Network Invasion as an Open Dynamical System: Response to Rossberg and Barabás

Ecosystems facing biological invasions are ideal models for studying the structure and functioning of open dynamic networks. We thus called for closer synergy between invasion science and network science and, as a first step, proposed using the major axis of the network interaction matrix as a ‘weather vane’ to signal, for both native and alien species, the short-term species-specific responses to invasion [ 1 Hui C. Richardson D.M. How to invade an ecological network. Trends Ecol. Evol. 2019; 34 : 121-131 Abstract Full Text Full Text PDF PubMed Scopus (0) Google Scholar ]. We illustrated this proposal using dynamical systems, d N/dt= F( N), with unspecified generic population growth function F( N) and added dimensions for prospective invaders. We divided the interaction matrix into four blocks: M, native-to-native interactions; P, impacts of invaders on natives; R, biotic resistance (and potential facilitation) from natives to invaders; and L, invasional meltdown from invader-to-invader interactions. Rossberg and Barabás [ 2 Rossberg A.G. Barabás G. How carefully executed network theory informs invasion ecology. Trends Ecol. Evol. 2019; 34 : 385-386 Google Scholar ] raised concerns about some mathematical features of our construct, especially regarding the structures of R and L of the aforementioned four blocks in the interaction matrix. Using a simple Lotka–Volterra (LV) competition model, d N/dt=diag( N)？ f( N), they showed that R is actually a zero matrix (no biotic resistance) and that L a diagonal matrix (no invasional meltdown), in contrast to the example in Figure 1 of our paper. Rossberg and Barabás then reported three results related to network invasion from their models, concluding that (i) abundances vary at random in response to invasion; (ii) invasion fitness should consider both habitat and biotic interactions in the invaded ecosystem; and (iii) there are equal rates of extinctions to invasions in saturated communities