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Periodic solution and oscillation in a competition model with diffusion and distributed delay effects. (English) Zbl 0862.35134
The paper deals with the coupled periodic-parabolic system with distributed delay \(((t,x)\in[0,\infty)\times\Omega)\), \[ {\partial u_1\over\partial t} (t,x)-A_1u_1(t,x)= u_1(t,x)\Biggl[a_1(t,x)-b_1(t,x)u_1(t,x)-\int^0_{-1} u_2(t+\tau(s),x)d_s\eta_1(t,s,x)\Biggr], \]
\[ {\partial u_2\over\partial t} (t,x)-A_2u_2(t,x)= u_2(t,x)\Biggl[a_2(t,x)- b_2(t,x)u_2(t,x)-\int^0_{-1}u_1(t+\tau(s),x)d_s\eta_2(t,s,x)\Biggr], \]
\[ B_1[u_1](t,x)=B_2[u_2](t,x)=0,\quad (t,x)\in [0,\infty)\times\partial\Omega, \] with initial conditions \[ u_1(s,x)=u_{1,0}(s,x),\quad u_2(s,x)=u_{2,0}(s,x),\quad (s,x)\in[-r,0]\times\Omega, \] where the operators \(A_1\), \(A_2\) are uniformly strongly elliptic operators, \(B[u]=\partial u/\partial\nu+\gamma(x)u\). The authors find conditions under which the trivial solution \((0,0)\) is globally asymptotically stable with respect to every nonnegative initial function \((u_{1,0},u_{2,0})\). Also, it is proved that this system has componentwise positive \(T\)-periodic solutions.
Reviewer: E.Minchev (Sofia)

MSC:
35R10 Partial functional-differential equations
92D25 Population dynamics (general)
35B10 Periodic solutions to PDEs
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