Tamtam Proceedings - lamsin

Bioeconomical fishing model 543where τ denotes the fast time, t the slow time with t = ετ. The fast model (3) is conservative.At the fast time scale, the total fish density n(t) = n 1 (t) + n 2 (t) + n 3 (t) remainsconstant. Similarly, setting ε = 0 into equations (2), we obtain:⎧dE 1⎪⎨ dτ = ( ˜mE 2 − mE 1 )dE 2dτ = (mE 1 + ˆmE 3 − ( ¯m + ˜m)E 2 )(4)⎪⎩ dE 3dτ = ( ¯mE 2 − ˆmE 3 )The fast model (4) is also conservative. At the fast time scale, the total fishing effortE(t) = E 1 (t) + E 2 (t) + E 3 (t) remains constant. A simple calculation shows that thereexists a single positive and stable equilibrium for any positive initial condition. This fastequilibrium for fish and for fishing efforts is given by the following expressions:⎧⎨⎩n ∗ 1 = ν ∗ 1 n, E ∗ 1 = µ ∗ 1En ∗ 2 = ν ∗ 2 n, E ∗ 2 = µ ∗ 2En ∗ 3 = ν ∗ 3 n, E ∗ 3 = µ ∗ 3E(5)where the fish proportions ν ∗ iin each zone i are constant and given by:ν ∗ 1 =11 + k˜k + k¯k˜kˆk, ν ∗ 2 =k˜k1 + k˜k + k¯k˜kˆkand ν ∗ 3 =k¯k˜kˆk1 + k˜k + k¯k .˜kˆkand the fish proportions µ ∗ i in each zone i are constant and given by:µ ∗ 1 =1 +1m ¯m+m˜m ˜m ˆm, µ ∗ 2 =1 +m˜mm ¯m+m˜m ˜m ˆmand µ ∗ 3 =1 +m ¯m˜m ˆmm ¯m+m˜m ˜m ˆmIt can easily be shown that the fast equilibrium for fish and boat is globally asymptoticallystable in the positive quadrant for any positive initial condition.2.2. The Aggregated modelThe next step consists in the substitution of the fast equilibria (5) into the equations ofthe complete model given by (1) and (2), and addition of fish and boat equations leadingto a reduced model, called the "aggregated model", which reads as follows:⎧⎪⎨dn= rn(n − n)(n − ¯n) − ãnEdtdE(6)⎪⎩dt = (˜bn − ˜c)ETAMTAM –Tunis– 2005