙ˆ₀ = −α₀Xˆ₀+

2

X

i=1

f_il_iXˆ_i−

2

X

i=1

p_iXˆ_iXˆ₀−p₀Xˆ₀²+ 3θα₁(q₂E+α₂) + 3θ²(q₂E+α₁+α₂) +θ³

α² ( y

q₂E −Xˆ₂) X˙ˆ₁ = αXˆ₀−α₁Xˆ₁+ 3θ(q₂E+α₂) + 3θ²

α ( y

q2E −Xˆ₂) X˙ˆ₂ = αXˆ₁−(α₂+q₂E) ˆX₂+ 3θ( y

q₂E −Xˆ₂)

This observer is particularly simple since it is only a copy of (2), together with a corrective term depending on θ. The gain S(θ)⁻¹C^t is fixed and increase with θ, thus the name ”High Gain observer”.

3.4 conclusion

We have tried to combine modern Automatic, Computer science and Mathematics to build state estimation for systems that model the dynamics of fish populations submitted to a fishing action. Indeed one of the important problems in fishery sciences is to estimate the state of the resource using the available data, in order to produce scientific opinions that can be helpful for developing management policies that need to have a good estimate of the available resource.

References

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