On tridiagonal predator–prey systems and a conjecture

Tridiagonal Kolmogorov systems of dimension nn can be seen as models describing the evolution of nn biological species living in a “linear environment”. We will say that such a system is predator–prey if each planar subsystem, composed of two neighbor species, is a predator–prey system with friction. Our conjecture is that if a predator–prey tridiagonal system is dissipative, then the system has a global attractor. Planar and Lotka–Volterra cases are proved to be true.