My research lies at the intersection of differential equations and mathematical ecology, with a strong focus on population dynamics to advance species conservation efforts and deepen our understanding of epidemiological patterns. I specialize in Dynamical Systems and Mathematical Biology, with both theoretical and applied interests, particularly in:
[1] Nonlinear dynamical systems, encompassing both ordinary (ODE) and partial differential equations (PDE).
[2] Mathematical biology, with an emphasis on population dynamics and spatial ecology.
[3] Epidemiological modeling to better understand disease spread and control.
I am deeply committed to leveraging mathematical and statistical methodologies to unravel complex population dynamics, contributing to biodiversity preservation and more effective epidemiological interventions.
FearEffect in Competition Systems: Theory and Applications to Avian Invasions
Non-consumptive effects, such as fear of depredation, can strongly influence predator-prey dynamics. There are several ecological and social motivations for these effects in competitive systems as well. In this work, we consider the classic two species ODE and PDE Lotka-Volterra competition models, where one of the competitors is "fearful" of the other. We find that the presence of fear can have several interesting dynamical effects on the classical competitive scenarios. Notably, for fear levels in certain regimes, we show novel bi-stability dynamics. Furthermore, in the spatially explicit setting, the effects of several spatially heterogeneous fear functions are investigated. In particular, we show that under certain integral restrictions on the fear function, a weak competition-type situation can change to competitive exclusion. Applications of these results to ecological as well as sociopolitical settings are discussed, which connect to the "landscape of fear" (LOF) concept in ecology. Using the test case of northern spotted and barred owl populations in the Pacific Northwest region of the United States, we evaluate if this fear (co-occurrence) model can generate more robust population estimates than previous models. We then evaluate if potential co-occurrence effects among barred and northern spotted owls are uni- or bi-directional. Lastly, we leverage the best-performing model to evaluate the degree to which a recently proposed barred owl culling program may help recover northern spotted owl populations.
Relevant publications
[2] Srivastava, V., N.J. Van Lanen, and R.D. Parshad. Modeling competition co-occurrence effects between the invasive barred owl and imperiled northern spotted owl. (In review)
[1] Srivastava, V., Takyi, E. M., & Parshad, R. D. The effect of "fear" on two species competition. Mathematical Biosciences and Engineering, 2023, 20(5): 8814-8855.
[PDF].
Societies at war are particularly affected by infectious disease outbreaks, necessitating the development of mathematical models tailored to the intricacies of war and disease dynamics as valuable tools for policy-makers. The frequently limited availability of public health resources, such as drugs or medical personnel, yields a fundamental optimal allocation problem. This study frames this problem in a generic, modifiable context and proposes model-informed solutions by identifying allocation strategies that minimize disease burden, measured by total deaths or infections. The desynchronization of epidemic peaks among a heterogeneous population emerges as a general disease mitigation strategy. Moreover, the level of contact heterogeneity proves to substantially affect disease spread and optimal control.
Relevant publications
[1] Srivastava, V. Sarkar, D., & Kadelka, C. (2024). Model-informed optimal allocation of limited resources to mitigate infectious disease outbreaks in societies at war. J. R. Soc. Interface (in press), 2024.
[Preprint]
This study examines a predator-prey model where the predator follows a generalist Leslie-Gower framework, and the prey employs group defense strategies. A key innovation introduced is the simultaneous occurrence of predator population "blow-up" (rapid growth) and prey population "quenching" (sharp decline) within a finite time. Despite this sharp decline, the prey population remains bounded. This simultaneous blow-up and quenching dynamic mirrors real-world phenomena, such as the surge in Burmese python populations in the Florida Everglades and the concurrent decline in mammalian prey species.
Relevant publications
[1] Srivastava, V., Antwi-Fordjour, K., & Parshad, R. D. (2024). Exploring unique dynamics in a predator-prey model with generalist predator and group defence in prey. Chaos , 34 (1): 013107.
[Link] [Preprint]
In this work, we explored the impact of both density-dependent (Allee effect) and non-consumptive, behavioral (fear effect) processes in a two-species competitive system using ODE and PDE models. We showed that while the Allee and fear effects alter extinction dynamics and the number of positive equilibria, they do not affect their stability. The combination of both effects can shift the system's competitive outcomes, especially for small fear levels, leading to qualitative changes like stronger competition or bi-stability. For large fear, the Allee effect reinforces fear-driven dynamics. The findings have implications for conservation, biological control, and biodiversity preservation.
Relevant publications
[1] Chen, S., Chen, F., Srivastava, V., & Parshad, R. D. (2023). Dynamical Analysis of a Lotka-Volterra Competition Model with both Allee and Fear Effect. International Journal of Biomathematics.
[Link] [Preprint]
This work focuses on the narrow escape problem of calculating the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps. The mean first-passage time (MFPT) is modeled by a Poisson partial differential equation and the objective is to perform direct numerical simulations of multiple particles and compare the results with the Poisson equation-based continuum model. The simulations validate the model and also provide additional insights into particle dynamics that cannot be captured in a continuum approach. The Poisson equation-based continuum model's validation is achieved through 104 particle runs. Additional insights into particle dynamics, such as isotropic vs. anisotropic diffusion effects, are gained through direct simulations.
Relevant publications
[2] Srivastava, V., & Cheviakov, A. (2021). Brownian dynamics simulations for the narrow escape problem in the unit sphere. Physical Review E, 104(6), 064113.
[Link]
This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity.
Relevant publications
[1] Chen, S., Chen, F., Srivastava, V., & Parshad, R. D. (2024). Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect. Qualitative Theory of Dynamical Systems , 23(4), 189
[Link] [Preprint]
In the current manuscript, a two-patch model with the Allee effect and nonlinear dispersal is presented. We study both the ordinary differential equation (ODE) case and the partial differential equation (PDE) case here. In the ODE model, the stability of the equilibrium points and the existence of saddle-node bifurcation are discussed. The phase diagram and bifurcation curve of our model are also given as a results of numerical simulation. Besides, the corresponding linear dispersal case is also presented. We show that, when the Allee effect is large, high intensity of linear dispersal is not favorable to the persistence of the species. We further show when the Allee effect is large, nonlinear diffusion is more beneficial to the survival of the population than linear diffusion. Moreover, the results of the PDE model extend our findings from discrete patches to continuous patches.
Relevant publications
[1] Xia, Y., Chen, L., Srivastava, V., & Parshad, R. D. (2023). Stability and bifurcation analysis of a two-patch model with Allee effect and dispersal. Mathematical Biosciences and Engineering, 2023, 20(11): 19781-19807.
[Link] [Preprint].
For my thesis, I investigated how strongly localized perturbations affect the solution of the eigenvalue problem, which has applications in understanding physical and biological phenomena such as the Narrow Escape problem. Specifically, I focused on analyzing the eigenvalues and eigenfunctions of this problem, with a particular interest in properties such as isolatedness, regularity, and simplicity that are associated with these eigenpairs.
Relevant publications
[1] Srivastava, V. The Qualitative Study of the Eigenvalue and Eigenfunctions of the Strong Localized Perturbed Eigenvalue Problem (A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics), (2020).
[1] SURI Program at the University of Saskatchewan(USASK).
In Summer of 2019, I got a chance to be an intern at University of Saskatchewan(USask) under Summer University Research Initiative (SURI-2019). I worked on the Brownian Dynamics Modelling for the Narrow Escape Problem in the case of the unit sphere under the guidance of Dr. Alexey Shevyakov(shevyakov@math.usask.ca).
[2] Harish-Chandra Research Institute, (HRI) Allahabad.
In 2018 I was selected for a prestigious summer research program, Summer Program in Mathematics (SPIM) in Mathematics at Harish-Chandra Research Institute, (HRI) Allahabad. The program involves intensive lectures on Algebra (Group Theory, Field Theory ), Analysis (Measure Theory, Basic Complex Analysis) and Topology (Set Topology up to homotopy theory) over a period of four weeks.
[3] Indian Academy of Sciences Summer Research Fellowship Program (IAS-SRFP)
In 2016, I got selected for one of the prestigious fellowships, Indian Academy of Sciences Summer Research Fellowship Program (IAS-SRFP). Under the guidance of Dr. Sanoli Gun at the Institute of Mathematical Sciences (IMSc), Chennai. In this project, we analyzed the structure of groups and rings.
[4] ANDC-304, Delhi University Innovation Project 2015-16.
In the project ANDC – 304 (Delhi University Innovation Project 2015-16), we developed a portable Electronic Nose prototype with autonomous and stand-alone operation for quantified Ambient Air Pollution (AAP) measurement using wireless data transfer protocol on Android enabled phone.