Fig. 7

Illustration of the approximations of population recovery times. a Exponential fitting model based on insights from (Yabe et al. 2019). The parameter that characterizes the model is τ_i. b Recovery curve fitted to linear functions. The parameter that characterizes the model is β_i. c Recovery time defined as the timing \(T_{i}^{\alpha }\) that M_i(t) exceeds \(M_{i}^{\alpha }\) which is characterized by constant parameter α