Online Lecture Series on Mathematics of Ecology: Predicting Collapse and Resilience
How can mathematics help us understand when an ecosystem is approaching collapse? Can changes in data warn us before an ecological system reaches a critical threshold? These questions were explored during the online lecture titled “Mathematics of Ecology: Predicting Collapse and Resilience,” organised by CHRIST (Deemed to be University), Pune Lavasa Campus, on 17 September 2026 as part of the Mathematics of Planet Earth Lecture Series.
The session was delivered by Prof. Vishwesha Guttal, Professor at the Centre for Ecological Sciences, Indian Institute of Science (IISc), Bengaluru. Bringing together mathematical theory, ecological observations and real-world data, Prof. Guttal demonstrated how mathematical models can help researchers understand the stability, transformation and resilience of ecological systems.
Prof. Guttal began by introducing the foundational principles of mathematical modelling in ecology. He explained that a mathematical model does not attempt to reproduce every detail of a natural system. Instead, it simplifies reality so that researchers can identify and study the processes that are most important to the question being investigated. Using population growth as an introductory example, he illustrated how differential equations can represent the effects of reproduction, mortality, resource limitations, negative feedback and carrying capacity on the growth of a population.
The central theme of the lecture was the concept of ecological tipping points. A tipping point is a critical threshold beyond which a small change in external conditions may cause a sudden and substantial transformation in an ecosystem. Prof. Guttal explained that ecological systems may possess multiple stable states. Once a system crosses a threshold and moves into a different state, simply reversing the original pressure may not be sufficient to restore it. Such changes can have serious ecological, social and economic consequences.
Through bifurcation diagrams and potential landscape representations, Prof. Guttal explained how nonlinear feedback mechanisms can generate abrupt transitions. He discussed how stable and unstable equilibria can be represented mathematically and how stochastic disturbances may push a system from one stable state to another. These mathematical ideas were connected to ecological phenomena such as changes in vegetation cover, the degradation of lakes and shifts in dryland ecosystems.
An important part of the lecture focused on critical slowing down. As an ecosystem approaches a tipping point, it may recover more slowly from disturbances. This slower recovery can leave detectable patterns in ecological data. Prof. Guttal discussed how statistical measures such as autocorrelation, variance, standard deviation and skewness may function as early-warning indicators of an approaching transition.
Experimental studies involving populations of Daphnia and yeast were presented to show how these indicators may increase before a population collapses. However, Prof. Guttal emphasised that early-warning signals should be treated as indicators rather than exact predictions. Real ecological data contain process noise, measurement error and external disturbances, making careful interpretation essential.
Another case study examined the historical transformation of Anchar Lake in Srinagar. By combining satellite-based land-cover information, nutrient records and climate data collected over several decades, researchers identified a persistent transition around 2010–2011. The lake shifted from a state dominated by open water and aquatic vegetation towards a marshland-dominated state, most likely influenced by nutrient enrichment. The example illustrated how mathematical and statistical methods can support the reconstruction and interpretation of long-term ecological change.
During the interactive session, Prof. Guttal addressed questions about balancing simple conceptual models with highly parameterised models, obtaining realistic parameter values and incorporating stochasticity into ecological research. He explained that simple process-based models are particularly useful for investigating broad conceptual questions, whereas highly parameterised models may be required for specific predictions about a particular ecosystem.
The discussion also highlighted the interdisciplinary relevance of tipping-point theory. Similar mathematical ideas have been applied to climatic systems, genetic circuits, financial markets and psychological disorders because these systems may also involve nonlinear feedback and multiple stable states. Prof. Guttal further discussed self-organising patterns in vegetation and fish schools that existing mathematical models are not yet able to explain completely, demonstrating that ecology continues to offer many fascinating mathematical research problems.
The session concluded with an important message for students: mathematics should be viewed not merely as a collection of formulas or problem-solving procedures, but as a language for understanding complex real-world systems. By identifying essential relationships and removing less relevant details, mathematical abstraction enables researchers to study ecological collapse, resilience and sustainability. The lecture provided participants with an engaging and thought-provoking illustration of how mathematics, ecology and data analysis can work together to address pressing environmental challenges.
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