Continuous dependence of the global attractors of a family of periodic Kolmogorov systems.
The authors investigate the periodic Kolmogorov system \[ x_i'=x_if_i(t,x_1,\dots,x_n),\quad 1\leq i\leq n, \] where \(f=(f_1,\dots,f_n):\mathbb R \times \mathbb R_+^n\to \mathbb R^n\) is a continuous function with the properties \((P_1) f(t+T,x)=f(t,x); (P_2)\) The partial derivative \(f_x(t,x)\) is continuous on \(\mathbb R\times \mathbb R_+^n.\) Moreover there hold the hypotheses \((H_1)\;c_i\frac{\partial f_i}{\partial x_i}(t,x)+\sum_{j\neq i} c_j| \frac{\partial f_j}{\partial x_i}(t,x)| \leq -m, 1\leq i\leq n\), with positive constants \(c_1,\dots,c_n,\;m\). \((H_2)\;\) The Kolmogorov system under consideration has a positive solution \(v\) defined and bounded on a terminal interval of \(\mathbb R.\) Under these assumptions, each positive solution \(x\) is defined on a terminal interval of \(\mathbb R\) and there exists a \(T-\)periodic solution \(U=U^f\) such that \(x(t)-U(t)\to 0\) as \(t\to +\infty.\) The function \(U^f\) is a global attractor. The main result here is the continuity of the function \(\mathcal U:\Lambda_T(c,m,n)\to \mathcal C(\mathbb R,\mathbb R^n);\;\mathcal U=U^f,\) where \(\Lambda_T(c,m,n)\) is a space of functions \(f\) fulfilling the above assumptions and hypotheses. The continuity theorem is applied to analyzing the system \(x_i'=x_if_i(t,x_1,\dots,x_n,y)\), \(i=1,\dots,n\), \(y'=yg(t,x_1,\dots ,x_n,y),\) where \(y\) is a predator and \(x_1,\dots,x_n\) are preys that in the absence of the predator behave as a cooperative system. Special attention is paid to the corresponding autonomous Lotka-Volterra system.
scientific article; zbMATH DE number 1915787