First Passage Times of Long Transient Dynamics in Ecology

Abstract

Long transient dynamics in ecological models are characterized by extended periods in one state or regime before an eventual, and often abrupt, transition. One mechanism leading to long transient dynamics is the presence of ghost attractors, states where system dynamics slow down and the system lingers before eventually transitioning to the true attractor. This transition results solely from system dynamics rather than external factors. This paper investigates the dynamics of a classical herbivore-grazer model with the potential for ghost attractors or alternative stable states. We propose an intuitive threshold for first passage time analysis applicable to both bistable and ghost attractor regimes. By formulating the first passage time problem as a backward Kolmogorov equation, we examine how the mean first passage time changes as parameters are varied from the ghost attractor regime to the bistable one, through a saddle-node bifurcation. Our results reveal that the mean and variance of first passage times vary smoothly across the bifurcation threshold, eliminating the deterministic distinction between ghost attractors and bistable regimes. This work suggests that first passage time analysis can be an informative way to classify the length of a long transient. A better understanding of the duration of long transients may contribute to greater ecological understanding and more effective environmental management.

Appendices

Appendix

A Supplementary figures

Fig. 5

Illustration of the nonlinearity of the thresholds, $\beta$, chosen for our FPT analysis as we vary a. A straight line (dashed) is plotted for comparison. Note that the two saddle-node bifurcations occur at $a_1\approx 0.0232$ and $a_2\approx 0.0252$. Other parameters are $r = 0.05, K = 2, h = 0.38$, and $q = 5$

Fig. 6

Exploration of how the MFPT changes with the noise constant $\sigma$. a How the MFPT varies as $\sigma$ is varied depends on the value of a. Curves correspond to 10 evenly spaced values of a between 0.0210 (bottom, darkest curve) and 0.0240 (top, lightest curve). The y axis uses a log10 scaling for ease of visualization. b–c A closer look at three of the curves in (a), illustrating that the MFPT may increase, decrease, or be non-monotonic as $\sigma$ is varied, depending on the value of a. Values of a in (b–c) are $a=0.021$, 0.022, and 0.024, respectively. Other parameters are $r = 0.05, K = 2, h = 0.38$, and $q = 5$

Fig. 7

Exploration of how the variance of FPTs changes as the noise constant $\sigma$ is varied around the value used elsewhere in the paper ($\sigma =0.02$). As in Fig. 6, the y axis uses a log10 scaling for ease of visualization and curves correspond to 10 evenly spaced values of a between 0.0210 (bottom, darkest curve) and 0.0240 (top, lightest curve). Other parameters are $r = 0.05, K = 2, h = 0.38$, and $q = 5$