Fig. 2
From: A theoretical framework for controlling complex microbial communities
Autonomous elements constrain the state of microbial communities, characterizing their driver species. a A three-species community with GLV dynamics \(\dot x_1 = x_1( - 1 + x_3)\), \(\dot x_2 = x_2(1 - x_3)\), \(\dot x_3 = x_3( - 0.5 + 1.5x_3)\). For actuating x3, we consider the impulsive control scheme with x3(t+) = x3(t) + u1(t) for \(t \in {\Bbb T}\). With this controlled population dynamics, our mathematical formalism reveals the autonomous element x1x2 that constraints the state of this microbial community to the low-dimensional manifold \(\{ x \in {\Bbb R}^3|x_1x_2 = x_1(0)x_2(0)\}\) (gray) for all control inputs. Five state trajectories (in colors) with random control inputs illustrate this fact. Hence, {x3} alone cannot be a set of driver species for this controlled population dynamics. b Including a second control input u2(t) actuating x1 (i.e., x1(t+) = x1(t) + u2(t) for \(t \in {\Bbb T}\)) eliminates the autonomous element, since the state of the microbial community (colors) can explore a three-dimensional space (gray). Hence {x1, x3} is a minimum set of driver species for this community with GLV dynamics. c We proved that, generically, increasing the complexity of the controlled population dynamics cannot create autonomous elements. In this example, increasing the deformation size C from the GLV in panel (a) (with C = 0) to the controlled population dynamics in Fig. 1 (with C > 0) eliminates the autonomous element that was present by actuating x3 alone (Example 1 in Supplementary Note 2). Therefore, increasing the complexity of the population dynamics makes {x3} a solo driver species
