A predator-prey model with inverse trophic relation and time delay

This paper proposes the following time-delay model for an inverse trophic relation where prey x(t) eaten by mature predators Y(t) can consume the immature predators y(t):x′(t)=x(t)[r1-a11x(t)+α1y(t)-a13Y(t)],y′(t)=-[r2+α2x(t)]y(t)+a31x(t)Y(t)-a31x(t-τ)Y(t-τ)e∫t-τt[-r2-α2x(s)]ds,Y′(t)=-r3Y2(t)+a31x(t-τ)Y(t-τ)e∫t-τt[-r2-α2x(s)]ds,where α1,α2 reflect an inverse trophic relation. Sufficient conditions are obtained for the global attractivity and local stability of an interior equilibrium of the model. It is also shown that α2 maintains the global properties of the interior equilibrium and that α1 can cause the existence of bi-stable interior equilibria.