Coexistence, permanence and resilience in biological reaction–diffusion systems

• Published in: IMA journal of applied mathematics Vol. 69; no. 2; pp. 111 - 129
• Main Author: ROWELL, Jonathan T
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.04.2004; Oxford Publishing Limited (England)
• Subjects: coexistence; competition; Exact sciences and technology; Mathematical analysis; Mathematics; Partial differential equations; reaction–diffusion system; resilience; Sciences and techniques of general use; Competition; Sufficient condition; Feedback; Applied mathematics; Ecology; Coexistence; Boundary condition; Reaction diffusion equation; Partial differential equation; Resilience; Biological system
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/69.2.111

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.