Coexistence, permanence and resilience in biological reaction–diffusion systems

Reaction–diffusion systems are widely used to describe spatio‐temporal phenomena in a variety of scientific fields, including population ecology. In this paper, I demonstrate that existing results for coexistence and permanence of general Lotka–Volterra systems with absorbing boundaries can be applied in a complementary manner to address a variety of boundary conditions, including the insulating problem. Furthermore, the condition is applicable even to systems containing positive feedback mechanisms in the dynamics. A single (vector) inequality, the first iterate condition, is derived which serves as a sufficient condition for coexistence, permanence and resilience. Additionally, I demonstrate that this inequality condition is but the first in a series of conditions that can be used to describe the behaviour of such systems. Finally, I provide a comparison between the iterate conditions and an alternative test for solution resiliency.