KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

A general model of hormesis in biological systems and its application to pest management

Hormesis, a phenomenon whereby exposure to high levels of stressors is inhibitory but low (mild, sublethal and subtoxic) doses are stimulatory, challenges decision-making in the management of cancer, neurodegenerative diseases, nutrition and ecotoxicology. In the latter, increasing amounts of a pesticide may lead to upsurges rather than declines of pests, ecological paradoxes that are difficult to predict. Using a novel re-formulation of the Ricker population equation, we show how interactions between intervention strengths and dose timings, dose–response functions and intrinsic factors can model such paradoxes and hormesis. A model with three critical parameters revealed hormetic biphasic dose and dose timing responses, either in a J-shape or an inverted U-shape, yielding a homeostatic change or a catastrophic shift and hormetic effects in many parameter regions. Such effects were enhanced by repeated pulses of low-level stimulations within one generation at different dose timings, thereby reducing threshold levels, maximum responses and inhibition. The model provides insights into the complex dynamics of such systems and a methodology for improved experimental design and analysis, with wide-reaching implications for understanding hormetic effects in ecology and in medical and veterinary treatment decision-making. We hypothesized that the dynamics of a discrete generation pest control system can be determined by various three-parameter spaces, some of which reveal the conditions for occurrence of hormesis, and confirmed this by fitting our model to both hormetic data from the literature and to a non-hormetic dataset on pesticidal control of mirid bugs in cotton.

Habitat loss-induced tipping points in metapopulations with facilitation

Habitat loss is known to pervade extinction thresholds in metapopulations. Such thresholds result from a loss of stability that can eventually lead to collapse. Several models have been developed to understand the nature of these transitions and how are they affected by the locality of interactions, fluctuations, or external drivers. Most models consider the impact of grazing or aridity as a control parameter that can trigger sudden shifts, once critical values are reached. Others explore instead the role played by habitat loss and fragmentation. Here we consider a minimal model incorporating facilitation along with habitat destruction, with the aim of understanding how local cooperation and habitat loss interact with each other. An explicit mathematical model is derived, along with a spatially explicit simulation model. It is found that a catastrophic shift is expected for increasing levels of habitat loss, but the breakpoint dynamics becomes continuous when dispersal is local. Under these conditions, spatial patchiness is found and the qualitative change from discontinuous to continuous results from a universal behaviour found in a broad class of nonlinear ecological systems (Weissmann and Shnerb, 2014; Martin et al. PNAS (2015) E1828-E1836). Our results suggest that species exhibiting facilitation and displaying short-range dispersal will be markedly more capable of dealing with habitat destruction, also avoiding catastrophic tipping points.

Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and contain a similar percentage of the human population. Theoretical models and palaeoclimatic data predict a grim future, with rapid shifts towards a desert state, with accelerated diversity losses and ecological collapses. These shifts are a consequence of the special nonlinearities resulting from ecological facilitation. Here, we investigate a simple model of semiarid ecosystems identifying the so-called ghost, which appears after a catastrophic transition from a vegetated to a desert state once a critical rate of soil degradation is overcome. The ghost involves a slowdown of transients towards the desert state, making the ecosystem seem stable even though vegetation extinction is inevitable. We use this model to show how to exploit the ecological ghosts to avoid collapse. Doing so involves the restoration of small fractions of desert areas with vegetation capable of maintaining a stable community once the catastrophic shift condition has been achieved. This intervention method is successfully tested under the presence of demographic stochastic fluctuations.

CATASTROPHIC SHIFT OF OBESITY IN THREE ZONES OF SUB-HIMALAYAN REGION, INDIA, AND ITS CORELATION WITH THE DIETARY INTAKE

Objective: The objective is to study the prevalence of obesity and overweight in three zones of sub-Himalayan Region and its corelation with the dietary intake.Methods: A survey was conducted in three zones of Uttrakhand (Dehradun, Rudraprayag, and Uttarkashi), India, representing the urban, semi-urban, and rural village population (18–45 years) of the state. For the study, 100 adults were selected from each of the 3 zones, respectively, to make a total sample size of 300 adults using purposive random sampling.Results: The high prevalence obesity and overweight occur among urban zone (Dehradun), followed by semi-urban zone, Rudraprayag, district of Uttrakhand. In Dehradun, 6.1% of males and 29.4% of females were obese, whereas 62.1% males and 58.8% of females were overweight. In Rudraprayag, 13.6% of males and 14.6% of females were obese, whereas 35.6% of males and 39% of females were overweight. In rural zone (Uttarkashi), there were no cases of obesity among both males and females.Conclusion: A high prevalence rate of obesity was depicted in the urban zone of Uttrakhand region. Urbanization seems to have a positive significant impact on the prevalence of obesity with women being at greater risk.

Trans-heteroclinic bifurcation: a novel type of catastrophic shift

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation , involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication–mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication–mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.

Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and a similar percentage of human population. As a consequence of global warming, these habitats face future potential shifts towards the desert state characterized by an accelerated loss of diversity and stability leading to collapse. Such possibility has been raised by several mathematical and computational models, along with several early warning signal methods applied to spatial vegetation patterns. Here we show that just after a catastrophic shift has taken place an expected feature is the presence of a ghost, i.e., a delayed extinction associated to the underlying dynamical flows. As a consequence, a system might exhibit for very long times an apparent stationarity hiding in fact an inevitable collapse. Here we explore this problem showing that the ecological ghost is a generic feature of standard models of green-desert transitions including facilitation. If present, the ghost could hide warning signals, since statistical patterns are not be expected to display growing fluctuations over time. We propose and computationally test a novel intervention method based on the restoration of small fractions of desert areas with vegetation as a way to maintain the fragile ecosystem beyond the catastrophic shift caused by a saddle-node bifurcation, taking advantage of the delaying capacity of the ghost just after the bifurcation.

Abrupt transitions to tumor extinction: A quasispecies phenotypic model

Background: The dynamics of heterogeneous tumor cell populations competing with healthy cells is an important topic in cancer research with deep implications in biomedicine. Multitude of theoretical and computational models have addressed this issue, especially focusing on the nature of the transitions governing tumor clearance as some relevant model parameters are tuned. In this contribution, we analyze a mathematical model of unstable tumor progression using the quasispecies framework. Our aim is to define a minimal model incorporating the dynamics of competition between healthy cells and a heterogeneous population of cancer cell phenotypes involving changes in replication-related genes (i.e., proto-oncogenes and tumor suppressor genes), in genes responsible for genomic stability, and in house-keeping genes. Such mutations or loss of genes result into different phenotypes with increased proliferation rates and/or increased genomic instabilities. Also, lethal phenotypes with mutations or loss of house-keeping genes are present in our model. Results: Despite bifurcations in the classical deterministic quasispecies model are typically given by smooth, continuous shifts (i.e., transcritical bifurcations), we here identify an novel type of abrupt transition causing tumor extinction. Such a bifurcation, named as trans-heteroclinic, is characterized by the exchange of stability between two distant fixed points (that do not collide) involving, respectively, tumor persistence and tumor clearance. The increase of mutation and/or the decrease of the replication rate of tumor cells involves this catastrophic shift of tumor cell populations. The transient times near bifurcation thresholds are also characterized, showing a power law dependence of exponent −1 of the transients as mutation is changed near the bifurcation value. Conclusions: An abrupt transition involving tumor clearance has been identified with a phenotypic quasispecies cancer model. This result is discussed in the context of targeted cancer therapy as a possible therapeutic strategy to force a catastrophic shift by delivering mutagenic and cytotoxic drugs inside tumor cells. Our model also reveals a novel mechanism causing a discontinuous transition given by the stability exchange of two distant fixed points, which we name as a trans-heteroclinic bifurcation.