Asymptotic stability of tri-trophic food chains sharing a common resource

• Local asymptotic stability of the interior equilibrium of tri-trophic food chains sharing a common resource is proved.
• In the case of two tri-trophic food chains the stability is global.
• Ecological resilience decreases while the return time increases with the number of food chains in the food web.

One of the key results of the food web theory states that the interior equilibrium of a tri-trophic food chain described by the Lotka–Volterra type dynamics is globally asymptotically stable whenever it exists. This article extends this result to food webs consisting of several food chains sharing a common resource. A Lyapunov function for such food webs is constructed and asymptotic stability of the interior equilibrium is proved. Numerical simulations show that as the number of food chains increases, the real part of the leading eigenvalue, while still negative, approaches zero. Thus the resilience of such food webs decreases with the number of food chains in the web.