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Permanence and extinction for dispersal population systems. (English) Zbl 1073.34052

The system of differential equations x ˙=f(t,x), x n , is said to be permanent if there exists a compact set Kint + n such that all solutions starting in int + n ultimately enter and remain in K.

The authors study the following predator-pray model in a patchy environment

x ˙ 1 =x 1 [b 1 (t)-a 1 (t)x 1 -yϕ(t,x 1 )]+ j=1 n (D 1j (t)x j -D j1 (t)x 1 ),x ˙ i =x i [b i (t)-a i (t)x i ]+ j=1 n (D ij (t)x j -D ji (t)x i ),i=2,...,n,y ˙=y[-d(t)+e(t)x 1 ϕ(t,x 1 )-f(t)y],

where x i denotes the species x in patch i; d(t)>0, e(t)>0, b i (t)>0, a i (t)>0, D ij (t)0 are continuous ω-periodic functions; D ij (t) is the dispersal coefficient of the species from patch j to patch i, D ii (t)0; the predator functional response x 1 ϕ(t,x 1 ) is bounded as x 1 , ϕ(t,x 1 )0, ϕ(t,x 1 )/x 1 0 and (x 1 ϕ(t,x 1 ))/x 1 0. Necessary and sufficient conditions for the permanence of the above system are presented. Sufficient conditions for the permanence of the single-species system in the absence of a predator (y=0) is also obtained. The approach is based on the well-known properties of the periodic logistic model z ˙=z(b(t)-a(t)z).

The biological interpretion of the main results is given.

34D05Asymptotic stability of ODE
34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)