Videos

Example trajectories of full model given in 'Coexistence of Competing Microbial Strains under Twofold Environmental Variability and Demographic Fluctuations' by Asker, M. et al (e-print: https://arxiv.org/abs/2307.06314). 

Specific parameters used in each case are given in the filename. Other parameters are: deltaK=0.0, K+=2000, and K-=400. A white background indicates the low toxin level state, while a grey background indicates high toxin level. Number of sensitive bacteria are plotted in red, number of resistant bacteria in blue, total number of bacteria in magenta, and carrying capacity (where it is not so large as to be self-averaging) in grey. In the cases where it is self-averaging we do not plot the carrying capacity to avoid cluttering the plot.

trajectory [s, nu_T, delta_T]=[0.5, 0.5, 0.0] nuK=0.2.gif: an unbiased fluctuating environment with intermediate T-switching rate (nu_T=0.5, delta_T=0) and K-switching rate (nu_K=0.2). Here we see the total population N rapidly attaining quasi-stationarity far prior to the fixation event occuring (S fixation in this example).

trajectory [s, nu_T, delta_T]=[0.5, 10.0, -0.15] nuK=5.0.gif: faster environmental T and K switching, with nu_T=10 and nu_K=5. There is also a bias towards the harsh T state favouring the strain R (delta_T=-0.15, high toxin level)

that is responsible for a robust offset in strain abundance eventually leading to extinction of S and fixation of R. K-EV switching rate is sufficiently high to ensure its self-averaging: in this fast K switching regime, the population size is distributed about the effective carrying capacity (here N approx 667).

trajectory [s, nu_T, delta_T]=[0.5, 10.0, 0.15] nuK=0.05.gif: fast T-EV (nu_T=10) with a bias towards the mild/low T state favouring the strain S (delta_T=0.15). In this example, the K-EV switching rate is low (nu_K=0.05) and the population size N follows K(t) and fluctuates about K_- and K_+. The T-EV bias (delta_T>0) is here responsible for a systematic offset in the strain abundance, with N_S>N_R, resulting in the fixation of S and extinction of R. In this example, K-EV is responsible for larger demographic fluctuations in the environmental state xi_K=-1, where N approx K_- and S fixation is more likely than when N approx K_+ (xi_K=+1).

trajectory [s, nu_T, delta_T]=[0.5, 10.0, 0.0] nuK=50.0.gif: long-lived coexistence phase in the case of unbiased fast switching T-EV and K-EV, with nu_T=10, delta_T=0 and nu_K=50. This set of parameters essentially corresponds to a point in the bright green region (see corresponding paper Fig. 3), where eta approx 1 and long-lived coexistence is almost certain. As in the previous case, the fast K switching leads to fluctuations of the total population size about N approx 667. In this example x*=1/2 and the number of R and S cells fluctuates about their averages: approx 333.5.

trajectory [s, nu_T, delta_T]=[0.5, 10.0, 0.1] nuK=5e-06 K- start.gif and trajectory [s, nu_T, delta_T]=[0.5, 10.0, 0.1] nuK=5e-06 K+ start.gif: fast T-EV (nu_T=10) and extremely slow K-switching rate  (nu_K=5x10^-6), with an initial carrying capacity K(0)=K_- in the K- start and K(0)=K_+ in the K+ start. There is also a small bias towards the mild T state favouring S (delta_T=0.1). This choice of parameters corresponds to a point in the faint green region (see corresponding paper Fig. 5(a)) , where 0=2K_+=4000 (for clarity, the time series here have been truncated at t=2400), we have long-lived coexistence.

Contact: mmmwa@leeds.ac.uk