arXiv subjects

Siddharth Sabharwal

Publications and source records attributed to Siddharth Sabharwal.

4 recordsLinked to original sources

arXiv2025

Population size in stochastic multi-patch ecological models

We look at the interaction of dispersal and environmental stochasticity in $n$-patch models. We are able to prove persistence and extinction results even in the setting when the dispersal rates are stochastic. As applications we look at Beverton-Holt and Hassell functional responses. We find explicit approximations for the total population size at stationarity when we look at slow and fast dispersal. In particular, we show that if dispersal is small then in the Beverton-Holt setting, if the carrying capacity is random, then environmental fluctuations are always detrimental and decrease the total population size. Instead, in the Hassell setting, if the inverse of the carrying capacity is made random, then environmental fluctuations always increase the population size. Fast dispersal can save populations from extinction and therefore increase the total population size. We also analyze a different type of environmental fluctuation which comes from switching environmental states according to a Markov chain and find explicit approximations when the switching is either fast or slow - in examples we are able to show that slow switching leads to a higher population size than fast switching. Using and modifying some approximation results due to Cuello, we find expressions for the total population size in the $n=2$ patch setting when the growth rates, carrying capacities, and dispersal rates are influenced by random fluctuations. We find that there is a complicated interaction between the various terms and that the covariances between the various random parameters (growth rate, carrying capacity, dispersal rate) play a key role in whether we get an increase or a decrease in the total population size. Environmental fluctuations turn to sometimes be beneficial -- this show that not only dispersal, but also environmental stochasticity can lead to an increase in population size.

q-bio.PE↗

arXiv2025

Population dynamics under random switching

Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often, environmental stochasticity is modeled by systems of stochastic differential equations. However, this type of stochasticity is not always the best suited for ecological modeling. Instead, biological systems can be modeled using piecewise deterministic Markov processes (PDMP). For a PDMP the process follows the flow of a system of ordinary differential equations for a random time, after which the environment switches to a different state, where the dynamics is given by a different system of differential equations. Then this is repeated. The current paper is devoted to the study of the dynamics of $n$ populations described by $n$-dimensional Kolmogorov PDMP. We provide sharp conditions for persistence and extinction, based on the invasion rates (Lyapunov exponents) of the ergodic probability measures supported on the boundary of the positive orthant. In order to showcase the applicability of our results, we apply the theory in some interesting ecological examples.

math.PR↗

arXiv2025

Population size in stochastic discrete-time ecological dynamics

We study how environmental stochasticity influences the long-term population size in certain one- and two-species models. The difficulty is that even when one can prove that there is persistence, it is usually impossible to say anything about the invariant probability measure which describes the persistent species. We are able to circumvent this problem for some important ecological models by noticing that the per-capita growth rates at stationarity are zero, something which can sometimes yield information about the invariant probability measure. For more complicated models we use a recent result by Cuello to explore how small noise influences the population size. We are able to show that environmental fluctuations can decrease, increase, or leave unchanged the expected population size. The results change according to the dynamical model and, within a fixed model, also according to which parameters (growth rate, carrying capacity, etc) are affected by environmental fluctuations. Moreover, we show that not only do things change if we introduce noise differently in a model, but it also matters what one takes as the deterministic `no-noise' baseline for comparison.

q-bio.PE↗

arXiv2014

Feature Selection through Minimization of the VC dimension

Feature selection involes identifying the most relevant subset of input features, with a view to improving generalization of predictive models by reducing overfitting. Directly searching for the most relevant combination of attributes is NP-hard. Variable selection is of critical importance in many applications, such as micro-array data analysis, where selecting a small number of discriminative features is crucial to developing useful models of disease mechanisms, as well as for prioritizing targets for drug discovery. The recently proposed Minimal Complexity Machine (MCM) provides a way to learn a hyperplane classifier by minimizing an exact (\boldmath{$Θ$}) bound on its VC dimension. It is well known that a lower VC dimension contributes to good generalization. For a linear hyperplane classifier in the input space, the VC dimension is upper bounded by the number of features; hence, a linear classifier with a small VC dimension is parsimonious in the set of features it employs. In this paper, we use the linear MCM to learn a classifier in which a large number of weights are zero; features with non-zero weights are the ones that are chosen. Selected features are used to learn a kernel SVM classifier. On a number of benchmark datasets, the features chosen by the linear MCM yield comparable or better test set accuracy than when methods such as ReliefF and FCBF are used for the task. The linear MCM typically chooses one-tenth the number of attributes chosen by the other methods; on some very high dimensional datasets, the MCM chooses about $0.6\%$ of the features; in comparison, ReliefF and FCBF choose 70 to 140 times more features, thus demonstrating that minimizing the VC dimension may provide a new, and very effective route for feature selection and for learning sparse representations.

cs.LG↗