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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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