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Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays. (English) Zbl 1031.35071
This paper is concerned with 3-species time-delayed Lotka-Volterra reaction-diffusion systems and their corresponding ordinary differential systems without diffusion; there are considered a two-prey one-predator model, a one-prey two-predator model and a three-species food-chain model. The model with two-prey and one-predator is the following: $\begin{gathered} \partial u/\partial t- L_1u= a_1(x) u(1-u- b_1 v-c_1 w- \beta_1 J_2* w- \gamma_1 J_3* w),\\ \partial v/\partial t- L_2v= a_2(x) v(1-v- b_2 u- c_2 w-\beta_2 J_1* u-\gamma_2 J_3* w),\\ \partial w/\partial t- L_3 w= a_3(x) w(1- w+ b_3 u+ c_3 v+\beta_3 J_1* u+\gamma_3 J_2* v)\quad (t> 0, x\in\Omega),\end{gathered}$ with Neumann boundary conditions and initial conditions. In the above system $$u$$, $$v$$ are the densities of the prey populations, $$w$$ is the density of the predator population, $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$, $$a_i(x)$$ is a positive $$C^\alpha$$-function on $$\overline\Omega= \Omega\cup \partial\Omega$$, $$b_i$$, $$c_i$$, $$\beta_i$$, $$\gamma_i$$ are nonnegative constants, $$L_i$$, $$i= 1,2,3$$ are uniformly elliptic operators, the function $$J_i* u_i$$, $$i= 1,2,3$$, are given either by $$J_i* u_i= u_i(t- r_i, x)$$ or by $$J_i* u_i= \int^t_{r_i} J_i(t- s)u_i(s,x) ds$$, $$J_i(t)$$ is a piecewise continuous function on $$[0,\infty)$$, $$I_i= [-r_i, 0]$$, $$r_i> 0$$ is a constant.
Some insufficient condition for the existence and global asymptotic stability of a positive solution for each of three model problems are given.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35R10 Partial functional-differential equations 92D25 Population dynamics (general) 35B35 Stability in context of PDEs
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