Ecological systems such as forest and lakes can exhibit multiple stable states, abrupt transitions and self-organization as a control parameter is varied. Understanding the dynamics of these systems and devising easily quantifiable measures with predictive capabilities using the theoretical tools of stochastic dynamics and nonequilibrium statistical physics form the focus of this thesis.
We study simple ecological models with no spatial degrees of freedom, that show a catastrophic transition as a control parameter is varied and propose a novel early warning signal that exploits two ubiquitous features of ecological systems: nonlinearity and large external fluctuations. It is shown that changes in asymmetry in the distribution of time series data, quantified by changing skewness, is an early warning signal of impending regime shifts. Using simple analytical calculations, model simulations that mimic field measurements and an analysis of real data from abrupt climate change in the Sahara, we study the feasibility of skewness calculations using data available from routine monitoring.
We consider a spatially explicit model of collapse of vegetation in one and two spatial dimensions. An analytical calculation based on the mean-field approximation shows that spatial variance and spatial skewness (with an appropriate sign) increase as one approaches the threshold of vegetation collapse. Our numerical calculations show that an increasing spatial variance in conjunction with a reversal in the initial changing trend of spatial skewness is a superior indicator of an impending spatial ecological regime shift when spatially explicit data are available. These results are shown to hold under several different dispersal kernels such as Gaussian, fat tailed and Cauchy.
Vegetation in semi-arid regions exhibits striking spatial patterns. Theoretical models often ignore the strong fluctuations in parameters such as those arising from seasonality. We present a fully seasonal rainfall model that produces vegetation patterns based on Turing mechanism. We present results for the mean-field and spatially extended versions of the model. We find that the patterns depend on the duration of the wet season even with fixed total annual precipitation (PPT) and our results of maximum vegetation cover as function of PPT is consistent with field observations.
In summary, using theoretical tools of stochastic and nonequilibrium dynamics, we have studied the dynamics of ecological systems showing catastrophic transitions and devised predictive measures which have the potential for practical applications in ecological systems.